# Thread: Optimization problem with cylinders

1. ## Optimization problem with cylinders

This is the question I am having trouble with:

A Cylindrical can is to hold 4pi cubic units of juice. The cost per square unit of constructing the metal top and bottom is twice (two times) the cost of constructing the cardboard side. What are the dimensions of the least expensive can?

I know the sides can be described by 2pi*r*h, and so in terms of cost I would write something like 2x*(2pi*r*h). Both the metal top and bottom combined can be described by its surface area as such: 2pi*r^2.

So here my cost equation looks like C = 2x*(2pi*r*h) + x*(2pi*r^2)

I don't know what to do with 4pi cubic units of juice, other than set that equal to pi*r^2*h, which is the volume of the cylinder.

I know I need to minimize cost, and find the dimensions of the cylinder at that value.

Which variables should I be relating? I have volume, surface area, cost and dimension(?). It seems like I need to create a cost equation and relate that to volume...

2. Originally Posted by cognoscente
This is the question I am having trouble with:

A Cylindrical can is to hold 4pi cubic units of juice. The cost per square unit of constructing the metal top and bottom is twice (two times) the cost of constructing the cardboard side. What are the dimensions of the least expensive can?

I know the sides can be described by 2pi*r*h, and so in terms of cost I would write something like 2x*(2pi*r*h). Both the metal top and bottom combined can be described by its surface area as such: 2pi*r^2.

So here my cost equation looks like C = 2x*(2pi*r*h) + x*(2pi*r^2)

I don't know what to do with 4pi cubic units of juice, other than set that equal to pi*r^2*h, which is the volume of the cylinder.

I know I need to minimize cost, and find the dimensions of the cylinder at that value.

Which variables should I be relating? I have volume, surface area, cost and dimension(?). It seems like I need to create a cost equation and relate that to volume...
The specified volume should give you a relationship between r and h. Use this to eliminate one or the other of these from the cost equation then minimise the cost in the usual manner.

CB