Final answer:
An architect is designing a triangular pool with constraints on length, width, and depth based on the pool's length x. The volume of the pool must not exceed 1680 cubic feet, leading to a mathematical equation involving x that needs to be solved to find the maximum dimensions of the pool.
Step-by-step explanation:
An architect is designing a swimming pool with a base in the shape of a right triangle. The pool's depth is to be 6 feet less than its length, represented as x, and its width should be 8 feet less than its length. To find the maximum length of the pool, we must ensure that the volume of water does not exceed 1680 cubic feet.
Since the pool is in the shape of a right triangle, we can calculate the volume with the formula V = 0.5 × base × height × depth. Let's denote the length of the pool as x, then the width will be x - 8 feet, and the depth will be x - 6 feet.
Based on the volume formula, the relationship can be expressed as:
V = 0.5 × (x) × (x - 8) × (x - 6) and for this scenario, V = 1680 cubic feet.
To find the actual dimensions of the pool that meet the volume criteria, we would need to solve for x in the equation 0.5 × (x) × (x - 8) × (x - 6) = 1680. This would require the use of algebraic methods such as expanding the expression and solving the resulting cubic equation for x.