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

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=(100)/(\pi r^2)`

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

`=3*2\pi r h+7*\pi r^2=\pi r(6h+7r)`

`=(600)/(r)+7\pi r^2`

for minimizing cost , putting
`(d\ C)/(d\ r)=0`

`\implies -(600)/(r^2)+44r=0 \Rightarrow r^3=(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.