Optimization

1. Two non-negative numbers sum to 12. How large can this product be?

2. Two non-negative numbers multiply to 49. How small can their sum be?

3. (a) An ordinary pop can has a volume of 355 mL. Find the dimensions (radius and height) that minimizes the surface area of such a can.

[A milliliter (mL) is the same as a cubic centimeter (cm), so using centimeters as your unit of length is probably best. Hit this button if you can't recall the formulae for volume and surface area of a cylinder.]

(b) [non-math question] Given the correct solution to the problem above, why are pop cans not constructed with those measurements?

4. Which point on the line y = 3 - x is closest to the origin?

5. Which point on the line y = 2 - 2x is closest to the origin?

6. Quicktime Express has limits for dimensions of parcels it will accept for shipping. The length plus girth (perimeter of the cross section perpendicular to the long side) may not exceed 10 feet.

(a) What is the edge length of the largest cube that can be shipped?

(b) What is the volume of the largest cylinder that can be shipped?

7. A certain truck driving on an open highway at a constant speed of 55 miles per hour gets 6 miles to the gallon. Fuel costs $1.17 per gallon. The truck loses a tenth of a mile per gallon in fuel efficiency for each mile per hour increase in speed. Drivers are paid $31.25 per hour in wages and benefits. Fixed costs for running the truck are $10.15 per hour. What speed minimizes operating expenses?

8. A person is on the shore of a circular lake of radius 100 feet. This person must be at a point directly opposite the present position in the shortest possible time.

The person can swim 20 feet per minute and run 50 feet per minute. At what point should this incredibly slow person swim to, and run the rest of the way, to minimize the time in transit?