Asked by tj
The 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
All Answers 1
Answered by
GPT-5 mini
AI
Solve x(x + 1)(x + 3) ≤ 200. Put all terms on one side:
f(x) = x^3 + 4x^2 + 3x − 200 ≤ 0.
The cubic has a single real root r of f(x) = 0, which lies between 4 and 5 (f(4) = −60, f(5) = 40). Numerically r ≈ 4.6406. Since f(x) < 0 for x < r and f(x) > 0 for x > r, the real solution set is
x ≤ r ≈ 4.6406.
Because x, x+1, x+3 are side lengths they must be positive, so x > 0. Thus the physically meaningful solutions are
0 < x ≤ 4.6406 (x ≈ 4.6406 being the upper bound).
(If a degenerate side of length 0 is allowed, include x = 0.)
f(x) = x^3 + 4x^2 + 3x − 200 ≤ 0.
The cubic has a single real root r of f(x) = 0, which lies between 4 and 5 (f(4) = −60, f(5) = 40). Numerically r ≈ 4.6406. Since f(x) < 0 for x < r and f(x) > 0 for x > r, the real solution set is
x ≤ r ≈ 4.6406.
Because x, x+1, x+3 are side lengths they must be positive, so x > 0. Thus the physically meaningful solutions are
0 < x ≤ 4.6406 (x ≈ 4.6406 being the upper bound).
(If a degenerate side of length 0 is allowed, include x = 0.)
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