# Worksheet: Optimization Using Derivatives

In this worksheet, we will practice applying derivatives to real-world problems to optimize a function under certain constraints.

**Q4: **

The sum of the sides of a rectangular parallelepiped with a square base is 12 cm. Find the dimensions that maximize the volume.

- A2 cm, 2 cm, and 2 cm
- B1 cm, 1 cm, and 1 cm
- C4 cm, 2 cm, and 6 cm
- D1 cm, 2 cm, and 2 cm

**Q7: **

A window is made of a semicircle on top of a rectangle, with the semicircle’s diameter equal to the rectangle’s width. Given that the window’s perimeter is 30 m, determine the semicircle’s radius that maximizes the window’s area.

- A m
- B m
- C m
- D m
- E m

**Q9: **

A rectangular piece of cardboard paper has two dimensions of 10 cm and 16 cm. If congruent squares of side length cm are cut from its four corners, and the projected parts are folded upward to form a box without a cover, calculate the dimension of the formed box when its volume is as maximum as possible.

- A2 cm, 6 cm, 12 cm
- B6 cm, 4 cm, 10 cm
- C6 cm, 2 cm, 4 cm
- D2 cm, 8 cm, 14 cm

**Q11: **

A rectangular corral is split into 3 pens with identical dimensions. Given that 500 feet of fencing was used to construct the corral and it was designed to have the maximum area possible, find the dimensions of the corral.

- ALength = 62.5 ft, width = 50 ft
- BLength = 62.5 ft, width = 62.5 ft
- CLength = 125 ft, width = 125 ft
- DLength = 125 ft, width = 62.5 ft
- ELength = 62.5 ft, width = 41.7 ft

**Q13: **

A vertical cylindrical silo of capacity m^{3} is to be built with a hemispherical dome top. If painting one square meter of the dome costs three times as much as painting one square meter of the side, what dimensions will minimize the painting cost?

- Aradius = 6 m, height = 36 m
- Bradius = 8 m, height = 24 m
- Cradius = 5 m, height = 8 m
- Dradius = 4 m, height = 24 m
- Eradius = 3 m, height = 30 m

**Q14: **

What is the maximum area of an isosceles triangle inscribed in a circle of radius 47 cm? Give your answer to the nearest hundredth.

**Q17: **

Find the dimensions of a cuboid of volume 1,000 cubic meters which has the least surface area.

- Awidth = 10 cm, height = 10 cm, and depth = 10 cm
- Bwidth = 25 cm, height = 5 cm, and depth = 8 cm
- Cwidth = 20 cm, height = 10 cm, and depth = 5 cm
- Dwidth = 5 cm, height = 8 cm, and depth = 25 cm
- Ewidth = 5 cm, height = 10 cm, and depth = 20 cm

**Q18: **

A ladder leans against a building and also touches the top of a fence. If the fence is 6 m high and 6.25 m away from the building, what is the shortest ladder that will do? Give your answer correct to the nearest thousandth.

**Q19: **

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

- A41 trees per acre
- B560 trees per acre
- C12 trees per acre
- D75 trees per acre
- E82 trees per acre

**Q20: **

A rectangular box with a square base is going to be built and it needs to have a volume of 20 cubic feet. The material for the base costs 30 cents per square foot. The material for the sides costs 10 cents per square foot. The material for the top costs 20 cents per square foot. Determine the dimensions that would yield the minimum cost.

- A5 ft by 5 ft by 5 ft
- B2 ft by 5 ft by 5 ft
- C2 ft by 2 ft by 2 ft
- D2 ft by 2 ft by 10 ft
- E2 ft by 2 ft by 5 ft

**Q22: **

A cuboid-shaped box has a square base. If the sum of all its edges equals 792 cm, calculate the dimensions of the box that will maximize its volume.

- A99 cm, 99 cm, 66 cm
- B198 cm, 198 cm, 99 cm
- C33 cm, 33 cm, 132 cm
- D66 cm, 66 cm, 66 cm

**Q23: **

A rectangular-shaped playground ends in two semicircles. Given that the perimeter of the playground is 594 m, determine its maximum area.

- A m
^{2} - B176,418
m
^{2} - C m
^{2} - D88,209 m
^{2}