This is the Multiple Choice Questions Part 1 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**

### Start Practice Exam Test Questions Part I of the Series

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

**Problem 1: ECE Board April 1999**

Find the minimum distance from the point (4, 2) to the parabola y^{2} = 8x.

- A. 4√3
- B. 2√2
- C. √3
- D. 2√3

**Problem 2: EE Board April 1990**

The sum of two positive numbers is 50. What are the numbers if their product is to be the largest possible.

- A. 24 and 26
- B. 28 and 22
- C. 25 and 25
- D. 20 and 30

**Problem 3: EE Board March 1998**

A triangle has variable sides x, y, z subject to the constraint such that the perimeter is fixed to 18 cm. What is the maximum possible area for the triangle?

- A. 15.59 cm
^{2} - B. 18.71 cm
^{2} - C. 17.15 cm
^{2} - D. 14.03 cm
^{2}

**Problem 4: EE Board October 1997**

A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose the maximum area?

- A. 25 m x 25 m
- B. 15 m x 35 m
- C. 20 m x 30 m
- D. 22.5 m x 27.5 m

**Problem 5: CE Board May 1997**

Find the minimum amount of tin sheet that can be made into a closed cylinder having a volume of 108 cu. inches in square inches.

- A. 125.50
- B. 127.50
- C. 129.50
- D. 123.50

**Problem 6: ME Board April 1998**

A box is to be constructed from a piece of zinc 20 sq. in by cutting equal squares from each corner and turning up the zinc to form the side. What is the volume of the largest box that can be constructed?

- A. 599.95 cu in.
- B. 592.59 cu in.
- C. 579.50 cu in.
- D. 622.49 cu in.

**Problem 7: EE Board April 1997**

A poster is to contain 300 (cm square) of printed matter with margins of 10 cm at the top and bottom and 5 cm at each side. Find the overall dimensions if the total area of the poster is minimum.

- A. 27.76 cm, 47.8 cm
- B. 20.45 cm, 35.6 cm
- C. 22.24 cm, 44.5 cm
- D. 25.55 cm, 46.7 cm

**Problem 8: CE Board November 1996**

A normal window is in the shape of a rectangle surmounted by a semi-circle. What is the ratio of the width of the rectangle to the total height so that it will yield a window admitting the most light for a given perimeter?

- A. 1
- B. 1/2
- C. 2
- D. 2/3

**Problem 9: CE Board November 1996**

Determine the diameter of a closed cylindrical tank having a volume of 11.3 cu m. to obtain minimum surface area.

- A. 1.22
- B. 1.64
- C. 2.44
- D. 2.68

**Problem 10: EE Board April 1997**

The cost fuel in running a locomotive is proportional to the square of the speed and is $25 per hour for a speed of 25 miles per hour. Other costs amount to $100 per hour, regardless of the speed. What is the speed which will make the cost per mile a minimum?

- A. 40 mph
- B. 55 mph
- C. 50 mph
- D. 45 mph

**Problem 11: ME Board April 1996**

The cost of C of a product is a function of the quantity x of the product: C(x) = x^{2} – 400x + 50. Find the quantity for which the cost is minimum.

- A. 1000
- B. 1500
- C. 2000
- D. 3000

**Problem 12: **

An open top rectangular tank with square bases is to have a volume of 10 cu. m. The materials for its bottom is to cost P15 per square meter and that for the sides, P6 per square meter. Find the most economical dimensions for the tank.

- A. 1.5 m x 1.5 m x 4.4 m
- B. 2 m x 2 m x 2.5 m
- C. 4 m x 4 m x 0.6 m
- D. 3 m x 3 m x 1.1 m

**Problem 13: ME Board October 1996**

What is the maximum profit when the profit-versus-production function is as given below? P is profit and x is unit of production.

- A. 285,000
- B. 200,000
- C. 250,000
- D. 305,000

**Problem 14: EE Board October 1993**

A boatman is at A which is 4.5 km from the nearest point B on a straight shore BM. He wishes to reach in minimum time at point C situated on the shore 9 km from B. How far from C should he land if he can row at the rate of 6 kph and can walk at the rate of 7.5 kph.

- A. 4.15 km
- B. 3.0 km
- C. 3.25 km
- D. 4.0 km

**Problem 15: EE Board March 1998**

A fencing is limited to 20 ft. length. What is the maximum rectangular area that can be fenced in using two perpendicular corner sides of an existing wall?

- A. 120 ft
^{2} - B. 100 ft
^{2} - C. 140 ft
^{2} - D. 190 ft
^{2}

**Problem 16: EE Board October 1992**

The cost per hour of running a motor boat is proportional to the cube of the speed. At what speed will the boat run against a current of 8 km/hr in order to go a given distance most economically?

- A. 10 kph
- B. 13 kph
- C. 11 kph
- D. 12 kph

**Problem 17: ECE Board November 1998**

Given a cone of diameter x and altitude of h. What percent is the volume of the largest cylinder which can inscribed in the cone to the volume of the cone?

- A. 44%
- B. 46%
- C. 56%
- D. 65%

**Problem 18: EE Board October 1993**

At any distance x from the source of light, the intensity of illumination varies directly as the intensity of the source and inversely as the square of x. Suppose that there is a light at A, and another at B, the one at B having an intensity 8 times that of A. The distance AB is 4 m. At what point from A on the line AB will the intensity of illumination be least?

- A. 2.15 m
- B. 1.33 m
- C. 1.50 m
- D. 1.92 m

**Problem 19: CE Board May 1995**

A wall “h” meters high is 2 m away from the building. The shortest ladder that can reach the building with one end resting on the ground outside the wall is 6 m. How high is the wall in meters?

- A. 2.34
- B. 2.24
- C. 2.44
- D. 2.14

**Problem 20: EE Board April 1997**

The coordinates (x, y) in feet of a moving particle P are given by x = cost – 1 and y = 2sint – 1, where t is the time in seconds. At what extreme rates in fps is P moving along the curve?

- A. 3 and 2
- B. 3 and 1
- C. 2 and 0.5
- D. 2 and 1

**Problem 21: ECE Board April 1998**

A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum?

- A. 3.41 m
- B. 3.51 m
- C. 3.71 m
- D. 4.41 m

**Problem 22: **

A man walks across a bridge at the rate of 5 fps as a boat directly beneath him at 10 fps. If the bridge is 10 feet above the boat, how fast are the man and the boat separating 1 second later?

- A. 8 fps
- B. 8.25 fps
- C. 8.33 fps
- D. 8.67 fps

**Problem 23: **

An LRT train 6 m above the ground crosses a street at 9 m/s at the instant that a car approaching at a speed of 4 m/s is 12 m up the street. Find the rate of the LRT train and the car separating one second later.

- A. 3.64 m/s
- B. 3.94 m/s
- C. 4.24 m/s
- D. 4.46 m/s

**Problem 24: EE Board October 1993**

Water is flowing into a conical cistern at the rate of 8 m^{3}/min. If the height of the inverted cone is 12 m and the radius of its circular opening is 6 m. How fast is the water level rising when the water is 4 m deep?

- A. 0.64 m/min
- B. 0.56 m/min
- C. 0.75 m/min
- D. 0.45 m/min

**Problem 25: CE Board November 1998**

Water is pouring into conical vessel 15 cm deep and having a radius of 3.75 cm across the top. If the rate at which the water rises is 2 cm/sec, how fast is the water flowing into the conical vessel when the water is 4 cm deep?

- A. 2.37 m
^{3}/sec - B. 5.73 m
^{3}/sec - C. 6.28 m
^{3}/sec - D. 4.57 m
^{3}/sec

**Problem 26: ME Board October 1996**

Water is pouring into a swimming pool. After t hours, there are t + √t gallons in the pool. At what rate is the water pouring into the pool when t = 9 hours?

- A. 7/6 gph
- B. 8/7 gph
- C. 6/5 gph
- D. 5/4 gph

**Problem 27: **

A helicopter is rising vertically from the ground at a constant rate of 4.5 meters per second. When it is 75 m off the ground, a jeep passed beneath the helicopter traveling in a straight line at a constant rate of 80 kph. Determine how fast the distance between them changing after 1 second.

- A. 12.34 m/sec
- B. 11.10 m/sec
- C. 10.32 m/sec
- D. 9.85 m/sec

**Problem 28: ECE Board November 1991**

A balloon is released from the ground 100 meters from an observer. The balloon rises directly upward at the rate of 4 meters per second. How fast is the balloon receding from the observer 10 seconds later?

- A. 1.68 m/sec
- B. 1.36 m/sec
- C. 1.55 m/sec
- D. 1.49 m/sec

**Problem 29: ECE Board April 1998**

A balloon is rising vertically over a point A on the ground at the rate of 15 ft./sec. A point B on the ground level with and 30 ft. from A. When the balloon is 40 ft. from A, at what rate is its distance from B changing?

- A. 13 ft/sec
- B. 15 ft/sec
- C. 12 ft/sec
- D. 10 ft/sec

**Problem 30: CE Board May 1997**

Car A moves due East at 30 kph at the same instant car B is moving S 30° E , with a speed of 60 kph. The distance from A to B is 30 km. Find how fast is the distance between them separating after one hour.

- A. 36 kph
- B. 38 kph
- C. 40 kph
- D. 45 kph

**Problem 31: CE Board November 1996**

A car starting at 12:00 noon travels West at a speed of 30 kph. Another car starting from the same point at 2:00 PM travels North at 45 kph. Find how (in kph) fast the two are separating at 4:00 PM?

- A. 49 kph
- B. 51 kph
- C. 53 kph
- D. 55 kph

**Problem 32: CE Board May 1996**

Two railroad tracks are perpendicular to each other. At 12:00 PM there is a train at each track approaching the crossing at 50 kph, one being 100 km and the other 150 km away from the crossing. How fast in kph is the distance between the two trains changing at 4:00 PM?

- A. 67.08 kph
- B. 68.08 kph
- C. 69.08 kph
- D. 70.08 kph

**Problem 33: CE Board May 1995**

Water is running into a hemispherical bowl having a radius of 10 cm at a constant rate of 3 cm^{3}/min. When the water x cm. deep, the water levels is rising at the rate of 0.0149 cm/min. What is the value of x?

- A. 3
- B. 2
- C. 4
- D. 5

**Problem 34: ECE Board November 1998**

What is the allowable error in measuring the edge of the cube that is intended to hold 8 cu. m., if the error of the computed volume is not to exceed 0.03 cu. m?

- A. 0.002 m
- B. 0.003 m
- C. 0.0025 m
- D. 0.001 m

**Problem 35: EE Board October 1993**

A standard cell has an emf “E” of 1.2 volts. If the resistance “R” of the circuit is increasing at the rate of 0.03 ohm/sec, at what rate is the current “I” changing at the instant when the resistance is 6 ohms? Assume Ohm’s law E = IR.

- A. -0.002 amp/sec
- B. 0.004 amp/sec
- C. -0.001 amp/sec
- D. 0.003 amp/sec

36. A function is given below, what x value maximizes y?

**y ^{2 }+ y +^{ }x^{2 }– 2x = 5**

- A. 2.23
- B. -1
- C. 5
- D. 1

37. The number of newspaper copies distributed is given by **C = 50 t ^{2 }– 200 t + 10000**, where t is in years. Find the minimum number of copies distributed from 1995 to 2002.

- A. 9850
- B. 9800
- C. 10200
- D. 7500

38. Given the following profit-versus-production function for a certain commodity:

**P = 200000 – x – [1.1 / (1 + x)]**^{ 8 }

Where P is the profit and x is unit of production. Determine the maximum profit.

- A. 190000
- B. 200000
- C. 250000
- D. 550000

39. The cost C of a product is a function of the quantity x of the product is given by the relation: **C(x) = x ^{2 }– 4000x + 50**. Find the quantity for which the cost is a minimum.

- A. 3000
- B. 2000
- C. 1000
- D. 1500

40. If y = x to the 3^{rd} power – 3x. find the maximum value of y.

- A. 0
- B. -1
- C. 1
- D. 2

41. Divide 120 into two parts so that product of one and the square of the other is maximum. Find the numbers.

- A. 60 & 60
- B. 100 & 20
- C. 70 & 50
- D. 80 & 40

42. If the sum of two numbers is C, find the minimum value of the sum of their squares.

- A. C
^{2 }/ 2 - B. C
^{2 }/ 4 - C. C
^{2 }/ 6 - D. C
^{2 }/ 8

43. A certain travel agency offered a tour that will cost each person P 1500.00 if not more than 150 persons will join, however the cost per person will be reduced by P 5.00 per person in excess of 150. How many persons will make the profit a maximum?

- A. 75
- B. 150
- C. 225
- D. 250

44. Two cities A and B are 8 km and 12 km, respectively, north of a river which runs due east. City B being 15 km east of A. a pumping station is to be constructed (along the river) to supply water for the two cities. Where should the station be located so that the amount of pipe is a minimum?

- A. 3 km east of A
- B. 4 km east of A
- C. 9 km east of A
- D. 6 km east of A

45. A boatman is at A, which is 4.5 km from the nearest point B on a straight shore BM. He wishes to reach, in minimum time, a point C situated on the shore 9 km from B. How far from C should he land if he can row at the rate of 6 Kph and walk at the rate of 7.5 Kph?

- A. 1 km
- B. 3 km
- C. 5 km
- D. 8 km

46. The shortest distance from the point (5, 10) to the curve x^{2 }= 12y is:

- A. 4.331
- B. 3.474
- C. 5.127
- D. 6.445

47. A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum?

- A. 3.41 m
- B. 3.51 m
- C. 3.71 m
- D. 4.41 m

48. An iron bar 20 m long is bent to form a closed plane area. What is the largest area possible?

- A. 21.56 square meter
- B. 25.68 square meter
- C. 28.56 square meter
- D. 31.83 square meter

49. A Norman window is in the shape of a rectangle surmounted by a semi-circle. What is the ratio of the width of the rectangle to the total height so that it will yield a window admitting the most light for a given perimeter?

- A. 1
- B. 2/3
- C. 1/3
- D. ½

50. A rectangular field is to be fenced into four equal parts. What is the size of the largest field that can be fenced this way with a fencing length of 1500 feet if the division is to be parallel to one side?

- A. 65,200
- B. 62,500
- C. 64,500
- D. 63,500

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