## Optimization Problems to Try on Your
Own

1. The sum of two non-negative numbers is 20. Find the numbers if one number
plus the square root of the other is to be as large as possible.

2. A square plot of ground 100 feet on a side has vertices labeled A, B,
C, D clockwise. Pipe is to be laid in a straight line from A to point P
on BC (P may be one of the vertices), and from there to C. The cost of laying
the pipe is $20 a foot if it goes through the lot (since it must be laid
underground), and $10 a foot if it is laid along one of the sides of the
square. What is the most economical way to lay the pipe?

3. A truck driving over a flat interstate at a constant rate of 50 MPH gets
4 miles to the gallon. Fuel costs $.89 per gallon. For each mile per hour
increase in speed, the truck loses a tenth of a mile per gallon in its mileage.
Drivers get $27.50 per hour in wages, and fixed costs for running the truck
amount to $11.33 per hour. What constant speed (between 50 MPH and the speed
limit of 65 MPH) shoucl a dispatcher require on a straight run through 260
miles of Kansas interstate to minimize the total cost of operating the truck?

4. At Pizza Hut, they deliver for you, and in pizza
boxes formed from 47 cm by 95 cm rectangluar pieces of cardboard by cutting
out eight squares of equal size, four from each of the 95 cm edges: In the
figure below, one square from the corner on the left, one from the middle
of the edge, and two (side-by-side) from the corner on the right, to make
a "lip" for the box. The result is then folded in the obvious
fashion. (If it isn't obvious, order one with sausage and mushrooms to investigate.)
How does one obtain the box of largest volume?

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