Question

1. a rectangular solid (with a square base) has a surface area of 13.5 cm2. find the dimensions that will result in a solid with maximum volume.

Explanation:

Step1: Define variables

Let the side of the square base be \(x\) (in cm) and the height be \(h\) (in cm).
The surface - area formula \(S = 2x^{2}+4xh\). Given \(S = 13.5\), so \(13.5=2x^{2}+4xh\), and we can express \(h=\frac{13.5 - 2x^{2}}{4x}\).

Step2: Write the volume formula

The volume formula \(V=x^{2}h\). Substitute \(h\) into the volume formula: \(V=x^{2}\times\frac{13.5 - 2x^{2}}{4x}=\frac{13.5x-2x^{3}}{4}=\frac{13.5}{4}x-\frac{1}{2}x^{3}\).

Step3: Differentiate the volume function

Differentiate \(V(x)\) with respect to \(x\). \(V^\prime(x)=\frac{13.5}{4}-\frac{3}{2}x^{2}\).
Set \(V^\prime(x) = 0\), then \(\frac{13.5}{4}-\frac{3}{2}x^{2}=0\).
Multiply through by 4 to get \(13.5 - 6x^{2}=0\), so \(6x^{2}=13.5\), and \(x^{2}=\frac{13.5}{6}=2.25\), \(x = 1.5\) (since \(x>0\)).

Step4: Find the second - derivative

Differentiate \(V^\prime(x)\) to get \(V^{\prime\prime}(x)=-3x\).
When \(x = 1.5\), \(V^{\prime\prime}(1.5)=-3\times1.5=-4.5<0\), so \(V(x)\) has a maximum at \(x = 1.5\).

Step5: Find the height

Substitute \(x = 1.5\) into the formula for \(h\): \(h=\frac{13.5-2\times(1.5)^{2}}{4\times1.5}=\frac{13.5 - 4.5}{6}=\frac{9}{6}=1.5\).

Answer:

The dimensions of the rectangular solid (square base) for maximum volume are \(x = 1.5\) cm (side of the square base) and \(h = 1.5\) cm (height).