1. The building should be 10' x 20' with the short side facing the lake.

2. If the cube root of pi is x then, the radius of the can should be 5/x and its height should be 4/x.

3. They should plant 80 trees per acre to maximize yield.

4. 2' x 2' x 1' boxes will minimize the amount of wood used.

5. There isn't enough information. Assuming you are looking for a fixed volume (otherwise the cheapest is no can at all) then the height should be 4 times the radius.

6. The radius should be 2/3 times the square root of 6 and the height of the cylinder should be 4/sqrt(3).

7. The pen should be 25' x 50'.

8. The numbers are the square root of 12 and its additive inverse.

9. 20*sqrt(3) by 15* sqrt(3).

10. 200'x200'

11. The radius and height should both be 1".

12. You get a minimum area by cutting it into pieces of length approximately 37.39" and 22.61" and using the smaller piece for the circle. You get a maximum by using the entire wire for the circle. This is a good problem to use Maple on!

13. The radius should be approximately 1.51" the height should be about 3.02". I doubt that you would want to hold this can. So, I expect they were trying to make the can appealing rather than to minimize the material used when they designed the coke can.

14. The cone of maximum volume is the one inscribed so that its height is 4/3 times the radius of the sphere and its radius will be 2*sqrt(2)/3 times the radius of the sphere.

15. (2,3/2)