Chapter 4.8, Problem 55E

To determine

To calculate: The value of the height and the value of the maximum volume.

Answer to Problem 55E

The maximum value of h is $9.238$ and V is $1238.2$ .

Explanation of Solution

Given information:

The given diagram of the sphere is:

  

Formula used:

Volume of cylinder:

  $V=\pi r^{2}h$

Calculation:

Consider the radius from the diagram of the sphere:

  $r=\sqrt{8^2-{\left(\frac{h}{2}\right)}^2}$

Substitute the above value in the equation of volume $V=\pi r^{2}h$ .

  $\begin{array}{l}V\left(h\right)=\pi \left(8^2-{\left(\frac{h}{2}\right)}^2\right)h\\ V\left(h\right)=-\frac{h^3}{4}\pi +64h\pi \end{array}$

Use a graphing calculator to plot the function.

  

It is seen from the graph that the maximum value of h is $9.238$ and V is $1238.2$ .

Hence, the maximum value of h is $9.238$ and V is $1238.2$ .