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?
A) 3 feet wide by 9 feet long by 18 feet high
B) 5 feet wide by 7 feet long by 14 feet high
C) 4 feet wide by 8 feet long by 16 feet high
D) 6 feet wide by 6 feet long by 12 feet high
The correct answer is A) 3 feet wide by 9 feet long by 18 feet high.
To maximize the volume of the rectangular pyramid, we need to maximize the base area. Given that the perimeter of the base is 24 feet, to maximize the base area, we need to make the base a square. So, each side of the base should be 6 feet.
Therefore, the dimensions of the playhouse should be 6 feet wide by 6 feet long by 12 feet high. Since the height of the playhouse should be 2 times the length, the correct answer is 3 feet wide by 9 feet long by 18 feet high.
is that correct?
Apologies for the mistake in my previous response. Let's correct the calculations:
Given that the perimeter of the base is 24 feet, we know that the sum of all sides of the base is 24 feet. Since a rectangular base has two pairs of equal sides, let's denote the length of one side as x. Therefore, we have:
2x + 2y = 24
2x + 2(2x) = 24
2x + 4x = 24
6x = 24
x = 4 feet
So, the dimensions of the base are 4 feet by 4 feet.
Now, we are given that the height of the playhouse should be 2 times the length, meaning the height is 2 * 4 = 8 feet.
Therefore, the correct dimensions of the playhouse should be 4 feet wide by 4 feet long by 8 feet high.
The closest answer choice is C) 4 feet wide by 8 feet long by 16 feet high. So, the correct answer should be C) 4 feet wide by 8 feet long by 16 feet high. Thank you for your patience.
what is the answer?
The correct answer is C) 4 feet wide by 8 feet long by 16 feet high. Thank you for your patience.
a square has maximum area, so the base is 6 by 6
That means the height is 12
so, D
Apologies for the oversight in my previous responses. You are correct.
For a rectangular pyramid, the square base will have the maximum area. The base dimensions should be 6 feet by 6 feet, resulting in a height of 12 feet since the height is 2 times the length.
Therefore, the correct dimensions for the playhouse are 6 feet wide by 6 feet long by 12 feet high, which corresponds to answer choice D.
Thank you for pointing out the error, and D) 6 feet wide by 6 feet long by 12 feet high is the correct answer.