Thread: Optimization question

In an optimization problem, how do you know which variable to try and eliminate?

For example -- A cylindrical soup can is meant to hold 460 cubic centimeters of soup.

The equation is (pi)(r^2)(h) = V
460 = (pi)(r^2)(h)

You try and eliminate h, but why?

Thanks!

What are you optimising? There's not much information here. Are you trying to get the biggest radius, smallest surface area?

sorry -- trying to minimize volume.

What is the smallest such soup can (in terms of amount of metal used) to hold this much soup?

That means that you're trying to minimise the surface area. They've already given you the volume. The eqaution for surface area for a cylinder is:
SA = 2(pi)r^2 + 2(pi)rh

Where r is radius and h is height.

If you're trying to minimise the surface area you should differentiate the SA equation, however first you have to eliminate one of the variables.

With your first question you don't have to eliminate h specifically. You should try to rearrange the Volume equation that you have so that you get r in terms of h or vice versa:
460 = (pi)r^2h
then you can get:
h = 460/[(pi)r^2]
or:
r^2 = 460/[(pi)h]

You can then substitute one of these equations into your SA equation and differentiate.
You can choose to eliminate h (by using the first equation) and it's probably easier that way. However you can use the second equation if you want.