  ### Home > APCALC > Chapter 5 > Lesson 5.1.4 > Problem5-40

5-40.

A rectangle is bounded by the function y = –x2 + 5 and the x-axis as shown below. Homework Help ✎ 1. If the base of the rectangle is 2x, what is the height?

Notice that the rectangle is symmetric across the y-axis.

Because of the symmetry, the endpoints of the rectangle can be found at −x and +x... after all, the distance between −x and x is 2x, the length of the base.

Consequently, the height of the rectangle can be evaluated at f(x):
height = y = −x2 + 5

2. What is the maximum area that the rectangle can enclose?

Area = (base)(height)
Use the base and height you found in part (a).Then optimize the Area function.

To find the maximum Area:
1. Let A'(x) = 0.
2. Solve for x.
3. Use the x and your expressions for base & height to calculate the maximum area.

3. If you make the rectangle from part (b) into a flag and spin it around the x-axis, what is the resulting volume?

A cylinder with height __________ and radius __________.

4. Determine the value of x so that the volume of the rotated rectangle is a maximum.

Volume of a cylinder: V = πr2h
If you need guidance optimizing the volume, refer to steps in part (b).