ChE Exam

Engineering Calculus

Question 1 (20 points)

A ball is thrown vertically upward from the top of a 96-foot tower. Two seconds later, the velocity of the ball is 16 feet per second. Find:
(a) the maximum height that the ball reaches;
(b) the speed of the ball when it hits the ground.

Answer: (a) 196 ft; (b) 112 ft/sec

Question 2 (10 points)

A cylindrical tank of radius 10 feet is being filled with wheat at the rate of 314 cubic feet per minute. How fast is the depth of the wheat increasing?
[Hint: The volume of a cylinder is πr2h, where r is its radius and h its height.]

Answer: 3.14/π ≈ 1 ft/min

Question 3 (10 points)

For what value of k will x − kx-1 have a relative maximum at x = -2?

Answer: k = -4

Question 4 (20 points)

Find the absolute maxima and minima (if any) of f(x) = (x2 + 4)/(x − 2) on the interval [0, 2).

Answer: Absolute max = -2 (at x = 0), no absolute min

Question 5 (20 points)

A rectangular bin, open at the top, is required to contain a volume of 128 cubic meters. If the bottom is to be a square, at a cost of $2 per square meter, whereas the sides cost $0.50 per square meter, what dimensions will minimize the cost?

Answer: Height = 8m; side of base = 4m