1. Extra Optimization Problems “Enrichment Problems”

3. _ _ +

4. 21. Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius 5. r 0.5h R h – height of cylinder r – radius of cylinder R – Given radius of sphere Therefore a max

5. 4. An open-top box with a square bottom and rectangular sides is to have a volume of 256 cubic inches. Find the dimensions that require the minimum amount of material. therefore a min 8 x 8 x 4 y x x

6. 6. A right triangle of hypotenuse 5 is rotated about one of its legs to generate a right circular cone. Find the cone of greatest volume. x y 5 Therefore max

7. 8. (calculator required) A poster is to contain 100 square inches of picture surrounded by a 4 inch margin at the top and bottom and a 2 inch margin on each side. Find the overall dimensions that will minimize the total area of the poster. Since f’ changes from neg to pos, we have a minimum 4 4 2 2 x y

8. + _ Therefore max

9. 11.(calculator required) Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10. Since f’ changes from pos to neg, we have a maximum

10. CALCULATOR REQUIRED Minimum since f ‘ (x) changes from neg to pos at –0.426

11. Since A ‘ changes from neg to pos, min area at t = 2

12. + _ Therefore max