A rectangle is bounded by the function $y = −x^2 + 5$ and the $x$-axis as shown below.

1. If the base of the rectangle is $2x$, what is the height?

   Hint 1(a):

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

   Hint 2(a):

   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.

   Answer (a):

   Consequently, the height of the rectangle can be evaluated at $f(x)$:
   height $= y = −x^2 + 5$

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

   Hint (b):

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

   Steps (b):

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

3. If you made the rectangle from part (b) into a flag and spun it around the $x$-axis, what would be the resulting volume?

   Hint (c):

   A cylinder with height __________ and radius __________.

4. Find $x$ so that the volume of the rotated rectangle is a maximum.

   Hint (d):

   Volume of a cylinder: $V = πr^2h$
   If you need guidance optimizing the volume, refer to steps in part (b).