Question

A rectangular solid (with a square base) has a surface area of 337.5 square centimeters. Find the dimensions that will result in a solid with maximum volume.

Answer 1

We are given the following information:

rectangular solid with a square base

surface area = 337.5 cm^2

We are asked to find the dimensions that will give us the maximum volume.

First, we need to represent the dimensions of the given solid. If x = the side of the square base, then we can represent the height as:

`\begin{gathered} SA=2(lw+lh+wh) \\ SA=2(x^2+xh+xh) \\ SA=2(x^2+2xh) \\ 337.5=2(x^2+2xh) \\ 168.75=x^2+2xh \\ 168.75-x^2=2xh \\ (168.75-x^2)/(2x)=h \end{gathered}`

So we can express the volume of the solid figure as:

`Volume=f(x)=(x)(x)((168.75-x^2)/(2x))`

Simplifying the equation, we get:

`\begin{gathered} f(x)=(x(168.75-x^2))/(2) \\ f(x)=(1)/(2)(168.75x-x^3) \\ \\ f(x)=84.375x-0.5x^3 \end{gathered}`

To find the maximum value of x, we will calculate the derivative of f(x).

`f^(\prime)(x)=84.375-1.5x^2`

Then, we will find the value of x that will make f'(x) = 0.

`\begin{gathered} 0=84.375-1.5x^2 \\ -84.375=-1.5x^2 \\ 56.25=x^2 \\ x=\pm7.5 \end{gathered}`

But because x represents the side of the square base, then we can only accept x = 7.5.

That gives us the height of:

`\begin{gathered} h=(168.75-7.5^2)/(2(7.5)) \\ \\ h=(168.75-56.25)/(15) \\ \\ h=(112.5)/(15) \\ \\ h=7.5 \end{gathered}`

So, the dimensions of the solid figure must be 7.5 cm x 7.5 cm x 7.5 cm.