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:
[tex]\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 \\ \frac{168.75-x^2}{2x}=h \end{gathered}[/tex]So we can express the volume of the solid figure as:
[tex]Volume=f(x)=(x)(x)(\frac{168.75-x^2}{2x})[/tex]Simplifying the equation, we get:
[tex]\begin{gathered} f(x)=\frac{x(168.75-x^2)}{2} \\ f(x)=\frac{1}{2}(168.75x-x^3) \\ \\ f(x)=84.375x-0.5x^3 \end{gathered}[/tex]To find the maximum value of x, we will calculate the derivative of f(x).
[tex]f^{\prime}(x)=84.375-1.5x^2[/tex]Then, we will find the value of x that will make f'(x) = 0.
[tex]\begin{gathered} 0=84.375-1.5x^2 \\ -84.375=-1.5x^2 \\ 56.25=x^2 \\ x=\pm7.5 \end{gathered}[/tex]But because x represents the side of the square base, then we can only accept x = 7.5.
That gives us the height of:
[tex]\begin{gathered} h=\frac{168.75-7.5^2}{2(7.5)} \\ \\ h=\frac{168.75-56.25}{15} \\ \\ h=\frac{112.5}{15} \\ \\ h=7.5 \end{gathered}[/tex]So, the dimensions of the solid figure must be 7.5 cm x 7.5 cm x 7.5 cm.
