Question

For its beef stew, Betty Moore Company uses aluminum containers that have the form of right circular cylinders. Find the radius and height of a container if it has a capacity of 44 in.3 and is constructed using the least amount of metal. (Round your answers to two decimal places.)

Answer 1

`1.91\ \text{in}`

`3.84\ \text{in}`

Explanation:

V = Volume of cylinder =
`44\ \text{in}^3`

h = Height of cylinder

r = Radius of cylinder

Volume of cylinder is given by

`V=\pi r^2h\\\Rightarrow h=(V)/(\pi r^2)\\\Rightarrow h=(44)/(\pi r^2)`

Total surface area of a cylinder is given by

`S=2\pi r^2+2\pi rh\\\Rightarrow S=2\pi r^2+2\pi r*(44)/(\pi r^2)\\\Rightarrow S=2\pi r^2+(88)/(r)`

Differentiating with respect to radius

`(dS)/(dr)=4\pi r-(88)/(r^2)`

Equating with zero

`4\pi r-(88)/(r^2)=0\\\Rightarrow 4\pi r=(88)/(r^2)\\\Rightarrow r^3=(88)/(4\pi)\\\Rightarrow r=((22)/(\pi))^{(1)/(3)}\\\Rightarrow r=1.91\ \text{in}`

Double derivative of S

`(d^2S)/(dr^2)=4\pi+176>0`

So
`r` is minimum at
`(dS)/(dr)=0`

`h=(44)/(\pi r^2)=(44)/(\pi 1.91^2)\\\Rightarrow h=3.84\ \text{in}`

So the radius and height of the cylinder is
`1.91\ \text{in}` and
`3.84\ \text{in}` respectively such that the least amount of metal is used.