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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