Question

A cylindrical oatmeal soda can has a capacity of 344 ml. Find the dimensions that will minimize the amount of material needed to construct the container.

Answer 1

r = 3.797 cm

h = 7.594 cm

Explanation:

The cylinder volume V = πr² h

So, given that the volume = 344 ml

Then,

344 = πr² h

Make h the subject of the formula:

`h = (344)/(\pi r^2)` ----- (1)

Similarly, the surface area of a cylinder is expressed by:

A = 2πr² h + 2 πr² ---- (2)

If we replace the value of h from above in (1) to (2)

Then;

`A = 2 \pi r ((344)/(\pi r^2))+2 \pi r^2`

`A = (688)/(r)+ 2 \pi r^2`

Taking the differential of A with respect to r, we have:

`(dA )/(dr) =- (688)/(r^2)+ 4 \pi r`

Set
`(dA )/(dr)=0` for the minimum surface area.

So,

`0 =- (688)/(r^2)+ 4 \pi r`

`(688)/(r^2) =4 \pi r`

Divide both sides by 4

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

`r^3 = (172)/(\pi)`

`r = \sqrt[3]{(172)/(\pi)}`

r = 3.797 cm

From (1), the height is:

`h = (344)/(\pi r^2)`

`h = (344)/(\pi (3.797)^2)`

h = 7.594 cm