1) A rancher has 200 feet of fencing with which to enclose ...



Ch. 5: Optimization & Related Rates Problems

A) A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum?

B) A spherical balloon is inflated with gas as the rate of 20 cube feet per minute. How fast is the radius of the balloon increasing at the instant the radius is a) 1 foot and b) 2 feet?

C) At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate of 10 cubic feet per min. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when it is 15 feet high?

**D) A Norman window is constructed by adjoining a semi-circle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

(Hint: write the equation in terms of length and radius)

E) A right circular cylinder is to be designed to hold 22 cubic inches of a soft drink (approximately 12 fluid ounces) using a minimum amount of material in its construction. What are the dimensions?

F) A physical fitness room consists of a rectangular region with a semi-circle on each end (note: looks like the indoor running track at Dimple Dell Fitness Center). If the perimeter of the room is to be a 200-meter running track, find the dimensions that will make the area of the rectangular region large as possible.

G) A dairy farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

H) A conical tank (with the vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at the rate of 10 cubic feet per minute, find the rate of change of the depth of the water the instant it is 8 feet deep.

I) A solid is formed by adjoining a hemisphere to each end of a right circular cylinder

(note: looks like a pill capsule). The total volume of the figure is 12 cubic inches. Find the radius of the cylinder that produces the minimum surface area.

J) A 25-foot ladder is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top moving down the wall when the base of the ladder is 7 ft., 15 ft., and 24 ft. from the wall? Find the rate at which the angle between the top of the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

A. 33[pic]ft. x 50 ft. F. [pic] x 50m

B. a) [pic] b) [pic] G. 300m x 600m

C. [pic] H. [pic]

D. r =[pic] x l =[pic] or 2.240 ft. x 2.240 ft. I. r = [pic]

E. r [pic]1.518 in. h [pic] 3.037 in. J. [pic]; [pic]

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