Question

The volume of a cylindrical tin can with a top and a bottom is to be 16π cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?

Answer 1

Answer:

h = 4 in

Explanation:

GIVEN DATA:

volume of tin$= 16 \pi$

we know that

volume of cylinder is $v = \pi r^2 h$

so,

$16 \pi = \pi r^2 h$

$16 = r^2 h$

$r = \sqrt{\frac{16}{h}}$

construct formula for surface area

$S = 2\pi r^2 + 2\pi rh$

$S = \frac{2v}{h} + 2 \sqrt{v \pi h}$

minimize the function wrt  h

$S' = \frac{2v}{h^2} + \sqrt{\frac{v \pi}{h} = 0$

solving for h we have

$h = [\frac{4 v}{\pi}]^{1/3}$

we kow $v = 16 \pi$   so

h = 4 in