Question

he volume of a right rectangular prism cannot exceed 200 cubic centimeters. The side lengths are given by
x, x + 1, and x + 3. Solve the following inequality to determine possible values of x.

x(x + 1)(x + 3) ≤ 200

Answer 1

We need x(x+1)(x+3) ≤ 200, with x representing a side length so x ≥ 0 (usually x>0).

Rewrite:
x^3 + 4x^2 + 3x − 200 ≤ 0.

The cubic has one real root at x ≈ 4.638 (found numerically). The cubic is increasing for x ≥ 0, so the inequality holds for 0 ≤ x ≤ 4.638 (or 0 < x ≤ 4.638 if you require strictly positive side lengths).

Thus the acceptable x-values are x ≤ approximately 4.638 (restricted to nonnegative x: 0 ≤ x ≤ 4.638).