Question

Select all the correct answers.

An architect is designing a swimming pool with a base in the shape of a right triangle. According to the architect, the pool’s depth should be 6 feet less than its length, x, and its width should be 8 feet less than its length. The volume of water in the pool cannot exceed 1,680 cubic feet. Which statements about the pool are true?

The water level in the pool cannot exceed 14 feet.
The water level in the pool cannot exceed 20 feet.
The inequality x3 − 14x2 + 48x − 1,680 ≥ 0 can be used to find pool’s length.
The inequality x3 − 14x2 + 48x − 1,680 ≤ 0 can be used to find pool’s length.
The inequality x3 − 14x2 + 48x − 3,360 ≤ 0 can be used to find pool’s length.

Answer 1

The inequality x3 − 14x2 + 48x − 1,680 ≤ 0 can be used to find pool’s length.

⇒The water level in the pool cannot exceed 14 feet.

Explanation:

The question is on inequalities

Given;

Length= x ft

depth= x-6 ft

Width= x-8 ft

volume ≤ 1680 ft³

Forming the inequality to find length x of the pool

Volume= base area × depth

base area × depth ≤ volume

x(x-8) × (x-6) = 1680

(x²-8x )(x-6) = 1680

x(x²-8x) -6 (x²-8x)=1680

x³-8x²-6x²+48x=1680

x³-14x²+48x=1680

x³-14x²+48x-1680 ≤ 0

⇒The water level in the pool cannot exceed 14 feet...why?

taking the value of x at maximum to be 17 according to the graph, then maximum depth will be;

d=x-6 = 17-6=11 ft

⇒11 ft is less than 14ft