Aug 2007

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.
Apr 2009
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...
Jan 2010
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.
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