This is the Multiple Choice Questions Part 2 of the Series in Differential Calculus (Maxima/Minima and Time Rates) topic in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

### Multiple Choice Questions Topic Outline

- MCQs in Maxima | MCQs in Minima | MCQs in Time Rates | MCQs in Relation between the variables | MCQs in Maxima/Minima values

### Online Questions and Answers in Differential Calculus (Limits and Derivatives) Series

Following is the list of multiple choice questions in this brand new series:

**Differential Calculus (Maxima/Minima and Time Rates) MCQs**

**MCQs from Number 1 – 50**Answer key:

**PART I**

**MCQs from Number 2 – 100**Answer key:

**PART II**

### Continue Practice Exam Test Questions Part II of the Series

**Choose the letter of the best answer in each questions.**

51. Find two numbers whose sum is 20, if the product of one by the cube of another is to be the maximum.

- a. 4 and 16
- b. 10 and 10
- c. 5 and 15
- d. 8 and 12

52. The sum of two numbers is 12. Find the minimum value of the sum of their cubes.

- a. 644
- b. 432
- c. 346
- d. 244

53. A printed page must contain 60 sq.m. of printed material. There are to be margins of 5cm. on either side and the margins of 3 cm. on top and bottom. How long should the printed lines be in order to minimize the amount of paper used?

- a. 10
- b. 18
- c. 12
- d. 15

54. A school sponsored trip will cost each student 15 pesos if not more than 150 students make the trip. However, the cost will be reduced by 5 centavos for each student in excess of 150. How many students should make the trip in order for the school to receive the largest group income?

- a. 250
- b. 225
- c. 200
- d. 195

55. A rectangular box with square base and open at the top is to have a capacity of 16823 cu.cm. Find the height of the box that requires minimum amount of material required.

- a. 16.14cm
- b. 14.12cm
- c. 12.14cm
- d. 10.36cm

56. A closed cylindrical tank has a capacity of 576.56 cubic meters. Find the minimum surface area of the tank.

- a. 218.60 cubic meters
- b. 412.60 cubic meters
- c. 516.32 cubic meters
- d. 383.40 cubic meters

For Problems 57-59:

Two vertices of a rectangle are on the x axis. The other two vertices are on the lines whose equations are y = 2x and 3x + y = 30.

.57. If the area of the rectangle is maximum, find the value of y.

- a. 8
- b. 7
- c. 9
- d. 6

58. Compute the maximum area of the rectangle.

- a. 30 sq. units
- b. 70 sq. units
- c. 90 sq. units
- d. 40 sq. units

59. At what point from the intersection of the x and y axes will the farthest vertex of the rectangle be located along the x axis so that its area is maximum.

- a. 9 units
- b. 7 units
- c. 8 units
- d. 6 units

60. A wall 2.245 m high, is “x” meters away from a building. The shortest ladder that can reach the building with one end resting on the ground outside the wall is 6 m. What is the value of x?

- a. 2 m
- b. 4 m
- c. 6 m
- d. 8 m

61. With only 381.7 square meter of materials, a closed cylindrical tank of maximum volume. What is to be the height of the tank in m?

- a. 7 m
- b. 9 m
- c. 11 m
- d. 13 m

62. If the hypotenuse of a right triangle is known, what is the ratio of the base and the altitude of the right triangle when its area is maximum?

- a. 1:1
- b. 1:2
- c. 1:3
- d. 1:4

63. What is the maximum length of the perimeter if the hypotenuse of a right triangle is 5 m long?

- a. 12.08 m
- b. 15.09 m
- c. 20.09 m
- d. 8.99 m

64. An open top rectangular tank with square bases is to have a volume of 10 cubic meters. The material for its bottom is to cost 15 cents per square meter and that for the sides 6 cents per square meter. Find the most economical dimension for the tank.

- a. 2 x 2 x 2.5
- b. 2 x 5 x 2.5
- c. 2 x 3 x 2.5
- d. 2 x 4 x 2.5

65. A trapezoidal gutter is to be made from a strip of metal 22 m wide by bending up the sides. If the base is 14 m, what width across the top gives the greatest carrying capacity?

- a. 10
- b. 22
- c. 16
- d. 27

66. Divide the number 60 into two parts so that the product P of one part and the square of the other is the maximum. Find the smallest part.

- a. 20
- b. 10
- c. 22
- d. 27

67. The edges of a rectangular box are to be reinforced with narrow metal strips. If the box will have a volume of 8 cubic meters, what would its dimension be to require the least total length of strips?

- a. 2 x 2 x 2
- b. 4 x 4 x 4
- c. 3 x 3 x 3
- d. 2 x 2 x 4

68. A rectangular window surmounted by a right isosceles triangle has a perimeter equal to 54.14 m. Find the height of the rectangular window so that the window will admit the most light.

- a. 10
- b. 22
- c. 12
- d. 27

69. A normal widow is in the shape of a rectangle surrounded by a semi-circle. If the perimeter of the window is 71.416, what is the radius and the height of the rectangular portion so that it will yield a window admitting the most light.

- a. 12
- b. 20
- c. 22
- d. 27

70. Find the radius of a right circular cone having a lateral area of 544.12 sq. m. to have a maximum value.

- a. 10
- b. 20
- c. 17
- d. 19

71. A gutter with trapezoidal cross section is to be made from a long sheet of tin that is 15 cm. wide by turning up one third of its width on each side. What is the width across the top that will give a maximum capacity?

- a. 10
- b. 20
- c. 15
- d. 13

72. A piece of plywood for a billboard has an area of 24 sq. feet. The margins at the top and bottom are 9 inches and at the sides are 6 in. Determine the size of the plywood for maximum dimensions of the painted area.

- a. 3x8
- b. 3x4
- c. 4x8
- d. 4x6

73. A manufacturer estimates that the cost of production of “x” units of a certain item is C = 40x – 0.02x^{2} – 600. How many units should be produced for minimum cost?

- a. 10 units
- b. 100 units
- c. 1000 units
- d. 10000 units

74. If the sum of the two numbers is 4, find the minimum value of the sum of their cubes.

- a. 10
- b. 18
- c. 16
- d. 32

75. If x units of a certain item are manufactured, each unit can be sold for 200 – 0.01x pesos. How many units can be manufactured for maximum revenue? What is the corresponding unit price?

- a. 10000,P100
- b. 10500,P300
- c. 20000,P200
- d. 15000,P400

76. A certain spare parts has a selling price of P150 if they would sell 8000 units per month. If for every P1.00 increase in selling price, 80 units less will be sold out per month. If the production cost is P100 per unit, find the price per unit for maximum profit per month.

- a. P150
- b. P250
- c. P175
- d. P225

77. The highway department is planning to build a picnic area for motorist along a major highway. It is to be rectangular with an area of 5000 sq. m. is to be fenced off on the three sides not adjacent to the highway. What is the least amount of fencing that will be needed to complete the job?

- a. 200 m.
- b. 300 m.
- c. 400 m.
- d. 500 m.

78. A rectangular lot has an area of 1600 sq. m. find the least amount of fence that could be used to enclose the area.

- a. 100 m.
- b. 160 m.
- c. 200 m.
- d. 300 m.

79. A student club on a college campus charges annual membership dues of P10, less 5 centavos for each member over 60. How many members would give the club the most revenue from annual dues?

- a. 130 members
- b. 420 members
- c. 240 members
- d. 650 members

80. A monthly overhead of a manufacturer of a certain commodity is P6000 and the cost of the material is P1.0 per unit. If not more than 4500 units are manufactured per month, labor cost is P0.40 per unit, but for each unit over 4500, the manufacturer must pay P0.60 for labor per unit. The manufacturer can sell 4000 units per month at P7.0 per unit and estimates that monthly sales will rise by 100 for each P0.10 reduction in price. Find the number of units that should be produced each month for maximum profit.

- a. 2600 units
- b. 4700 units
- c. 6800 units
- d. 9900 units

81. A company estimates that it can sell 1000 units per week if it sets the unit price at P3.00, but it’s weekly sales will rise by 100 units for each P0.10 decrease in price. Find the number of units sold each week and its unit price per maximum revenue.

- a. 1500 ; P1.50
- b. 1000 ; P3.00
- c. 2500 ; P2.50
- d. 2000 ; P2.00

82. In manufacturing and selling “x” units of a certain commodity, the selling price per unit is P = 5 – 0.002x and the production cost in pesos is C = 3 + 1.10x. Determine the production level that will produce the maximum profit and what would this profit be?

- a. 975, P1898.25
- b. 800, P1750.75
- c. 865, P1670.50
- d. 785, P1920.60

83. ABC company manufactures computer spare parts. With its present machines, it has an output of 500 units annually. With the addition of the new machines, the company could boost its yearly production to 750 units. If it produces “x: parts it can set a price of P = 200 – 0.15x pesos per unit and will have a total yearly cost of C = 6000 + 6x + 0.003x^{2} in pesos. What production level maximizes total yearly profit?

- a. 660 units
- b. 237 units
- c. 560 units
- d. 243 units

84. The hypotenuse of a right triangle is 20 cm. What is the maximum possible area of the triangle in square centimeters?

- a. 100
- b. 170
- c. 120
- d. 160

85. Sand is falling off a conveyor onto a conical pile at the rate of 15cm^{3}/min. The base of the cone is approximately twice the altitude. Find the height of the pile if the height of the pile is changing at the rate 0.047746 cm/min.

- a. 12 cm
- b. 10 cm
- c. 8 cm
- d. 6 cm

86. A machine is rolling a metal cylinder under pressure. The radius of the cylinder is decreasing at the rate of 0.05cm per second and the volume V is 128Ï€ cu.cm. At what rate is the length “h” changing when the radius is 2.5 cm.

- a. 0.8192 cm/sec
- b. 0.7652 cm/sec
- c. 0.6178 cm/sec
- d. 0.5214 cm/sec

87. Two sides of a triangle are 15 cm and 20 cm long respectively. How fast is the third side increasing if the angle between the given sides is 60Âº and is increasing at the rate of 2Âº/sec.

- a. 3.60 cm/sec
- b. 2.70 cm/sec
- c. 1.20 cm/sec
- d. 0.05 cm/sec

88. Two sides of a triangle are 30 cm and 40 cm respectively. How fast is the area of the triangle increasing if the angle between the given sides is 60Âº and is increasing at the rate of 4Âº/sec.

- a. 20.94 m
^{2}/sec - b. 29.34 m
^{2}/sec - c. 14.68 m
^{2}/sec - d. 24.58 m
^{2}/sec

89. A man 6 ft. tall is walking toward a building at the rate of 5 ft/sec. If there is a light on the ground 50 ft. from the building, how fast is the man/s shadow on the building growing shorter when he is 30 ft. from the building?

- a. -3.75 fps
- b. -7.35 fps
- c. -5.37 fps
- d. -4.86 fps

90. The volume of the sphere is increasing at the rate of 6cm^{3}/hr. At what rate is its surface area increasing when the radius is 50 cm(in cm^{3}/hr)

- a. 20.94 m
^{2}/sec - b. 29.34 m
^{2}/sec - c. 14.68 m
^{2}/sec - d. 24.58 m
^{2}/sec

91. A particle moves in a plane according to the parametric equations of motions: x = t^{2}, y = t^{3}. Find the magnitude of the acceleration when the t = 0.6667.

- a. 6.12
- b. 5.10
- c. 4.90
- d. 4.47

92. The acceleration of the particle is given by a = 2 + 12t in m/s^{2} where t is the time in minutes. If the velocity of this particle is 11 m/s after 1 min, find the velocity after 2 mins.

- a. 26 m/sec
- b. 31 m/sec
- c. 37 m/sec
- d. 45 m/sec

93. A particle moves along a path whose parametric equations are x = t^{3} and y = 2t^{2} . What is the acceleration when t = 3sec?

- a. 15.93 m/sec
^{2} - b. 18.44 m/sec
^{2} - c. 23.36m/sec
^{2} - d. 10.59 m/sec
^{2}

94. A vehicle moves along a trajectory having coordinates given as x = t^{3} and y = 1 – t^{2}. The acceleration of the vehicle at any point of the trajectory is a vector, having magnitude and direction. Find the acceleration when t = 2.

- a. 13.20
- b. 12.17
- c. 15.32
- d. 12.45

95. Y = x^{3} – 3x. Find the maximum value of y.

- a. 2
- b.1
- c. 0
- d. 3

96. Find the radius of curvature of the curve y = 2x^{3 }+ 3x^{2} at (1,5).

- a. 90
- b. 84
- c. 95
- d. 97

97. Compute the radius of curvature of the curve x = 2y^{3 }– 3y^{2} at (4, 2).

- a. -99.38
- b. - 97.15
- c. -95.11
- d. -84.62

98. Find the radius of curvature of a parabola y^{2} – 4x = 0 at point (4, 4).

- a. 25.78
- b. 22.36
- c. 20.33
- d. 15.42

99. Find the radius of curvature of the curve x = y^{3} at point (1, 1).

- a. -1.76
- b. -1.24
- c. 2.19
- d. 2.89

100. Find the point of inflection of the curve y = x^{3 }– 3x^{2 }+ 6.

- a. (0, 2)
- b. (1,3)
- c. (1, 4)
- d. (2, 1)

101. Three sides of a trapezoid are each 8 cm long. How long is the 4^{th} side, when the area of the trapezoid has the greatest value?

- A. 16 cm
- B. 15 cm
- C. 12 cm
- D. 10 cm

102. An open top rectangular tank with square bases is to have a volume of 10 cubic meters. The material for its bottom cost P 150.00 per square meter, and that for the sides is P 60.00 per square meter. The most economical height is:

- A. 2 meters
- B. 2.5 meters
- C. 3 meters
- D. 3.5 meters

103. A rectangular box having a square base and open at the top is to have a capacity of 16823 cc. Find the height of the box to use the least amount of material.

- A. 16.14 cm
- B. 32.28 cm
- C. 18.41 cm
- D. 28.74 cm

104. The altitude of a cylinder of maximum volume that can be inscribed in a right circular cone of radius r and height h is:

- A. h/3
- B. 2h/3
- C. 3h/2
- D. h/4

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