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

Answer:

r = 3.797 cm

h = 7.594 cm

Step-by-step 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:

$h = \dfrac{344}{\pi r^2}$  ----- (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;

$A = 2 \pi r (\dfrac{344}{\pi r^2})+2 \pi r^2$

$A = \dfrac{688}{r}+ 2 \pi r^2$

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

$\dfrac{dA }{dr} =- \dfrac{688}{r^2}+ 4 \pi r$

Set $\dfrac{dA }{dr}=0$ for the minimum surface area.

So,

$0 =- \dfrac{688}{r^2}+ 4 \pi r$

$\dfrac{688}{r^2} =4 \pi r$

Divide both sides by 4

$\dfrac{177}{r^2} = \pi r$

$r^3 = \dfrac{172}{\pi}$

$r = \sqrt[3]{\dfrac{172}{\pi}}$

r = 3.797 cm

From (1), the height is:

$h = \dfrac{344}{\pi r^2}$  

$h = \dfrac{344}{\pi (3.797)^2}$

h = 7.594 cm