FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

Карточки в этой колоде

1. Карточка 1

   Вопрос

   SI base unit of length?

   Ответ

   The metre, symbol $m$.

2. Карточка 2

   Вопрос

   Slope between two points?

   Ответ

   $$m=\frac{y_2-y_1}{x_2-x_1}$$

   This requires $x_2\ne x_1$; slope is the change in the dependent variable per unit change in the independent variable.

3. Карточка 3

   Вопрос

   Net force relationship for constant mass in an inertial frame?

   Ответ

   $$\sum\mathbf{F}=m\mathbf{a}$$

   Use a coherent unit system and include only external forces on the chosen body.

4. Карточка 4

   Вопрос

   Cash-flow sign convention from the decision maker's viewpoint?

   Ответ

   Receipts are positive and disbursements are negative. Choose one viewpoint and keep it consistent across the cash-flow diagram and equations.

5. Карточка 5

   Вопрос

   Where should an FE candidate get the current reference handbook for practice?

   Ответ

   From the candidate's MyNCEES account. Use the current supplied file rather than an old indexed copy or a third-party mirror.

6. Карточка 6

   Вопрос

   Probability of the complement of event $A$?

   Ответ

   $$P(A^c)=1-P(A)$$

7. Карточка 7

   Вопрос

   An engineering judgment conflicts with pressure to protect schedule or cost. First duty?

   Ответ

   Protect public health, safety, and welfare. Document the technical concern and escalate through appropriate channels.

8. Карточка 8

   Вопрос

   SI base unit of mass?

   Ответ

   The kilogram, symbol $kg$. It is a mass unit, not a force unit.

9. Карточка 9

   Вопрос

   Linear equation from slope and intercept?

   Ответ

   $$y=mx+b$$

   Here $m$ is slope and $b$ is the value of $y$ where $x=0$.

10. Карточка 10

    Вопрос

    Velocity after time $t$ under constant acceleration?

    Ответ

    $$v=v_0+at$$

    This one-dimensional form requires constant $a$ and a consistent positive direction.

11. Карточка 11

    Вопрос

    Where is an end-of-period cash flow placed?

    Ответ

    At the boundary marking the end of that period. Interest factors assume consistent period lengths and cash-flow timing.

12. Карточка 12

    Вопрос

    How is the FE reference handbook available during the exam?

    Ответ

    It is supplied onscreen as a searchable PDF. Practice locating information in the same current reference rather than relying on memory of a printed layout.

13. Карточка 13

    Вопрос

    Probability of $A$ or $B$ for possibly overlapping events?

    Ответ

    $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

14. Карточка 14

    Вопрос

    SI base unit of time?

    Ответ

    The second, symbol $s$.

15. Карточка 15

    Вопрос

    Linear equation from one point and a slope?

    Ответ

    $$y-y_1=m(x-x_1)$$

    Use it when the line passes through $(x_1,y_1)$ and has slope $m$.

16. Карточка 16

    Вопрос

    Position after time $t$ under constant acceleration?

    Ответ

    $$x=x_0+v_0t+\frac{1}{2}at^2$$

    The acceleration is constant and all signed components use one coordinate direction.

17. Карточка 17

    Вопрос

    A project assignment lies outside your competence. Proper response?

    Ответ

    Accept only the parts you are qualified to perform, or obtain qualified supervision or support. Do not present unfamiliar work as your own competent judgment.

18. Карточка 18

    Вопрос

    SI base unit of electric current?

    Ответ

    The ampere, symbol $A$.

19. Карточка 19

    Вопрос

    Future amount under simple interest?

    Ответ

    $$F=P(1+in)$$

    Here $P$ is principal, $i$ is interest per period, and $n$ is the number of periods; interest is earned only on principal.

20. Карточка 20

    Вопрос

    Distance between two Cartesian points?

    Ответ

    $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

    The coordinate components must use the same length unit.

21. Карточка 21

    Вопрос

    Constant-acceleration relation without elapsed time?

    Ответ

    $$v^2=v_0^2+2a(x-x_0)$$

    Use signed one-dimensional components and constant $a$.

22. Карточка 22

    Вопрос

    Where do you search the supplied reference material in the exam interface?

    Ответ

    Use the exam software's search box beside the supplied reference material.

23. Карточка 23

    Вопрос

    Conditional probability of $A$ given $B$?

    Ответ

    $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$

    This requires $P(B)>0$.

24. Карточка 24

    Вопрос

    SI base unit of thermodynamic temperature?

    Ответ

    The kelvin, symbol $K$. Kelvin is written without a degree sign.

25. Карточка 25

    Вопрос

    A client asks you to remove an unfavorable test result from a report. Proper response?

    Ответ

    Keep the material result and explain it accurately. Engineering statements should be objective, complete enough to avoid deception, and supported by evidence.

26. Карточка 26

    Вопрос

    Midpoint of two Cartesian points?

    Ответ

    $$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

    Average corresponding coordinates.

27. Карточка 27

    Вопрос

    Linear momentum of a particle?

    Ответ

    $$\mathbf{p}=m\mathbf{v}$$

    For constant mass, SI units are $kg\cdot m/s$.

28. Карточка 28

    Вопрос

    Future amount under compound interest?

    Ответ

    $$F=P(1+i)^n$$

    The rate period must match the period used for $n$.

29. Карточка 29

    Вопрос

    Should an FE search workflow depend on Ctrl+F?

    Ответ

    No. The May 2026 NCEES Examinee Guide says Ctrl+F isn't available; practice with the exam interface's own search box.

30. Карточка 30

    Вопрос

    SI base unit of amount of substance?

    Ответ

    The mole, symbol $mol$.

31. Карточка 31

    Вопрос

    Joint probability expressed with a conditional probability?

    Ответ

    $$P(A\cap B)=P(A\mid B)P(B)$$

    This elementary formula requires $P(B)>0$.

32. Карточка 32

    Вопрос

    Standard equation of a circle centered at $(h,k)$?

    Ответ

    $$(x-h)^2+(y-k)^2=r^2$$

    The coordinate units and radius unit must match.

33. Карточка 33

    Вопрос

    Impulse-momentum relationship?

    Ответ

    $$\int_{t_1}^{t_2}\sum\mathbf{F}\,dt=\mathbf{p}_2-\mathbf{p}_1$$

    The force is the net external force on the chosen system.

34. Карточка 34

    Вопрос

    You discover a financial interest that could affect your recommendation. Proper response?

    Ответ

    Disclose the conflict promptly to affected parties and obtain appropriate consent or withdraw. Hidden personal interests undermine independent judgment.

35. Карточка 35

    Вопрос

    SI base unit of luminous intensity?

    Ответ

    The candela, symbol $cd$.

36. Карточка 36

    Вопрос

    Present worth of one future amount?

    Ответ

    $$P=\frac{F}{(1+i)^n}$$

    Use an effective rate $i$ per period consistent with $n$.

37. Карточка 37

    Вопрос

    Roots of $ax^2+bx+c=0$?

    Ответ

    $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

    This form requires $a\ne0$.

38. Карточка 38

    Вопрос

    When is total linear momentum conserved?

    Ответ

    When net external impulse on the system is zero. Internal forces can redistribute momentum but do not change the system total.

39. Карточка 39

    Вопрос

    Why avoid memorizing FE handbook page numbers?

    Ответ

    Pagination and layout can change. Remember section names, concepts, and strong search terms instead.

40. Карточка 40

    Вопрос

    Probability condition that defines independence of $A$ and $B$?

    Ответ

    $$P(A\cap B)=P(A)P(B)$$

    Independence is not the same as mutual exclusivity.

41. Карточка 41

    Вопрос

    SI unit of frequency in base-unit form?

    Ответ

    The hertz, $Hz=s^{-1}$, applies to periodic phenomena. One hertz is one cycle per second.

42. Карточка 42

    Вопрос

    Present worth of an end-of-period uniform series $A$?

    Ответ

    $$P=A\frac{(1+i)^n-1}{i(1+i)^n}$$

    This requires $i\ne0$; at $i=0$, $P=An$.

43. Карточка 43

    Вопрос

    What does $b^2-4ac$ reveal for a real-coefficient quadratic?

    Ответ

    It determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives a complex-conjugate pair.

44. Карточка 44

    Вопрос

    Work done by a constant force through displacement?

    Ответ

    $$W=\mathbf{F}\cdot\mathbf{s}=Fs\cos\theta$$

    Use the angle between force and displacement; SI units are joules.

45. Карточка 45

    Вопрос

    A vendor offers a valuable gift while you evaluate bids. Proper response?

    Ответ

    Decline it and follow the organization's disclosure policy. Even an intended courtesy can create an actual or apparent conflict.

46. Карточка 46

    Вопрос

    SI unit of force in base-unit form?

    Ответ

    The newton, $N=kg\cdot m/s^2$.

47. Карточка 47

    Вопрос

    First search cue for a named quantity such as coefficient of thermal expansion?

    Ответ

    Search the exact technical noun phrase. Distinctive nouns usually narrow results better than a whole question sentence.

48. Карточка 48

    Вопрос

    Bayes' rule for $P(A\mid B)$?

    Ответ

    $$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$$

    This requires $P(B)>0$.

49. Карточка 49

    Вопрос

    Product of powers with the same positive base and real exponents?

    Ответ

    $$a^m a^n=a^{m+n}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

50. Карточка 50

    Вопрос

    Translational kinetic energy of a particle?

    Ответ

    $$K=\frac{1}{2}mv^2$$

    Mass and speed must use a coherent unit system.

51. Карточка 51

    Вопрос

    SI unit of pressure in force-area form?

    Ответ

    The pascal, $Pa=N/m^2$. It is the SI derived unit for pressure and stress.

52. Карточка 52

    Вопрос

    Equivalent annual amount for present worth $P$?

    Ответ

    $$A=P\frac{i(1+i)^n}{(1+i)^n-1}$$

    For the displayed formula, $i\ne0$; at $i=0$, $A=P/n$. Payments occur at period ends and $i$ matches that period.

53. Карточка 53

    Вопрос

    May confidential client information be shared for personal benefit?

    Ответ

    No. Protect it unless the client authorizes disclosure or law, safety, or professional duty requires it.

54. Карточка 54

    Вопрос

    A handbook search term returns too many results. Best refinement?

    Ответ

    Add one meaningful qualifier, such as the phenomenon, geometry, distribution, or material property you need.

55. Карточка 55

    Вопрос

    Quotient of powers with the same positive base and real exponents?

    Ответ

    $$\frac{a^m}{a^n}=a^{m-n}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

56. Карточка 56

    Вопрос

    Net work-energy relationship for a particle?

    Ответ

    $$W_{net}=K_2-K_1$$

    The net work includes the work of all forces represented in the particle model.

57. Карточка 57

    Вопрос

    Ordered selections of $r$ objects from $n$ distinct objects?

    Ответ

    $$P(n,r)=\frac{n!}{(n-r)!}$$

    Use this when order matters and selection is without replacement.

58. Карточка 58

    Вопрос

    Future worth of an end-of-period uniform series $A$?

    Ответ

    $$F=A\frac{(1+i)^n-1}{i}$$

    This requires $i\ne0$; at $i=0$, $F=An$.

59. Карточка 59

    Вопрос

    Power of a power for a positive base and real exponents?

    Ответ

    $$(a^m)^n=a^{mn}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

60. Карточка 60

    Вопрос

    SI unit of energy in force-distance form?

    Ответ

    The joule, $J=N\cdot m$.

61. Карточка 61

    Вопрос

    A problem gives only an unfamiliar symbol. Best search preparation?

    Ответ

    Translate the symbol into its likely quantity name and discipline context before searching; symbol-only searches are often ambiguous.

62. Карточка 62

    Вопрос

    Confidential information reveals an imminent public-safety risk. What comes first?

    Ответ

    Take steps required to protect people, using appropriate legal and organizational channels. Share only what is necessary and document the basis for escalation.

63. Карточка 63

    Вопрос

    Unordered selections of $r$ objects from $n$ distinct objects?

    Ответ

    $$C(n,r)=\frac{n!}{r!(n-r)!}$$

    Use this when order does not matter and selection is without replacement.

64. Карточка 64

    Вопрос

    Change in gravitational potential energy near Earth's surface?

    Ответ

    $$\Delta U_g=mg\Delta h$$

    Use approximately constant $g$ and a clearly chosen height datum.

65. Карточка 65

    Вопрос

    Negative-exponent rule for a positive base and real exponent?

    Ответ

    $$a^{-n}=\frac{1}{a^n}$$

    This form assumes $a>0$ and real exponent $n$.

66. Карточка 66

    Вопрос

    Uniform end-of-period deposit needed for future amount $F$?

    Ответ

    $$A=F\frac{i}{(1+i)^n-1}$$

    For the displayed formula, $i\ne0$; at $i=0$, $A=F/n$. The first deposit is one period from now and the last occurs at period $n$.

67. Карточка 67

    Вопрос

    A handbook search for an acronym fails. Next move?

    Ответ

    Try the expanded term and a common alternate name. Acronyms vary by section and discipline.

68. Карточка 68

    Вопрос

    SI unit of power in energy-time form?

    Ответ

    The watt, $W=J/s$.

69. Карточка 69

    Вопрос

    Logarithm of a product?

    Ответ

    $$\log_b(xy)=\log_b x+\log_b y$$

    For real logarithms, $b>0$, $b\ne1$, and both arguments are positive.

70. Карточка 70

    Вопрос

    Conditions for a binomial model?

    Ответ

    A fixed number of independent trials, two outcomes per trial, and the same success probability on every trial.

71. Карточка 71

    Вопрос

    Using another engineer's analysis in your deliverable?

    Ответ

    Credit the source and verify the analysis before relying on it. Do not present someone else's work as your own.

72. Карточка 72

    Вопрос

    SI unit of electric charge in current-time form?

    Ответ

    The coulomb, $C=A\cdot s$.

73. Карточка 73

    Вопрос

    Average power from work over elapsed time?

    Ответ

    $$P_{avg}=\frac{W}{\Delta t}$$

    Use consistent energy and time units; watts are joules per second.

74. Карточка 74

    Вопрос

    Annual equivalent of cost $P$ with salvage $S$ after $n$ periods?

    Ответ

    $$A=P\frac{i(1+i)^n}{(1+i)^n-1}-S\frac{i}{(1+i)^n-1}$$

    The first factor converts present cost to an annual amount; the second converts future salvage. Use the same effective rate and period basis. At $i=0$, $A=(P-S)/n$.

75. Карточка 75

    Вопрос

    Logarithm of a positive quantity raised to a power?

    Ответ

    $$\log_b(x^p)=p\log_b x$$

    For real logarithms, $x>0$, $b>0$, and $b\ne1$.

76. Карточка 76

    Вопрос

    SI unit of electric potential difference in energy-charge form?

    Ответ

    The volt, $V=J/C$.

77. Карточка 77

    Вопрос

    Instantaneous mechanical power delivered by a force?

    Ответ

    $$P=\mathbf{F}\cdot\mathbf{v}$$

    Use the velocity of the force's point of application.

78. Карточка 78

    Вопрос

    A precise handbook term gives no useful match. Next move?

    Ответ

    Try a standard synonym or the broader parent concept, then narrow within the relevant section.

79. Карточка 79

    Вопрос

    Mean of $X\sim\operatorname{Binomial}(n,p)$?

    Ответ

    $$E[X]=np$$

80. Карточка 80

    Вопрос

    SI unit of electrical resistance in voltage-current form?

    Ответ

    The ohm, $\Omega=V/A$.

81. Карточка 81

    Вопрос

    Moment of a force about a point?

    Ответ

    $$\mathbf{M}=\mathbf{r}\times\mathbf{F}$$

    Vector $\mathbf{r}$ runs from the moment point to a point on the force's line of action.

82. Карточка 82

    Вопрос

    Cash-flow pattern described by an arithmetic gradient $G$?

    Ответ

    A series that changes by the same amount $G$ each period. In the usual convention, the first gradient increment appears after the first base payment.

83. Карточка 83

    Вопрос

    Inverse relationship between the natural log and exponential?

    Ответ

    $$\ln(e^x)=x,\qquad e^{\ln x}=x$$

    The second identity requires $x>0$.

84. Карточка 84

    Вопрос

    SI prefix giga?

    Ответ

    $G=10^9$.

85. Карточка 85

    Вопрос

    Angular velocity after time $t$ under constant angular acceleration?

    Ответ

    $$\omega=\omega_0+\alpha t$$

    Use consistent angular sign convention and radians for angular measures.

86. Карточка 86

    Вопрос

    When should an engineer sign or seal technical work?

    Ответ

    Only when authorized and when the work was prepared by the engineer or under the level of responsible charge required by the applicable jurisdiction.

87. Карточка 87

    Вопрос

    Best memory target for returning to a handbook formula?

    Ответ

    The section or concept name plus a distinctive search phrase. That survives layout changes better than a page number.

88. Карточка 88

    Вопрос

    SI prefix mega?

    Ответ

    $M=10^6$. Uppercase matters: lowercase $m$ means milli.

89. Карточка 89

    Вопрос

    Convert degrees to radians?

    Ответ

    $$\theta_{rad}=\theta_{deg}\frac{\pi}{180}$$

    Use radians in calculus formulas unless a formula explicitly says otherwise.

90. Карточка 90

    Вопрос

    Variance of $X\sim\operatorname{Binomial}(n,p)$?

    Ответ

    $$\operatorname{Var}(X)=np(1-p)$$

91. Карточка 91

    Вопрос

    Cash-flow pattern described by a geometric gradient rate $g$?

    Ответ

    A series in which each payment is the previous payment multiplied by $1+g$. Keep $g$ on the same period basis as the interest rate.

92. Карточка 92

    Вопрос

    Where do licensure and sealing requirements come from?

    Ответ

    The applicable jurisdiction's current laws and board rules. Confirm local requirements rather than assuming one state's rules apply everywhere.

93. Карточка 93

    Вопрос

    Rotational kinetic energy about a fixed axis?

    Ответ

    $$K_{rot}=\frac{1}{2}I\omega^2$$

    $I$ is the mass moment of inertia about that axis.

94. Карточка 94

    Вопрос

    SI prefix kilo?

    Ответ

    $k=10^3$.

95. Карточка 95

    Вопрос

    Right-triangle definition of sine?

    Ответ

    Sine is opposite over hypotenuse. This definition applies to an acute angle in a right triangle.

96. Карточка 96

    Вопрос

    After search lands near a result, what should you scan first?

    Ответ

    The section heading and nearby subheading. They establish whether the result belongs to the right model and subject.

97. Карточка 97

    Вопрос

    Rotational analogue of Newton's second law about a fixed principal axis?

    Ответ

    $$\sum M=I\alpha$$

    This scalar form assumes the stated fixed-axis model and a consistent sign convention.

98. Карточка 98

    Вопрос

    SI prefix milli?

    Ответ

    $m=10^{-3}$. Context distinguishes the prefix from the metre symbol.

99. Карточка 99

    Вопрос

    Meaning of a nominal annual rate compounded $m$ times per year?

    Ответ

    It is a quoted annual rate equal to $m$ times the periodic rate. It is not itself the effective annual rate unless compounding is annual.

100. Карточка 100

     Вопрос

     Standardize an observation $x$ from a distribution with mean $\mu$ and standard deviation $\sigma$?

     Ответ

     $$z=\frac{x-\mu}{\sigma}$$

     This requires $\sigma>0$; using a normal probability table also assumes an appropriate normal model.

101. Карточка 101

     Вопрос

     Right-triangle definition of cosine?

     Ответ

     Cosine is adjacent over hypotenuse. This definition applies to an acute angle in a right triangle.

102. Карточка 102

     Вопрос

     Force-displacement relation for an ideal linear spring?

     Ответ

     $$F=-kx$$

     The minus sign shows restoring direction; the model applies within the spring's linear range.

103. Карточка 103

     Вопрос

     May qualifications or project responsibility be exaggerated in a proposal?

     Ответ

     No. State education, licensure, experience, and responsibility accurately, including the actual roles of collaborators.

104. Карточка 104

     Вопрос

     SI prefix micro?

     Ответ

     $\mu=10^{-6}$.

105. Карточка 105

     Вопрос

     A handbook symbol conflicts with your usual notation. What controls?

     Ответ

     The definition beside the selected relationship. Symbols can represent different quantities in different sections.

106. Карточка 106

     Вопрос

     Periodic rate from nominal annual rate $r$ compounded $m$ times per year?

     Ответ

     $$i=\frac{r}{m}$$

     Use consistent decimal units, and pair this rate with the number of compounding periods.

107. Карточка 107

     Вопрос

     Right-triangle definition of tangent?

     Ответ

     Tangent is opposite over adjacent, or $\sin\theta/\cos\theta$ where $\cos\theta\ne0$.

108. Карточка 108

     Вопрос

     Average normal stress under centered axial load?

     Ответ

     $$\sigma=\frac{F}{A}$$

     $F$ is normal to area $A$; the average ignores local stress concentrations.

109. Карточка 109

     Вопрос

     Mean, median, and mode of a symmetric normal distribution?

     Ответ

     They are equal and located at $\mu$. The distribution is symmetric about that center.

110. Карточка 110

     Вопрос

     SI prefix nano?

     Ответ

     $n=10^{-9}$.

111. Карточка 111

     Вопрос

     Fundamental Pythagorean trigonometric identity?

     Ответ

     $$\sin^2\theta+\cos^2\theta=1$$

112. Карточка 112

     Вопрос

     Engineering normal strain?

     Ответ

     $$\varepsilon=\frac{\Delta L}{L_0}$$

     It is dimensionless and uses original gauge length $L_0$.

113. Карточка 113

     Вопрос

     A candidate retrieves a formula from the handbook. What must be checked with it?

     Ответ

     Its applicability assumptions and variable definitions. A familiar equation used under the wrong model is still wrong.

114. Карточка 114

     Вопрос

     Technical duty to an employer or client?

     Ответ

     Act as a faithful professional within lawful, ethical limits, communicate material risks, and preserve independent technical judgment.

115. Карточка 115

     Вопрос

     How does a length conversion factor apply to area?

     Ответ

     Square it. If $1\,u=a\,v$, then $1\,u^2=a^2\,v^2$.

116. Карточка 116

     Вопрос

     Effective annual rate from nominal rate $r$ compounded $m$ times per year?

     Ответ

     $$i_{eff}=\left(1+\frac{r}{m}\right)^m-1$$

117. Карточка 117

     Вопрос

     When is a Poisson count model reasonable?

     Ответ

     For counts in a fixed interval when events occur independently at an approximately constant average rate and simultaneous events are negligible.

118. Карточка 118

     Вопрос

     Relationship for the third side of a non-right triangle?

     Ответ

     $$c^2=a^2+b^2-2ab\cos C$$

     Angle $C$ lies between sides $a$ and $b$, opposite side $c$.

119. Карточка 119

     Вопрос

     Linear elastic stress-strain relation in uniaxial loading?

     Ответ

     $$\sigma=E\varepsilon$$

     Use it in the material's linear elastic range; $E$ has stress units.

120. Карточка 120

     Вопрос

     How does a length conversion factor apply to volume?

     Ответ

     Cube it. If $1\,u=a\,v$, then $1\,u^3=a^3\,v^3$.

121. Карточка 121

     Вопрос

     What should you record for every variable in a retrieved equation?

     Ответ

     Its meaning and expected unit or dimension. This catches swapped symbols and incompatible substitutions.

122. Карточка 122

     Вопрос

     Effective annual rate for continuous compounding at nominal rate $r$?

     Ответ

     $$i_{eff}=e^r-1$$

     Here $r$ is expressed as a decimal per year.

123. Карточка 123

     Вопрос

     A supervisor directs an illegal engineering action. Proper response?

     Ответ

     Refuse the illegal action, document the direction, and escalate through appropriate legal or organizational channels.

124. Карточка 124

     Вопрос

     Magnitude of a three-component vector $\mathbf{v}$?

     Ответ

     $$\lVert\mathbf{v}\rVert=\sqrt{v_x^2+v_y^2+v_z^2}$$

     All components must represent the same physical quantity in compatible units.

125. Карточка 125

     Вопрос

     Mass density?

     Ответ

     $$\rho=\frac{m}{V}$$

     Use mass, not weight, in the numerator.

126. Карточка 126

     Вопрос

     Inches in one foot?

     Ответ

     $12\,in=1\,ft$.

127. Карточка 127

     Вопрос

     Mean and variance of $X\sim\operatorname{Poisson}(\lambda)$?

     Ответ

     Both equal $\lambda$. The standard deviation is $\sqrt{\lambda}$.

128. Карточка 128

     Вопрос

     How should subscripts be handled in a retrieved relationship?

     Ответ

     Read what each subscript identifies—state, direction, phase, location, or reference—before substituting values.

129. Карточка 129

     Вопрос

     Dot product from vector magnitudes and included angle?

     Ответ

     $$\mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta$$

     The result is a scalar.

130. Карточка 130

     Вопрос

     Average normal pressure on an area?

     Ответ

     $$p=\frac{F_n}{A}$$

     $F_n$ is the normal force; pressure is scalar in a fluid at rest.

131. Карточка 131

     Вопрос

     Feet in one yard?

     Ответ

     $3\,ft=1\,yd$.

132. Карточка 132

     Вопрос

     When are two interest rates equivalent?

     Ответ

     When they produce the same accumulation over the same time interval. Convert both rates to a common effective period before comparing them.

133. Карточка 133

     Вопрос

     You discover a material error in issued work. Proper response?

     Ответ

     Notify responsible parties promptly, assess the safety impact, correct the record, and document the response. Do not hide the error.

134. Карточка 134

     Вопрос

     Expected value of a discrete random variable?

     Ответ

     $$E[X]=\sum_x xP(X=x)$$

     Sum over all possible values with a valid probability mass function.

135. Карточка 135

     Вопрос

     Dot-product test for perpendicular nonzero vectors?

     Ответ

     Their dot product is zero. The zero vector also has zero dot product with every vector, so require both vectors to be nonzero before concluding perpendicularity.

136. Карточка 136

     Вопрос

     Hydrostatic pressure change with vertical depth in a constant-density fluid?

     Ответ

     $$\Delta p=\rho g\Delta h$$

     Depth increases downward and the fluid is at rest.

137. Карточка 137

     Вопрос

     Feet in one statute mile?

     Ответ

     $5280\,ft=1\,mi$.

138. Карточка 138

     Вопрос

     Present-worth acceptance rule for an independent investment at the minimum attractive rate of return?

     Ответ

     Accept it when net present worth is nonnegative. All cash flows must be discounted to the same time using the chosen comparison rate.

     

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139. Карточка 139

     Вопрос

     A retrieved formula contains signed quantities. What should you confirm?

     Ответ

     The stated sign convention and positive direction. Do not mix magnitudes with signed components silently.

140. Карточка 140

     Вопрос

     Magnitude of a vector cross product?

     Ответ

     $$\lVert\mathbf{a}\times\mathbf{b}\rVert=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\sin\theta$$

     The direction follows the right-hand rule.

141. Карточка 141

     Вопрос

     Exact inch-to-metre relationship?

     Ответ

     $1\,in=0.0254\,m$ exactly.

142. Карточка 142

     Вопрос

     Volumetric flow rate through a cross-section?

     Ответ

     $$Q=Av_{avg}$$

     Velocity is the area-average component normal to the section.

143. Карточка 143

     Вопрос

     Variance from the mean $\mu$?

     Ответ

     $$\operatorname{Var}(X)=E[(X-\mu)^2]$$

     Variance has squared units; standard deviation has the original units.

144. Карточка 144

     Вопрос

     Future-worth acceptance rule for an independent investment?

     Ответ

     Accept it when net future worth at the study horizon is nonnegative, using the minimum attractive rate of return consistently.

145. Карточка 145

     Вопрос

     What coordinate detail should be checked before using a vector or mechanics formula?

     Ответ

     Axis directions and the reference point or origin. Components and moments depend on those choices.

146. Карточка 146

     Вопрос

     Exact foot-to-metre relationship?

     Ответ

     $1\,ft=0.3048\,m$ exactly.

147. Карточка 147

     Вопрос

     How should important uncertainty be communicated?

     Ответ

     State the assumptions, limits, and plausible consequences clearly enough for the decision maker to understand the risk.

148. Карточка 148

     Вопрос

     When is the matrix product $AB$ defined?

     Ответ

     It is defined when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$.

149. Карточка 149

     Вопрос

     Steady one-inlet, one-outlet mass continuity?

     Ответ

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     Here $v_1$ and $v_2$ are cross-sectional average velocity components normal to their sections, with density treated as uniform over each section. For incompressible flow with constant density, this reduces to $A_1v_1=A_2v_2$.

150. Карточка 150

     Вопрос

     Sample mean of observations $x_1,\ldots,x_n$?

     Ответ

     $$\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i$$

151. Карточка 151

     Вопрос

     Square feet in one acre?

     Ответ

     $1\,acre=43{,}560\,ft^2$.

152. Карточка 152

     Вопрос

     Determinant of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$?

     Ответ

     $$ad-bc$$

153. Карточка 153

     Вопрос

     Bernoulli relationship along a streamline for ideal steady flow?

     Ответ

     $$\frac{p}{\rho g}+\frac{v^2}{2g}+z=\text{constant}$$

     This basic form assumes steady, incompressible, negligible-viscosity flow with no shaft work or heat effects between points.

154. Карточка 154

     Вопрос

     May inconvenient measurements be discarded without a technical basis?

     Ответ

     No. Use a documented, technically justified rule for excluding data and preserve the original record.

155. Карточка 155

     Вопрос

     Annual-worth acceptance rule for an independent investment?

     Ответ

     Accept it when equivalent annual worth is nonnegative. Compare alternatives over compatible service assumptions.

156. Карточка 156

     Вопрос

     Before using a handbook equation, what unit-system check matters?

     Ответ

     Confirm that every input and any embedded constant use one compatible unit system. Convert first when necessary.

157. Карточка 157

     Вопрос

     Cubic metres in one litre?

     Ответ

     $1\,L=10^{-3}\,m^3$.

158. Карточка 158

     Вопрос

     What does a zero determinant mean for a square matrix?

     Ответ

     The matrix is singular and has no inverse. A corresponding linear system cannot have one unique solution.

159. Карточка 159

     Вопрос

     Reynolds number for internal flow using characteristic length $L$?

     Ответ

     $$Re=\frac{\rho vL}{\mu}=\frac{vL}{\nu}$$

     It is dimensionless; use consistent properties and the geometry-appropriate characteristic length.

160. Карточка 160

     Вопрос

     Unbiased sample variance for an independent sample from one population?

     Ответ

     $$s^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2$$

     This requires at least two observations, so $n\ge2$. The $n-1$ denominator estimates population variance after fitting the sample mean.

161. Карточка 161

     Вопрос

     How should mutually exclusive alternatives be compared?

     Ответ

     Use one consistent economic measure over a common study period and select the economically preferred feasible alternative; do not accept every alternative with a positive individual worth.

162. Карточка 162

     Вопрос

     Cubic centimetres in one millilitre?

     Ответ

     $1\,mL=1\,cm^3$.

163. Карточка 163

     Вопрос

     Why keep contemporaneous engineering records?

     Ответ

     They preserve assumptions, decisions, calculations, approvals, and changes so the work can be checked and responsibilities understood.

164. Карточка 164

     Вопрос

     Fast check after substituting into a handbook formula?

     Ответ

     Verify that the resulting dimensions match the requested quantity and that additive terms share dimensions.

165. Карточка 165

     Вопрос

     Matrix solution of $A\mathbf{x}=\mathbf{b}$ when $A$ is invertible?

     Ответ

     $$\mathbf{x}=A^{-1}\mathbf{b}$$

     In numerical work, direct factorization is usually preferable to explicitly forming the inverse.

166. Карточка 166

     Вопрос

     Sensible heat for constant specific heat and no phase change?

     Ответ

     $$Q=mc\Delta T$$

     $c$ must match the material, temperature range, mass unit, and temperature-difference unit.

167. Карточка 167

     Вопрос

     Standard error of the mean for an i.i.d. sample with population standard deviation $\sigma$?

     Ответ

     $$SE(\bar{x})=\frac{\sigma}{\sqrt{n}}$$

     The observations are independent and identically distributed with common variance $\sigma^2$. Use $s$ for unknown $\sigma$ only when the selected inference method allows it.

168. Карточка 168

     Вопрос

     Safe setup for converting $25\,ft^2$ to square metres?

     Ответ

     $$25\,ft^2\left(\frac{0.3048\,m}{1\,ft}\right)^2$$

     The squared length factor cancels $ft^2$ and leaves $m^2$.

169. Карточка 169

     Вопрос

     Purpose of incremental analysis for mutually exclusive alternatives?

     Ответ

     It tests whether the additional investment in the higher-cost option earns the required return compared with the lower-cost option.

170. Карточка 170

     Вопрос

     Physical meaning of $dy/dx$?

     Ответ

     It is the instantaneous rate of change of $y$ with respect to $x$, with units of the $y$ unit divided by the $x$ unit.

171. Карточка 171

     Вопрос

     One-dimensional steady conduction through a plane wall?

     Ответ

     $$\dot Q=-kA\frac{dT}{dx}$$

     The minus sign indicates heat flows toward lower temperature; $k$ is thermal conductivity.

172. Карточка 172

     Вопрос

     When should you algebraically rearrange a handbook relationship?

     Ответ

     After confirming the correct equation, definitions, assumptions, and sign convention. Then isolate the requested variable without changing its domain restrictions.

173. Карточка 173

     Вопрос

     Safe setup for converting $2\,in^3$ to cubic metres?

     Ответ

     $$2\,in^3\left(\frac{0.0254\,m}{1\,in}\right)^3$$

     Cube the exact length conversion because the original quantity is volume.

174. Карточка 174

     Вопрос

     A client changes a safety-critical design requirement verbally. Proper response?

     Ответ

     Confirm the change in writing, analyze its consequences, and obtain the required approvals before relying on it.

175. Карточка 175

     Вопрос

     Derivative of $x^n$ for constant $n$?

     Ответ

     $$\frac{d}{dx}x^n=nx^{n-1}$$

     Apply domain restrictions when $n$ is non-integer or negative.

176. Карточка 176

     Вопрос

     Convective heat-transfer rate at a surface?

     Ответ

     $$\dot Q=hA(T_s-T_\infty)$$

     $h$ depends on the flow and geometry; the sign follows the chosen temperature difference.

177. Карточка 177

     Вопрос

     What does the central limit theorem say about a sample mean?

     Ответ

     For sufficiently large independent, identically distributed samples with finite variance, the sampling distribution of the properly standardized mean approaches normality. It does not make the raw data normal.

178. Карточка 178

     Вопрос

     Internal rate of return?

     Ответ

     The discount rate that makes net present worth zero. Multiple sign changes in cash flow can produce multiple or no meaningful real rates.

179. Карточка 179

     Вопрос

     Several similar handbook equations appear together. How do you choose?

     Ответ

     Match each equation's stated geometry, boundary conditions, material model, and variable definitions to the problem.

180. Карточка 180

     Вопрос

     Convert Celsius temperature to kelvin?

     Ответ

     $$T_K=T_{^\circ C}+273.15$$

     This conversion is for absolute temperature, not temperature difference.

181. Карточка 181

     Вопрос

     Derivative of a product $u(x)v(x)$?

     Ответ

     $$(uv)'=u'v+uv'$$

182. Карточка 182

     Вопрос

     Internal escalation does not resolve a serious public-safety threat. Next step?

     Ответ

     Use the next appropriate authority or reporting channel, consistent with law and the urgency of the hazard. Keep a factual record.

183. Карточка 183

     Вопрос

     Two-sided confidence interval for a population mean with known $\sigma$?

     Ответ

     $$\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$$

     Use the normal-model conditions and the critical value for confidence level $1-\alpha$.

184. Карточка 184

     Вопрос

     Net thermal radiation from a small gray surface to large surroundings?

     Ответ

     $$\dot Q=\varepsilon\sigma A(T_s^4-T_{sur}^4)$$

     Use absolute temperatures; the model assumes the stated gray-surface surroundings.

185. Карточка 185

     Вопрос

     Basic benefit-cost acceptance rule for an independent public project?

     Ответ

     Accept when the consistently defined benefit-cost ratio is at least one. For mutually exclusive choices, use an incremental comparison rather than ranking by individual ratios alone.

186. Карточка 186

     Вопрос

     Derivative of a composite function $f(g(x))$?

     Ответ

     $$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$

187. Карточка 187

     Вопрос

     Why inspect boundary or initial conditions near a formula?

     Ответ

     They determine which solution form applies and can eliminate constants or terms. A formula without its conditions may be unusable.

188. Карточка 188

     Вопрос

     Convert Celsius temperature to Fahrenheit?

     Ответ

     $$T_{^\circ F}=\frac{9}{5}T_{^\circ C}+32$$

189. Карточка 189

     Вопрос

     Ethical basis for expert technical testimony?

     Ответ

     Offer opinions only within your competence, disclose important assumptions and limitations, and distinguish facts from professional judgment.

190. Карточка 190

     Вопрос

     Meaning of a breakeven value?

     Ответ

     The value of a selected variable that makes two alternatives economically equivalent or makes net worth zero. Other assumptions are held fixed.

191. Карточка 191

     Вопрос

     Derivative of $e^x$?

     Ответ

     $$\frac{d}{dx}e^x=e^x$$

192. Карточка 192

     Вопрос

     First-law energy balance for a closed system with work positive out?

     Ответ

     $$\Delta E=Q-W$$

     $Q$ is heat into the system and $W$ is work done by the system; state a different sign convention before using it.

193. Карточка 193

     Вопрос

     Frequentist meaning of a 95% confidence procedure?

     Ответ

     Across repeated samples, about 95% of intervals built by the procedure would contain the fixed population parameter. It is not a 95% probability assigned to this already-computed interval.

194. Карточка 194

     Вопрос

     A needed value falls between handbook table entries. What is the cue?

     Ответ

     Check whether interpolation is appropriate, keep both axes and units straight, and avoid extrapolating unless the source supports it.

195. Карточка 195

     Вопрос

     Derivative of $\ln x$ over the real domain?

     Ответ

     $$\frac{d}{dx}\ln x=\frac{1}{x}$$

     This real-valued form requires $x>0$.

196. Карточка 196

     Вопрос

     Relationship between Celsius and kelvin temperature differences?

     Ответ

     $1\,K$ of temperature difference equals $1\,^{\circ}C$ of temperature difference. Do not add $273.15$ to a difference.

197. Карточка 197

     Вопрос

     How should credit for team engineering work be assigned?

     Ответ

     Accurately recognize each contributor's work and responsibility. Do not claim sole credit for a collaborative result.

198. Карточка 198

     Вопрос

     Main limitation of simple payback period?

     Ответ

     It ignores the time value of money and usually ignores cash flows after payback. Discounted payback fixes the first issue, not the second.

199. Карточка 199

     Вопрос

     Roles of the null and alternative hypotheses?

     Ответ

     The null states the baseline parameter claim tested; the alternative states the competing direction or difference supported when evidence is strong enough.

200. Карточка 200

     Вопрос

     Ideal-gas equation of state?

     Ответ

     $$pV=nRT$$

     Use absolute temperature and a gas constant consistent with pressure, volume, amount, and energy units.

201. Карточка 201

     Вопрос

     First check when reading a handbook table?

     Ответ

     Read the title, column headings, units, notes, and basis conditions before selecting a value.

202. Карточка 202

     Вопрос

     Candidates for absolute extrema of a continuous function on a closed interval?

     Ответ

     The endpoints and every interior point where the first derivative is zero or does not exist. Evaluate the function at each candidate; the largest and smallest values are the absolute extrema.

203. Карточка 203

     Вопрос

     How should a sunk cost affect a current engineering-economy decision?

     Ответ

     It should not affect the choice because it cannot be changed by the decision. Use future incremental cash flows instead.

204. Карточка 204

     Вопрос

     Convert Fahrenheit absolute temperature to Rankine?

     Ответ

     $$T_{^\circ R}=T_{^\circ F}+459.67$$

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Карточка 205

     Вопрос

     Ohm's law for an ideal resistor?

     Ответ

     $$V=IR$$

     Use a consistent passive sign convention; $R$ is constant only over the assumed operating range.

206. Карточка 206

     Вопрос

     You have confidential details about a competitor's bid. Proper response?

     Ответ

     Do not use or disclose improperly obtained information. Notify the appropriate procurement or ethics contact.

207. Карточка 207

     Вопрос

     Physical meaning of a definite integral $\int_a^b f(x)\,dx$?

     Ответ

     It is signed accumulation of $f$ from $a$ to $b$. Its units are the units of $f$ multiplied by the units of $x$.

208. Карточка 208

     Вопрос

     Is a radian dimensionless?

     Ответ

     Yes. It is the ratio of arc length to radius, so the length units cancel; keep $rad$ when it clarifies angular meaning.

209. Карточка 209

     Вопрос

     Electrical power absorbed under the passive sign convention?

     Ответ

     $$P=VI$$

     For a resistor, this also gives $P=I^2R=V^2/R$ when Ohm's law applies.

210. Карточка 210

     Вопрос

     Meaning of a p-value?

     Ответ

     Assuming the null hypothesis and model are correct, it is the probability of a result at least as extreme as the observed result. It is not the probability that the null is true.

211. Карточка 211

     Вопрос

     First check when reading a handbook graph?

     Ответ

     Identify both axes, scales, units, curve labels, and any log scale before estimating a value.

212. Карточка 212

     Вопрос

     Difference between mass and weight?

     Ответ

     Mass measures inertia; weight is gravitational force on that mass. Their dimensions and units are different.

213. Карточка 213

     Вопрос

     Kirchhoff's current law at a node?

     Ответ

     The algebraic sum of currents at a node is zero. Choose one sign convention, such as currents entering positive, and apply it consistently.

214. Карточка 214

     Вопрос

     Evaluate $\int_a^b f(x)\,dx$ from an antiderivative $F$?

     Ответ

     $$\int_a^b f(x)\,dx=F(b)-F(a)$$

     This applies when $F'=f$ on the interval under the usual continuity conditions.

215. Карточка 215

     Вопрос

     Opportunity cost?

     Ответ

     The value of the best forgone alternative. Include it when choosing one use of a scarce resource prevents another valuable use.

216. Карточка 216

     Вопрос

     Weight from mass in a gravitational field?

     Ответ

     $$W=mg$$

     Use compatible units; $g$ is local gravitational acceleration, not a universal constant for every location.

217. Карточка 217

     Вопрос

     Kirchhoff's voltage law around a closed loop?

     Ответ

     The algebraic sum of voltage rises and drops around the loop is zero. Traverse the loop once with a consistent polarity convention.

218. Карточка 218

     Вопрос

     Decision rule using significance level $\alpha$?

     Ответ

     Reject the null when the p-value is at most $\alpha$; otherwise fail to reject it. Failing to reject does not prove the null.

219. Карточка 219

     Вопрос

     Can proprietary methods from a former employer be reused freely?

     Ответ

     Only if you have the legal right or permission to use them. General skill and experience may transfer; confidential proprietary material does not.

220. Карточка 220

     Вопрос

     Pound-force to newtons?

     Ответ

     $1\,lbf\approx4.44822\,N$. Use more digits only when the required precision justifies them.

221. Карточка 221

     Вопрос

     When does a definite integral equal ordinary geometric area?

     Ответ

     When the integrand is nonnegative over the interval. If it changes sign, the integral is signed area; integrate the absolute value or split the interval for total area.

222. Карточка 222

     Вопрос

     Charge-voltage relation for an ideal capacitor?

     Ответ

     $$q=CV$$

     $C$ is capacitance in farads; $V$ is the voltage across the capacitor under the chosen polarity.

223. Карточка 223

     Вопрос

     Searching for a material or fluid property: what qualifiers often matter?

     Ответ

     Material identity, phase, temperature, pressure, and composition. Use only the qualifiers relevant to that property source.

224. Карточка 224

     Вопрос

     Force produced when one slug accelerates at $1\,ft/s^2$?

     Ответ

     $1\,lbf$. Equivalently, $1\,lbf=1\,slug\cdot ft/s^2$.

225. Карточка 225

     Вопрос

     Relationship among nominal rate $i$, real rate $r$, and inflation $f$?

     Ответ

     $$1+i=(1+r)(1+f)$$

     Keep cash-flow dollars and discount rates on the same real-or-nominal basis.

226. Карточка 226

     Вопрос

     Antiderivative of $x^n$ for $n\ne-1$?

     Ответ

     $$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$$

227. Карточка 227

     Вопрос

     Time constant of a first-order resistor-capacitor circuit?

     Ответ

     $$\tau=RC$$

     Use the resistance seen by the capacitor in the applicable first-order linear circuit.

228. Карточка 228

     Вопрос

     What physical quantity does pound-mass measure?

     Ответ

     Mass. Do not substitute $lbm$ directly where a force unit such as $lbf$ is required.

229. Карточка 229

     Вопрос

     Type I error?

     Ответ

     Rejecting a true null hypothesis. Its probability is controlled by the chosen significance level under the test assumptions.

230. Карточка 230

     Вопрос

     A search hit contains the right words. Is that enough?

     Ответ

     No. Read the surrounding definition, equation label, units, and notes to confirm the hit answers the actual question.

231. Карточка 231

     Вопрос

     Annual straight-line depreciation from basis $B$ to salvage $S$ over $n$ years?

     Ответ

     $$D=\frac{B-S}{n}$$

     This is an accounting allocation, not a market-value forecast.

232. Карточка 232

     Вопрос

     Purpose of $g_c$ in some US customary force-mass equations?

     Ответ

     It reconciles chosen $lbm$, $lbf$, foot, and second units. Include it only when the equation and unit convention require it; coherent SI equations do not use it.

233. Карточка 233

     Вопрос

     Recognizable form of a separable first-order differential equation?

     Ответ

     $$\frac{dy}{dx}=g(x)h(y)$$

     Rearrange to place the $y$ terms with $dy$ and the $x$ terms with $dx$, while checking for solutions lost by division.

234. Карточка 234

     Вопрос

     Using unverified software output in a safety-relevant calculation?

     Ответ

     Check inputs, assumptions, units, and results with an independent method appropriate to the risk. The engineer remains responsible for professional judgment.

235. Карточка 235

     Вопрос

     Equivalent resistance of ideal resistors in series?

     Ответ

     $$R_{eq}=\sum_i R_i$$

     Series elements carry the same current with no branching node between them.

236. Карточка 236

     Вопрос

     Pressure conversion from psi to pascals?

     Ответ

     $1\,psi\approx6894.76\,Pa$. Keep enough significant figures for the problem's data.

237. Карточка 237

     Вопрос

     Type II error?

     Ответ

     Failing to reject a false null hypothesis. Its probability is $\beta$ for a specified alternative.

238. Карточка 238

     Вопрос

     Useful flashcard to make after a failed handbook search?

     Ответ

     A short scenario-to-search-cue card recording the section name and terms that eventually worked—not a copied problem or page number.

239. Карточка 239

     Вопрос

     Standard form of a first-order linear differential equation?

     Ответ

     $$\frac{dy}{dx}+P(x)y=Q(x)$$

     An integrating factor is $e^{\int P(x)\,dx}$.

240. Карточка 240

     Вопрос

     Equivalent resistance of ideal resistors in parallel?

     Ответ

     $$\frac{1}{R_{eq}}=\sum_i\frac{1}{R_i}$$

     Parallel elements share the same two nodes and therefore the same voltage.

241. Карточка 241

     Вопрос

     Book value after $k$ years of straight-line depreciation?

     Ответ

     $$BV_k=B-kD$$

     Apply it within the depreciable life and do not depreciate below the assumed salvage value.

242. Карточка 242

     Вопрос

     Standard atmosphere in pascals?

     Ответ

     $1\,atm=101{,}325\,Pa$ exactly.

243. Карточка 243

     Вопрос

     Using AI-generated technical content in engineering work?

     Ответ

     Treat it as unverified input: protect confidential data, check sources and calculations, disclose use when required, and keep accountable human review.

244. Карточка 244

     Вопрос

     Trial form for a homogeneous linear ODE with constant coefficients?

     Ответ

     Use $y=e^{rx}$ to obtain the characteristic polynomial in $r$. Repeated or complex roots require the corresponding standard solution forms.

245. Карточка 245

     Вопрос

     Wave-speed relationship among frequency and wavelength?

     Ответ

     $$v=f\lambda$$

     Use frequency in cycles per second and wavelength in compatible length units.

246. Карточка 246

     Вопрос

     Statistical power?

     Ответ

     The probability of rejecting the null when a specified alternative is true: $1-\beta$. Larger effects, larger samples, or lower variability commonly increase power.

247. Карточка 247

     Вопрос

     Bar in pascals?

     Ответ

     $1\,bar=100{,}000\,Pa$ exactly.

248. Карточка 248

     Вопрос

     How should handbook navigation be practiced before the FE exam?

     Ответ

     Use the current handbook from MyNCEES while solving original or properly licensed practice problems, and rehearse the exact search terms that find definitions and assumptions.

249. Карточка 249

     Вопрос

     Is depreciation itself a cash outflow?

     Ответ

     No. It is a noncash accounting expense, though it can affect taxes and therefore after-tax cash flow.

250. Карточка 250

     Вопрос

     First-order linear approximation of $f(x)$ near $x=a$?

     Ответ

     $$f(x)\approx f(a)+f'(a)(x-a)$$

     It is most accurate for $x$ close to $a$ when higher derivatives stay bounded.

251. Карточка 251

     Вопрос

     Relationship between period and frequency?

     Ответ

     $$T=\frac{1}{f}$$

     This requires $f>0$; $T$ has time units.

252. Карточка 252

     Вопрос

     How should foreseeable environmental harm enter an engineering decision?

     Ответ

     Identify and communicate it, follow applicable requirements, and consider practicable ways to reduce harm without compromising public safety.

253. Карточка 253

     Вопрос

     Foot-pound-force in joules?

     Ответ

     $1\,ft\cdot lbf\approx1.35582\,J$. This is a work or energy conversion; keep torque labeled as force times distance even though its dimensions match energy.

254. Карточка 254

     Вопрос

     Allowed range and sign meaning of Pearson correlation $r$?

     Ответ

     $-1\le r\le1$. The sign gives the direction of linear association, and magnitude measures its linear strength.

255. Карточка 255

     Вопрос

     When should you search instead of relying on recall?

     Ответ

     Search when a constant, coefficient, convention, specialized relationship, or applicability detail could change the result. Recall can start the route; the reference should confirm details.

256. Карточка 256

     Вопрос

     After-tax effect of a deductible expense $E$ at marginal tax rate $t$?

     Ответ

     The net cost is $E(1-t)$ under the simplified assumption that the deduction can be used fully in that period.

257. Карточка 257

     Вопрос

     Centered finite-difference approximation to $f'(x)$?

     Ответ

     $$f'(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$

     For smooth $f$, truncation error is proportional to $h^2$ before roundoff dominates.

258. Карточка 258

     Вопрос

     General material balance on a control volume?

     Ответ

     $$\text{accumulation}=\text{in}-\text{out}+\text{generation}-\text{consumption}$$

     At steady state, accumulation is zero; generation and consumption depend on reactions or sources.

259. Карточка 259

     Вопрос

     Approximate joules in one International Table Btu?

     Ответ

     $1\,Btu_{IT}\approx1055.06\,J$. Identify the Btu convention when precision matters.

260. Карточка 260

     Вопрос

     What is the defender in a replacement analysis?

     Ответ

     The currently owned asset being compared with a proposed replacement. Its current market value is an opportunity cost; its original purchase price is sunk.

261. Карточка 261

     Вопрос

     Speaking publicly about a technical matter outside your expertise?

     Ответ

     Limit claims to what you can support, identify the boundary of your expertise, and avoid implying qualifications you do not have.

262. Карточка 262

     Вопрос

     Single-interval trapezoidal approximation to an integral?

     Ответ

     $$\int_a^b f(x)\,dx\approx\frac{b-a}{2}[f(a)+f(b)]$$

     It is exact for linear $f$.

263. Карточка 263

     Вопрос

     Energy efficiency of a device?

     Ответ

     $$\eta=\frac{\text{useful output energy}}{\text{input energy}}$$

     Use the same energy basis in numerator and denominator; a real passive device has $0\le\eta\le1$.

264. Карточка 264

     Вопрос

     Purpose of sensitivity analysis in engineering economics?

     Ответ

     It shows how the decision changes when an uncertain input changes. Vary inputs over plausible ranges while preserving units and model relationships.

265. Карточка 265

     Вопрос

     Can a strong correlation establish causation?

     Ответ

     No. Correlation alone cannot rule out confounding, reverse direction, selection effects, or coincidence.

266. Карточка 266

     Вопрос

     Where should you confirm the tested topics for your FE discipline?

     Ответ

     Use the discipline specification linked from the official NCEES FE exam hub. This common deck does not replace that discipline-specific boundary.

267. Карточка 267

     Вопрос

     Mechanical horsepower in watts?

     Ответ

     $1\,hp\approx745.7\,W$, based on $550\,ft\cdot lbf/s$. Distinguish it from metric horsepower.

268. Карточка 268

     Вопрос

     Combine independent standard uncertainties $u_1,\ldots,u_n$ in a sum or difference?

     Ответ

     $$u_c=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}$$

     This root-sum-square form assumes uncorrelated inputs and coefficients of magnitude one; otherwise include sensitivities and covariance.

269. Карточка 269

     Вопрос

     Newton iteration for a root of $f(x)=0$?

     Ответ

     $$x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}$$

     It requires $f'(x_k)\ne0$ and a suitable starting value; convergence is not guaranteed.

270. Карточка 270

     Вопрос

     Meaning of slope in simple linear regression of $y$ on $x$?

     Ответ

     It is the estimated change in mean response $y$ for a one-unit increase in $x$, under the fitted linear model. Its units are $y$ units per $x$ unit.

271. Карточка 271

     Вопрос

     Kilowatt-hour in joules?

     Ответ

     $1\,kW\cdot h=3.6\times10^6\,J$. It is energy, not power.

272. Карточка 272

     Вопрос

     Percent error relative to an accepted nonzero value?

     Ответ

     $$\%\,error=\left|\frac{x_{measured}-x_{accepted}}{x_{accepted}}\right|100\%$$

     The accepted value must be nonzero and both values must use compatible units.

273. Карточка 273

     Вопрос

     Does legal compliance alone settle an engineering ethics question?

     Ответ

     No. Law sets mandatory boundaries, while professional duties can require stronger action, especially when safety, honesty, or conflicts are involved.

274. Карточка 274

     Вопрос

     Does handbook fluency replace discipline-specific problem practice?

     Ответ

     No. Pair navigation and common recall with the current discipline specification and timed, rights-safe problems that match your selected FE discipline.

275. Карточка 275

     Вопрос

     Linear interpolation between $(x_1,y_1)$ and $(x_2,y_2)$?

     Ответ

     $$y\approx y_1+(y_2-y_1)\frac{x-x_1}{x_2-x_1}$$

     Use it between distinct $x$ values when local linear behavior is reasonable.

276. Карточка 276

     Вопрос

     Dimensional requirement for a physically meaningful additive equation?

     Ответ

     Every term being added or equated must have the same dimensions. A dimensionally consistent equation can still be physically wrong, but an inconsistent one is wrong.