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Chapter 9: Problem 14

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is \(337.5\) square centimeters.

Short Answer

Expert verified
The dimensions of the rectangular solid with maximum volume are base side length \(x = sqrt(337.5 / 6) cm\) and height \(h = (337.5 - 2*sqrt(337.5 / 6)^2) / (4*sqrt(337.5 / 6)) cm\).

Step by step solution

01

Set up the equations

First, let's denote the side length of the square base as \(x\) and the height of the box as \(h\). The surface area \(S\) and volume \(V\) of the rectangular solid can be represented as: \(S = 2x^2 + 4xh = 337.5\) cm^2 and \(V = x^2 * h\). From the surface area equation, we can isolate \(h\) to get \(h = (337.5 - 2x^2) / (4x)\).
02

Substitute h in the volume equation

Substitute \(h\) in the volume \(V\) equation to get \(V = x^2 * [(337.5 - 2x^2) / (4x)] = [(337.5x - 2x^3) / 4]\). This reduces the problem to a function of a single variable \(x\), which we can take derivatives of and find maximums.
03

Differentiate and Find Maximum Volume

Differentiate the volume function \(V'(x) = (337.5 - 6x^2) / 4\). Set this equal to zero and solve for \(x\) to get the critical points. The solutions are \(x = 0\) and \(x = sqrt(337.5 / 6)\). Plug these values into the volume function, as well as the end points (since \(x > 0\)), to find the maximum volume. The maximum volume will correspond to \(x = sqrt(337.5 / 6)\).
04

Find dimensions

Plug the value of \(x\) into the equation for \(h\) to find the height of the rectangular solid. So the dimensions of the rectangle with maximum volume are square base with side \(x = sqrt(337.5 / 6) cm\) and height \(h = (337.5 - 2*sqrt(337.5 / 6)^2) / (4*sqrt(337.5 / 6)) cm\).

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