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Applications of derivatives: maximizing area and revenue

4. A Norman window consists of a rectangle with a semi-circle mounted on top (see the figure). What are the dimensions of the Norman window with the largest area and a fixed perimeter of P meters?

5. A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each person pays $60. In larger groups, everyone's fare is reduced by $1 for each person in excess of 35.

(a) What is the revenue, if the bus is chartered to 35 people? 36 people?
37 people?
(b) Find a formula for the revenue in terms of the number of people
chartering the bus.
(c) Determine the size of the group(s) for which the bus company's
revenue will be the greatest.
(Note: you need to check that your answer makes physical sense and
that it is correct as the physical constraints allow.)

See attached file for full problem description.

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Solution Summary

It composed of two applications of derivative problems: maximizing the area with a fixed perimeter and maximzing the revenue of a bus company. The solution is well presented and detailed.