Isla wants to build a playhouse that is in the shape of a rectangular pyramid that has the maximum volume possible. The perimeter of the base must be 24 feet. The height of the playhouse should be 2 times the length. What should the dimensions of the playhouse be?

Answer 1

Let's denote the length of the base of the rectangular pyramid as l, the width as w, and the height as h.

Given that the perimeter of the base is 24 feet, we have:

2l + 2w = 24
l + w = 12
w = 12 - l

Since the height of the playhouse is 2 times the length, we have:

h = 2l

The volume V of a rectangular pyramid is given by:

V = (1/3) * base area * height
V = (1/3) * l * w * h

Substitute w = 12 - l and h = 2l into the volume formula to get it in terms of l:

V = (1/3) * l * (12 - l) * 2l
V = (2/3) * l * (12 - l) * l
V = (2/3) * l^2 * (12 - l)

To find the dimensions that give the maximum volume, we need to take the derivative of V with respect to l and set it equal to zero:

dV/dl = 0
(-4/3) * l + 8 = 0
-4l + 24 = 0
l = 6

Since w = 12 - l, w = 6

Therefore, the dimensions of the playhouse that will maximize the volume are:
Length (l) = 6 feet
Width (w) = 6 feet
Height (h) = 2l = 12 feet.