Chapter 2.8, Problem 42PS

Solution

a.

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

  $l=\frac{600-\pi r^2}{\pi r}$

Given information:

You have 600 square feet to material for building a greenhouse that is shaped like half a cylinder. The surface area S of the greenhouse is given by $S=\pi r^2+\pi rl$ .

Calculation:

Substitute $S=600$ in the surface area as shown below:

  $\begin{array}{c}600=\pi r^2+\pi rl\\ 600-\pi r^2=\pi r^2-\pi r^2+\pi rl\left(\text{Subract }\pi r^2\text{ from both sides}\right)\\ \frac{600-\pi r^2}{\pi r}=\frac{\pi rl}{\pi r}\left(\text{Divide both sides by }\pi r\right)\\ \frac{600-\pi r^2}{\pi r}=l\end{array}$

Therefore, the required expression would be $\frac{600-\pi r^2}{\pi r}$ .

b.

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

  $V=300r-\pi r^3$

Given information:

The volume V of the greenhouse is given by $V=\frac{1}{2}\pi r^{2}l$ .

Calculation:

To obtain the required equation use $l=\frac{600-\pi r^2}{\pi r}$ in the volume formula as shown below:

  $\begin{array}{c}V=\frac{1}{2}\pi r^{2}l\\ V=\frac{1}{2}\pi r^2\left(\frac{600-\pi r^2}{\pi r}\right)\left(\text{Use }l=\frac{600-\pi r^2}{\pi r}\right)\\ V=\frac{1}{2}r\left(600-\pi r^2\right)\left(\text{Cancel out }\pi r\right)\\ V=300r-\pi r^3\end{array}$

Therefore, the required equation would be $V=300r-\pi r^3$ .

c.

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

The maximum volume of the greenhouse would be approximately 1596 cubic feet.

Given information:

The volume V of the greenhouse is given by $V=\frac{1}{2}\pi r^{2}l$ .

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 $1596$ cubic feet and the value of r is approximately 8 feet.

  $\begin{array}{c}l=\frac{600-\pi {\left(8\right)}^2}{\pi \left(8\right)}\\ =15.87\end{array}$

Therefore, the value of l would be 15.87 feet.