$\begingroup$I have a problem which should be very easy (as the rest of them are on this worksheet) but this one has me stumped. The question reads:
A metal can is in the form of a cylinder. It has a bottom but no top. The surface area is a constant S square meters. Find the dimensions of the can which maximize the volume.
For the surface area of a cylinder, I know the equation is $2\pi rh + 2\pi r^2$. Since we have no top of the can, the formula changes to $2\pi rh + \pi r^2$. The volume of a cylinder is $V=\pi r^2h$.
In all of the other problems on the sheet we've basically solved for one variable like r or h using a given surface area or volume. This problem, however says the surface area is $S$ instead of actually giving a numeric value.
Is there any way to solve this problem?
$\endgroup$32 Answers
$\begingroup$Use the equation
$$ S= 2\pi r h + \pi r^2 $$
to get $h$ as
$$ h=\frac{S-\pi r^2 }{2\pi r } $$
then substitute in the volume equation V for $h$ to get an equation in $r$
$$ V(r) = \pi r^2 \left(\frac{S-\pi r^2 }{2\pi r } \right) $$
and then use derivative techniques to maximize.
$\endgroup$2$\begingroup$You are close.
Since the surface area is $S$, from your formula, $S =2\pi rh+\pi r^2 $. From this, you can solve for either $r$ or $h$ as a function of $S$ and the other dimension.
It would probably be easiest to solve for $h$ as a function of $S$ and $r$ since $h$ occurs linearly.
Now, substitute the formula for $h$ into the equation for $V$. This will be an expression involving $S$ and $r$. You want to find the value of $r$ that maximizes this expression, since that will produce the largest volume.
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