Question

Suppose you want to make a cylindrical pen for your cat to play in (with open top) and you want the volume to be 100 cubic feet. Suppose the material for the side costs $3 per square foot, and the material for the bottom costs $7 per square foot. What are the dimensions of the pen that minimize the cost of building it

Answer 1

Answer:

Step-by-step explanation:

GIVEN: Suppose you want to make a cylindrical pen for your cat to play in (with open top) and you want the volume to be $100$ cubic feet. Suppose the material for the side costs $\ 3$ per square foot, and the material for the bottom costs $\ 7$ per square foot.

TO FIND: What are the dimensions of the pen that minimize the cost of building it.

SOLUTION:

Let height and radius of pen be $r\text{ and }h$

Volume $=\pi r^2h=100\implies h=\frac{100}{\pi r^2}$

total cost of building cylindrical pen $C=3\times \text{lateral area}+7\times\text{bottom area}$

                                                                $=3\times2\pi r h+7\times\pi r^2=\pi r(6h+7r)$

                                                                $=\frac{600}{r}+7\pi r^2$

for minimizing cost , putting $\frac{d\ C}{d\ r}=0$

$\implies -\frac{600}{r^2}+44r=0 \Rightarrow r^3=\frac{600}{44}\Rightarrow r=2.39\text{ feet}$

$\implies h=5.57\text{ feet}$

Hence the radius and height of cylindrical pen are $2.39\text{ feet}$ and $5.57\text{ feet}$ respectively.