#1. A triangular construction lot has vertices at (0,0), (18,1), and (18,20) in a Cartesian site grid measured in m. Using the shoelace method, what is the lot area?
#2. A triangular construction lot has vertices at (0,0), (24,2), and (11,14) in a Cartesian site grid measured in ft. Using the shoelace method, what is the lot area?
#3. A triangular construction lot has vertices at (0,0), (19,2), and (12,18) in a Cartesian site grid measured in m. Using the shoelace method, what is the lot area?
#4. A triangular construction lot has vertices at (0,0), (24,-1), and (7,14) in a Cartesian site grid measured in ft. Using the shoelace method, what is the lot area?
#5. A triangular construction lot has vertices at (0,0), (14,3), and (4,13) in a Cartesian site grid measured in m. Using the shoelace method, what is the lot area?
#6. Two retaining-wall faces are modeled by parallel lines 8x + 3y + (-13) = 0 and 8x + 3y + 20 = 0, with coordinates in ft. What is the perpendicular separation between the faces?
#7. Two retaining-wall faces are modeled by parallel lines 6x + 2y + (-22) = 0 and 6x + 2y + 25 = 0, with coordinates in m. What is the perpendicular separation between the faces?
#8. Two retaining-wall faces are modeled by parallel lines 3x + 6y + (-12) = 0 and 3x + 6y + 13 = 0, with coordinates in ft. What is the perpendicular separation between the faces?
#9. Two retaining-wall faces are modeled by parallel lines 6x + 4y + (-9) = 0 and 6x + 4y + 11 = 0, with coordinates in m. What is the perpendicular separation between the faces?
#10. Two retaining-wall faces are modeled by parallel lines 5x + 4y + (-14) = 0 and 5x + 4y + 13 = 0, with coordinates in ft. What is the perpendicular separation between the faces?
#11. A horizontal curve is represented by x^2 + y^2 + (-6)x + (2)y + (-134) = 0. What is the radius of the circle?
#12. A horizontal curve is represented by x^2 + y^2 + (2)x + (-12)y + (-44) = 0. What is the radius of the circle?
#13. A horizontal curve is represented by x^2 + y^2 + (6)x + (-6)y + (-18) = 0. What is the radius of the circle?
#14. A horizontal curve is represented by x^2 + y^2 + (10)x + (-12)y + (12) = 0. What is the radius of the circle?
#15. A horizontal curve is represented by x^2 + y^2 + (-4)x + (8)y + (-61) = 0. What is the radius of the circle?
#16. For the circle (x–1)^2 + (y-4)^2 = 81 and the line y = -0.5x + 12.67, how many real intersection points are there?
#17. For the circle (x-3)^2 + (y-4)^2 = 49 and the line y = -1.5x + -4.12, how many real intersection points are there?
#18. For the circle (x–4)^2 + (y-4)^2 = 25 and the line y = -0.5x + -4.71, how many real intersection points are there?
#19. For the circle (x–1)^2 + (y–3)^2 = 16 and the line y = 1x + -6.53, how many real intersection points are there?
#20. For the circle (x-3)^2 + (y–1)^2 = 64 and the line y = -1x + 13.31, how many real intersection points are there?
#21. Lines 2x + 4y = -6 and 4x + (-4)y = 5 intersect at P. What is the equation of the line through P that is perpendicular to y = -2x + 1?
#22. Lines 3x + 4y = -12 and 4x + (-5)y = 14 intersect at P. What is the equation of the line through P that is perpendicular to y = -2x + 1?
#23. Lines 3x + 1y = -15 and 2x + (-4)y = 12 intersect at P. What is the equation of the line through P that is perpendicular to y = 3x + 1?
#24. Lines 2x + 3y = -11 and 2x + (-5)y = 11 intersect at P. What is the equation of the line through P that is perpendicular to y = -2x + 1?
#25. Lines 1x + 2y = -16 and 4x + (-2)y = 11 intersect at P. What is the equation of the line through P that is perpendicular to y = -0.5x + 1?
#26. A dimensionless loading index is modeled by f(x) = 3x^2 + 6/x for x in [1, 7]. Compute I = integral from 1 to 7 of f(x) dx.
#27. A dimensionless loading index is modeled by f(x) = 4x^2 + 6/x for x in [3, 7]. Compute I = integral from 3 to 7 of f(x) dx.
#28. A dimensionless loading index is modeled by f(x) = 4x^2 + 4/x for x in [1, 4]. Compute I = integral from 1 to 4 of f(x) dx.
#29. A dimensionless loading index is modeled by f(x) = 2x^2 + 3/x for x in [3, 7]. Compute I = integral from 3 to 7 of f(x) dx.
#30. A dimensionless loading index is modeled by f(x) = 2x^2 + 7/x for x in [1, 5]. Compute I = integral from 1 to 5 of f(x) dx.
#31. In an earthwork cross-section, boundaries are y = 12 – x and y = 3x^2. What is the enclosed area between the two curves?
#32. In an earthwork cross-section, boundaries are y = 10 – x and y = 1x^2. What is the enclosed area between the two curves?
#33. In an earthwork cross-section, boundaries are y = 6 – x and y = 3x^2. What is the enclosed area between the two curves?
#34. In an earthwork cross-section, boundaries are y = 7 – x and y = 2x^2. What is the enclosed area between the two curves?
#35. In an earthwork cross-section, boundaries are y = 8 – x and y = 1x^2. What is the enclosed area between the two curves?
#36. A design penalty function is C(x) = 7x + 215/x + 14 for x>0. At what x is C(x) minimized?
#37. A design penalty function is C(x) = 8x + 68/x + 5 for x>0. At what x is C(x) minimized?
#38. A design penalty function is C(x) = 2x + 187/x + 11 for x>0. At what x is C(x) minimized?
#39. A design penalty function is C(x) = 3x + 139/x + 7 for x>0. At what x is C(x) minimized?
#40. A design penalty function is C(x) = 4x + 90/x + 19 for x>0. At what x is C(x) minimized?
#41. Use first-order linearization about x0 = 2 to estimate sqrt(7x + 35) at x = 2.12.
#42. Use first-order linearization about x0 = 2 to estimate sqrt(4x + 18) at x = 2.08.
#43. Use first-order linearization about x0 = 3 to estimate sqrt(5x + 19) at x = 3.12.
#44. Use first-order linearization about x0 = 4 to estimate sqrt(7x + 25) at x = 4.08.
#45. Use first-order linearization about x0 = 2 to estimate sqrt(9x + 18) at x = 2.08.
#46. Apply one Newton-Raphson iteration to estimate the root of x^4 – 102 = 0 starting from x0 = 7. What is x1?
#47. Apply one Newton-Raphson iteration to estimate the root of x^2 – 77 = 0 starting from x0 = 2. What is x1?
#48. Apply one Newton-Raphson iteration to estimate the root of x^2 – 209 = 0 starting from x0 = 6. What is x1?
#49. Apply one Newton-Raphson iteration to estimate the root of x^5 – 85 = 0 starting from x0 = 2. What is x1?
#50. Apply one Newton-Raphson iteration to estimate the root of x^2 – 203 = 0 starting from x0 = 3. What is x1?
#51. For force vectors A=[-3, 5, 7], B=[-2, 1, -1], and C=[-1, 7, -6], compute the magnitude of resultant R = A – 2B + 3C in kN.
#52. For force vectors A=[7, 1, 2], B=[-1, -3, 6], and C=[-5, 9, 1], compute the magnitude of resultant R = A – 2B + 3C in kip.
#53. For force vectors A=[5, 3, 1], B=[1, -6, 0], and C=[6, 4, 2], compute the magnitude of resultant R = A – 2B + 3C in kN.
#54. For force vectors A=[5, 6, 4], B=[-6, -3, 2], and C=[-1, 2, -5], compute the magnitude of resultant R = A – 2B + 3C in kip.
#55. For force vectors A=[5, 4, 7], B=[-3, 6, 0], and C=[2, -5, 7], compute the magnitude of resultant R = A – 2B + 3C in kN.
#56. Two direction vectors are u=[6, -1, 2] and v=[6, 2, 6]. What is the angle between them in degrees?
#57. Two direction vectors are u=[5, 1, 5] and v=[7, -2, 7]. What is the angle between them in degrees?
#58. Two direction vectors are u=[3, -4, 7] and v=[7, 3, 3]. What is the angle between them in degrees?
#59. Two direction vectors are u=[7, 1, 1] and v=[5, -3, 4]. What is the angle between them in degrees?
#60. Two direction vectors are u=[8, 0, 8] and v=[4, 1, 8]. What is the angle between them in degrees?
#61. A bracing panel is spanned by vectors u=[5, -4, -1] m and v=[3, 5, 5] m. The panel area equals |u x v|. What is that area?
#62. A bracing panel is spanned by vectors u=[-4, 7, -2] m and v=[5, -1, -2] m. The panel area equals |u x v|. What is that area?
#63. A bracing panel is spanned by vectors u=[-5, -5, -2] m and v=[2, 4, 7] m. The panel area equals |u x v|. What is that area?
#64. A bracing panel is spanned by vectors u=[1, 5, 4] m and v=[-2, 6, 6] m. The panel area equals |u x v|. What is that area?
#65. A bracing panel is spanned by vectors u=[1, -2, -3] m and v=[5, 6, -5] m. The panel area equals |u x v|. What is that area?
#66. Three edge vectors of a skewed material cell are a=[-1, -2, 4], b=[3, -4, 4], c=[-1, -3, 3] (ft). What is the cell volume |a·(b×c)|?
#67. Three edge vectors of a skewed material cell are a=[6, 4, 4], b=[5, 1, 3], c=[5, 4, 2] (ft). What is the cell volume |a·(b×c)|?
#68. Three edge vectors of a skewed material cell are a=[-2, 3, 3], b=[0, -1, 6], c=[0, 4, 3] (ft). What is the cell volume |a·(b×c)|?
#69. Three edge vectors of a skewed material cell are a=[0, 3, -3], b=[0, -1, 0], c=[1, 1, 4] (ft). What is the cell volume |a·(b×c)|?
#70. Three edge vectors of a skewed material cell are a=[-2, -1, 2], b=[-2, -1, -3], c=[2, 2, 1] (ft). What is the cell volume |a·(b×c)|?
#71. Given vectors a=[7, -5, 0] and b=[7, 7, 1], what is the scalar projection of a onto b (comp_b a)?
#72. Given vectors a=[9, -6, 5] and b=[5, 7, 7], what is the scalar projection of a onto b (comp_b a)?
#73. Given vectors a=[9, 1, 2] and b=[7, 8, 1], what is the scalar projection of a onto b (comp_b a)?
#74. Given vectors a=[6, -1, 8] and b=[3, 9, 1], what is the scalar projection of a onto b (comp_b a)?
#75. Given vectors a=[-4, 7, -2] and b=[8, 3, 1], what is the scalar projection of a onto b (comp_b a)?
#76. Material test results occur at values 20, 29, and 55 MPa with frequencies 6, 7, and 7. What is the weighted mean strength?
#77. Material test results occur at values 18, 39, and 43 MPa with frequencies 7, 5, and 9. What is the weighted mean strength?
#78. Material test results occur at values 21, 29, and 43 MPa with frequencies 10, 10, and 4. What is the weighted mean strength?
#79. Material test results occur at values 16, 39, and 41 MPa with frequencies 8, 6, and 5. What is the weighted mean strength?
#80. Material test results occur at values 13, 35, and 47 MPa with frequencies 7, 10, and 6. What is the weighted mean strength?
#81. Assume X is normally distributed with mean 68 and standard deviation 10. What is P(63 <= X <= 73)?
#82. Assume X is normally distributed with mean 36 and standard deviation 10. What is P(21 <= X <= 51)?
#83. Assume X is normally distributed with mean 55 and standard deviation 5. What is P(47.5 <= X <= 60)?
#84. Assume X is normally distributed with mean 73 and standard deviation 10. What is P(58 <= X <= 78)?
#85. Assume X is normally distributed with mean 72 and standard deviation 6. What is P(63 <= X <= 81)?
#86. For a process with known sigma = 6.4, sample size n = 25, and z* = 2.576 for a 99% confidence level, what is the margin of error E for the mean?
#87. For a process with known sigma = 4.5, sample size n = 64, and z* = 1.96 for a 95% confidence level, what is the margin of error E for the mean?
#88. For a process with known sigma = 7.1, sample size n = 81, and z* = 2.576 for a 99% confidence level, what is the margin of error E for the mean?
#89. For a process with known sigma = 7.1, sample size n = 49, and z* = 2.576 for a 99% confidence level, what is the margin of error E for the mean?
#90. For a process with known sigma = 5.8, sample size n = 64, and z* = 1.96 for a 95% confidence level, what is the margin of error E for the mean?
#91. A linear model y = b0 + b1x has summary values: xbar=41, ybar=31, Sxx=206, Sxy=288.4. What is the predicted y at x=44?
#92. A linear model y = b0 + b1x has summary values: xbar=42, ybar=81, Sxx=146, Sxy=204.4. What is the predicted y at x=47?
#93. A linear model y = b0 + b1x has summary values: xbar=55, ybar=30, Sxx=212, Sxy=233.2. What is the predicted y at x=58?
#94. A linear model y = b0 + b1x has summary values: xbar=51, ybar=65, Sxx=141, Sxy=239.7. What is the predicted y at x=59?
#95. A linear model y = b0 + b1x has summary values: xbar=48, ybar=80, Sxx=84, Sxy=67.2. What is the predicted y at x=53?
#96. A geotechnical alert model flags instability (event B). Prior probability of true instability A is 0.2. Sensitivity P(B|A)=0.85, false-positive rate P(B|not A)=0.2. What is P(A|B)?
#97. A geotechnical alert model flags instability (event B). Prior probability of true instability A is 0.35. Sensitivity P(B|A)=0.85, false-positive rate P(B|not A)=0.15. What is P(A|B)?
#98. A geotechnical alert model flags instability (event B). Prior probability of true instability A is 0.25. Sensitivity P(B|A)=0.9, false-positive rate P(B|not A)=0.12. What is P(A|B)?
#99. A geotechnical alert model flags instability (event B). Prior probability of true instability A is 0.2. Sensitivity P(B|A)=0.7, false-positive rate P(B|not A)=0.1. What is P(A|B)?
#100. A geotechnical alert model flags instability (event B). Prior probability of true instability A is 0.35. Sensitivity P(B|A)=0.8, false-positive rate P(B|not A)=0.1. What is P(A|B)?
