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A rectangular box has a square base. If the sum of the height and the perimeter of the square base is 16 in, what is the maximum possible volume?

(a) 8 in³
(b) 16 in³
(c) 32 in³
(d) 64 in³

User Smiles
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Final answer:

The maximum possible volume of the box is found by using the equation for the volume of a box with a square base (V = x²h). By substituting the height with the expression derived from the given perimeter and height sum (16 inches), we find the maximum volume is 32 in³, corresponding to answer choice (c).

Step-by-step explanation:

The question asks to find the maximum possible volume of a rectangular box with a square base when the sum of the height and the perimeter of the base is 16 inches. Let's denote the side of the square base as x inches and the height as h inches. Since the base is square, its perimeter is 4x inches. Given the sum of height and perimeter equals 16 inches, we have h + 4x = 16. To find the maximum volume, we need to create a function for the volume (V = x²h) and optimize it.

First, we solve for h: h = 16 - 4x. Then we substitute into the volume formula: V = x² (16 - 4x). Now we have the volume in terms of x alone. We can calculate the derivative to find the maximum volume, but we can also recognize that this is a quadratic equation which represents an inverted parabola, which reaches its maximum when x = 2, as this will be at the vertex of the parabola. At x = 2, h becomes 16 - 4(2) = 8. Thus, the maximum volume is V = (2 in) ² (8 in) = 4 in² × 8 in = 32 in³, answer choice (c) 32 in³.

User Malar Kandasamy
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