A rain gutter is made by bending up the edges of a piece of metal of width 36 in. If the length of the piece of metal is 24 in., find the height that maximizes the volume.

I need all of the steps to get to the answer.

To find the height that maximizes the volume of the rain gutter, we need to use calculus and optimization techniques. Let's break down the problem into steps:

Step 1: Understand the problem
We have a rectangular piece of metal with a width of 36 in and a length of 24 in. We need to bend the edges to form a rain gutter, and we want to find the height that maximizes the volume of the gutter.

Step 2: Define variables
Let's denote the height of the gutter as h (in inches). We also need to determine the width and length of the gutter.

Step 3: Determine the width of the gutter
Since we are bending up the edges of the metal, the width of the gutter will be 36 inches minus twice the height (2h).

Width of the gutter = 36 - 2h

Step 4: Determine the length of the gutter
The length of the gutter remains the same as the original piece of metal, so it is 24 inches.

Length of the gutter = 24

Step 5: Determine the volume of the gutter
The volume of the gutter is given by the formula: Volume = length × width × height.

Volume of the gutter = length × width × height
= 24 × (36 - 2h) × h
= 24(36h - 2h^2)

Step 6: Maximize the volume
To find the height that maximizes the volume, we need to find the maximum point of the volume function by taking the derivative and setting it equal to zero.

dV/dh = 24(36 - 4h) (using power rule)
= 864 - 96h

Setting dV/dh equal to zero:
864 - 96h = 0
96h = 864
h = 9

Step 7: Validate the maximum
We found that h = 9 satisfies the condition for the maximum. To confirm it is the maximum, we can check if the second derivative is negative.

d²V/dh² = -96

Since the second derivative is negative, we can conclude that the volume is maximized when h = 9 inches.

Therefore, the height that maximizes the volume of the rain gutter is 9 inches.