# Lesson Worksheet: Optimization: Applications on Extreme Values Mathematics • Higher Education

In this worksheet, we will practice applying derivatives to real-world problems to find the maximum and the minimum values of a function under certain conditions.

**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 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?

- Ar = 6 m, h = 36 m
- Br = 8 m, h = 24 m
- Cr = 5 m, h = 8 m
- Dr = 4 m, h = 24 m
- Er = 3 m, h = 30 m

**Q12: **

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

**Q14: **

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.

**Q15: **

A rectangular-prism-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, and 66 cm
- B198 cm, 198 cm, and 99 cm
- C33 cm, 33 cm, and 132 cm
- D66 cm, 66 cm, and 66 cm

**Q16: **

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}

**Q18: **

The rectangular cross section of a block of wood is cut from a cylindrical log of diameter 67 cm. The resistance of this block is proportional to its width and the square of its length. What dimensions give the maximum resistance?

- A cm, cm
- B cm, cm
- C cm, cm
- D cm, cm

**Q19: **

In the figure, what is the minimum value of ? Give your answer to 3 decimal places.

**Q20: **

A rectangular parallelepiped has height twice the width of the base. If its volume is 7,375, what dimensions will minimize its surface area?

- A17.69, 35.38, 11.78
- B14.04, 28.08, 18.71
- C22.28, 44.56, 7.43

**Q21: **

An open-topped box is constructed by removing equal squares from the corners of a square sheet of side 12 cm, then turning up the sides. Find the side length of the removed squares that maximizes the volume of the box.

**Q22: **

In the figure, a semicircle is attached to a rectangle. What is the smallest perimeter if the area enclosed is 100? Give your answer exactly and in terms of .

- A
- B
- C
- D
- E

**Q23: **

If the length of a hypotenuse of a right triangle equals 33.5 cm, find the lengths of the two sides of the right angle to the nearest thousand when the area of the triangle is at its maximum.

- A23.688 cm, 41.029 cm
- B16.750 cm, 16.750 cm
- C23.688 cm, 7.089 cm
- D23.688 cm, 23.688 cm

**Q24: **

In the figure, given that , what is the maximum value of ? Give your answer to 3 decimal places.

**Q25: **

All the vertices of a certain rectangle lie on an equilateral triangle with sides of length 14 cm; one side of the rectangle lies on the base of the triangle and the vertices of the opposite side of the rectangle lie on the two other sides of the triangle. What is the maximum area this rectangle could have?

- A196 cm
^{2} - B cm
^{2} - C49 cm
^{2} - D cm
^{2}