Question

A new building in the shape of a square pyramid is to be constructed. The slant height will be five times the side length of the base. There will be between 20,000 square feet and 50,000 square feet of construction material used for the outside of the building. What would be the maximum possible side length of the base of the building? Round your answer to the nearest foot.

Please explain how to solve this problem, not just the answer.

Answer 1

Answer: 71 feet

Explanation:

Let's assume that the side length of the base of the square pyramid is x. Then, the slant height of the pyramid would be 5x, as given in the problem.

The lateral surface area of a square pyramid can be calculated using the formula:

Lateral surface area = (perimeter of base) × (slant height) / 2

The perimeter of the square base is 4x. So, the lateral surface area of the square pyramid can be written as:

Lateral surface area = 4x × (5x) / 2 = 10x^2

We are told that the construction material used for the outside of the building is between 20,000 square feet and 50,000 square feet. So, we can set up the following inequality:

20,000 ≤ 10x^2 ≤ 50,000

Dividing both sides by 10, we get:

2000 ≤ x^2 ≤ 5000

Taking the square root of both sides, we get:

44.7 ≤ x ≤ 70.7

Since we are looking for the maximum possible value of x, we can round 70.7 up to the nearest foot, which is 71 feet.

Therefore, the maximum possible side length of the base of the building is 71 feet.