1. The manager of a large apartment complex knows from experience that 80 units will be occupied if the rent is 320 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decrease in rent. What rent should the manager charge to maximize revenue?

2. A rancher wants to fence in an area of 2500000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?

3. A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 44 feet?

Solution Preview

1. We compare the two cases and find the maximum revenue.
Case1: Suppose x additional units remain vacant after the rent increases 8x dollars each. So there are 80-x units occupied and the monthly rent is 320+8x. The total revenue is:
f(x)=(320+8x)(80-x)=-8x^2+320x+25600.
When x=-(320)/(2*(-8))=20, f(x) reaches the maximum value which is:
f(20)=-8*(20)^2+320*20+25600=35200
Case2: Suppose x additional units are occupied after the rent decreases 8x dollars each. So there are 80+x units occupied and ...

Solution Summary

There are three maximization/minimization problems: one to charge rent to maximize revenue, one to minimize fence length for a given area, and one to maximize area for a given perimeter.

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