# Thread: optimization

1. ## optimization

A cylindrical can, open at the top, is to hold 1000cm^3 of liquid. The material for the side of the can costs $.50 per cm^2 to manufacture, and the material for the base costs$.75 per cm^2 to manufacture. Find the height and radius that minimize the cost to manufacture the can.

Does anyone know how to solve this problem? I'm a bit lost.

2. Originally Posted by bluebyte22
A cylindrical can, open at the top, is to hold 1000cm^3 of liquid. The material for the side of the can costs $.50 per cm^2 to manufacture, and the material for the base costs$.75 per cm^2 to manufacture. Find the height and radius that minimize the cost to manufacture the can.

Does anyone know how to solve this problem? I'm a bit lost.
volume information ...

$\pi r^2 h = 1000$

$h = \frac{1000}{\pi r^2}$

surface area ...

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

$A = \pi r^2 + 2\pi r \left(\frac{1000}{\pi r^2}\right)$

cost ...

$C = .75(\pi r^2) + .50\left[2\pi r \left(\frac{1000}{\pi r^2}\right)\right]$

clean up the algebra for the cost function, find $\frac{dC}{dr}$ and minimize the cost.

3. Thank you