# If a cylindrical can has a surface area of 60 square inches, how do you express the volume of the can as a function of the radius?

Feb 17, 2015

$v o l \left(r\right) = 60 r - 2 \Pi {r}^{3}$

How this was determined:
Volume of cylinder $= 2 \Pi {r}^{2} \times h$ where $h$ is the cylinder height

surface area
$= 2 \left(\Pi {r}^{2}\right) + 2 \Pi r \cdot h$
$= \left(2 \Pi r\right) \cdot \left(r + h\right)$
$= 60$ (given)

$r + h = \frac{60}{2 \Pi r}$

$h = \frac{60}{2 \Pi r} - r$

Plugging this back into the formula for the volume of cylinder:
$v o l \left(r\right)$
$= \frac{2 \Pi {r}^{2} \cdot \left(60 - 2 \Pi {r}^{2}\right)}{2 \Pi r}$

$= 60 r - 2 \Pi {r}^{3}$