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Chapter 3: 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 for maximum volume, with a surface area of 337.5 square centimeters, are approximately \(sqrt[3]{56.25}\) centimeters for each side. The maximum volume is approximately 471.48 cubic centimeters.

Step by step solution

01

Set Up the Equations

Let us assume the side of the rectangular solid to be \(x\). So the given total surface area \(6x^2 = 337.5\), solving this we get \(x^2 = 56.25\)
02

Find volume from given dimensions

The volume of a rectangular solid is given by \(x^3\). So substitute \(x^2 = 56.25\) in there to get \(V = x^3\) or \(V = \sqrt[3]{56.25^3}\)
03

Differentiate and Find Maximum Volume

Calculate the derivative of the volume with respect to the side length, set it equal to zero and solve for \(x\). The derivative \(V'=3x^2\). Setting this equal to zero, we get \(x = 0\). However, since a side length of zero does not make physical sense in this context, there must be a maximum at the critical point \(x = \sqrt[3]{56.25}\). So, differentiate again to find out whether this point gives local maxima or minima, \(V'' = 6x\). Substituting \(x = \sqrt[3]{56.25}\) will give a positive value thus confirming the point will give maximum volume.
04

Compute the Final Answer

Substitute \(x = \sqrt[3]{56.25}\) in the volume formula to get the maximum volume. This comes out to be \( (\sqrt[3]{56.25})^3\), which is approximately 471.48 cubic centimeters

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