Chapter 2.8, Problem 7MRPS

Solution

a.

To write: An expression for h in terms of r .

  $h=\frac{4000-\pi r^2}{2\pi r}$

Given information:

A zookeeper is building a cylindrical habitat for alligators. He has 4000 square feet of waterproof material use. The habitat will not need a roof. The surface area (in square feet) of the habitat is given by $S=\pi r^2+2\pi rh$ .

Calculation:

Substitute $S=4000$ in the surface area formula as shown below:

  $\begin{array}{c}S=\pi r^2+2\pi rh\\ 4000-\pi r^2=\pi r^2-\pi r^2+2\pi rh\left(\text{Subract }\pi r^2\text{ from both sides}\right)\\ \frac{4000-\pi r^2}{2\pi r}=\frac{2\pi rh}{2\pi r}\left(\text{Divide both sides by 2}\pi r\right)\\ \frac{4000-\pi r^2}{2\pi r}=h\end{array}$

Therefore, the required expression would be $\frac{4000-\pi r^2}{2\pi r}$ .

b.

To write: An equation that given V as a polynomial function of r alone.

  $V=-\frac{\pi }{2}{r}^3+2000r$

Given information:

The volume V of the habitat is given by $V=\pi r^{2}h$ .

Calculation:

To obtain the required equation use $h=\frac{4000-\pi r^2}{2\pi r}$ in the volume formula as shown below:

  $\begin{array}{c}V=\pi r^{2}h\\ V=\pi r^2\left(\frac{4000-\pi r^2}{2\pi r}\right)\left(\text{Use }h=\frac{4000-\pi r^2}{2\pi r}\right)\\ V=\frac{1}{2}r\left(4000-\pi r^2\right)\left(\text{Cancel out 2}\pi r\right)\\ V=2000r-\frac{\pi }{2}{r}^3\end{array}$

Therefore, the required function would be $V=-\frac{\pi }{2}{r}^3+2000r$ .

c.

To graph: The volume function from part (b). What are the dimensions of r and h that maximize the volume of the habitat? What is the maximum volume?

The maximum occurs, when $r\approx 20.6\text{ and }h\approx 20.6$ .

The maximum volume of the habitat would be approximately $27,468$ cubic feet.

Given information:

The volume V of the habitat is given by $V=\pi r^{2}h$ .

Calculation:

Upon graphing the volume function, the required graph would look like as shown below:

  

Upon looking at the graph, it is visible that the maximum volume is approximately $27,468$ cubic feet and the value of r is approximately $20.6$ feet.

  $\begin{array}{c}h=\frac{4000-\pi r^2}{2\pi r}\\ =\frac{4000-\pi {\left(20.6\right)}^2}{2\pi \left(20.6\right)}\\ =20.6\end{array}$

Therefore, the value of h would be $20.6$ feet.

d.

To find: Which value of r did you have to reject when finding the maximum volume? Why?

Any value less than 0 will be rejected while finding the maximum volume.

Given information:

A zookeeper is building a cylindrical habitat for alligators. He has 4000 square feet of waterproof material use. The habitat will not need a roof. The surface area (in square feet) of the habitat is given by $S=\pi r^2+2\pi rh$ .

Calculation:

The negative values or values less than 0 will be rejected while finding the maximum value. It is because radius cannot be negative.