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

A rectangle is bounded by the function

*y*= −*x*^{2}+ 5 and the*x*-axis as shown below. Homework Help ✎If the base of the rectangle is 2

*x*, what is the height?What is the maximum area that the rectangle can obtain?

If you made the rectangle from part (b) into a flag and spun it around the

*x*-axis, what would be the resulting volume?Find

*x*so that the volume of the rotated rectangle is a maximum.

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 2*x*, the length of the base.

Consequently, the height of the rectangle can be evaluated at *f*(*x*):

height = *y* = −*x*^{2} + 5

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.

A cylinder with height __________ and radius __________.

Volume of a cylinder: *V* = *πr*^{2}*h*

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