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

A rectangle is bounded by the function and the

*-axis as shown below.*

If the base of the rectangle is

, what is the height? Notice that the rectangle is symmetric across the

-axis.Because of the symmetry, the endpoints of the rectangle can be found at

and after all, the distance between and is, the length of the base. Consequently, the height of the rectangle can be evaluated at

:

heightWhat 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.

2. Solve for.

3. Use theand your expressions for base & height to calculate the maximum area.If you make the rectangle from part (b) into a flag and spin it around the

-axis, what is the resulting volume?A cylinder with height __________ and radius __________.

Determine the value of

so that the volume of the rotated rectangle is a maximum.Volume of a cylinder:

If you need guidance optimizing the volume, refer to steps in part (b).