# Volumes

#### name101

Hello,

I am stumped on this question. I think I'm missing something really obvious but here is goes.

"You have been asked to design a 1000cm^3 can shaped like a right cylinder. What dimensions will use the least material?"

So I have V = $$\displaystyle \pi r^2h$$
Which is given for obvious reasons. But I really don't know how to go from there.
If I get the derivative of the formula I will have 2 variables and then I start to confuse myself.

#### Diemo

The amount of material used would be the surface area. You want to minimise the amount of material used. (I.e. Set the derivative of the area equal to zero. I think it has to be the derivative with respect to the radius, because with respect to the height I get r=0 and h=infinity). This will give a relationship between your radius and your height. Put this into your equation for volume and viola...

#### drumist

You know that

$$\displaystyle V=\pi r^2 h=1000$$

Solving for $$\displaystyle h$$ gives

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

Determine an equation for surface area of the cylinder, then use the above equation to eliminate $$\displaystyle h$$ from the equation. You will be left with only one variable.

• name101

#### name101

Cheers for that , I got it working now 