Question

a rectangular solid (with a square base) has a surface area of 181.5 square centimeters. find the dimensions that will result in a solid with maximum volume. (enter your answers as a comma - separated list.)

Explanation:

Step1: Let the side length of the square base be \(x\) and the height be \(h\)

The surface area formula \(S = 2x^{2}+4xh\). Given \(S = 181.5\), so \(2x^{2}+4xh=181.5\), and \(h=\frac{181.5 - 2x^{2}}{4x}=\frac{363}{8x}-\frac{x}{2}\)

Step2: Write the volume formula

The volume formula \(V=x^{2}h\). Substitute \(h\) into it: \(V=x^{2}(\frac{363}{8x}-\frac{x}{2})=\frac{363}{8}x-\frac{1}{2}x^{3}\)

Step3: Take the derivative of \(V\) with respect to \(x\)

\(V^\prime=\frac{363}{8}-\frac{3}{2}x^{2}\)
Set \(V^\prime = 0\), then \(\frac{363}{8}-\frac{3}{2}x^{2}=0\)
\(\frac{3}{2}x^{2}=\frac{363}{8}\)
\(x^{2}=\frac{363}{8}\times\frac{2}{3}=\frac{121}{4}\)
\(x=\frac{11}{2} = 5.5\) (since \(x>0\))

Step4: Find \(h\)

Substitute \(x = 5.5\) into \(h=\frac{363}{8x}-\frac{x}{2}\)
\(h=\frac{363}{8\times5.5}-\frac{5.5}{2}=\frac{363}{44}- 2.75=8.25 - 2.75 = 5.5\)

Answer:

\(5.5,5.5,5.5\)