1. ## open top cylinder

A drum in the form of a circular cylinder and open at one of the circular ends, is to be made so as to contain one cubic yard. Find the dimensions of the drum (height h and base radius r) which minimizes the amount of material going into the drum. The surface area of the drum includes the area of the cylinder and the circles at the bottom.

r=
h=

So far what I have tried to do is use the formulas for area of the circle and the cylinder and the put h in terms of r :

1 cubic yard = pi*r^2 + 2*pi*r*h
1 - pi*r^2 = 2*pi*r*h
(1 - pi*r^2) / (2*pi*r) = h

Then I put this back into the original formula:

= pi*r^2 + 2*pi*r*(1 - pi*r^2) / (2*pi*r)

In trying to differentiate I ended up with a very convoluted piece of paper that ended up with r = 9/pi and this is wrong.

Can anyone tell me if the work I've posted so far is correct? It is possible that I just have a problem with the differentiation process, but also possible that I haven't set the problem up correctly.

Thanks!!

2. $\pi r^2h = 1$

$h = \frac{1}{\pi r^2}$

you want to minimize surface area ...

$A = \pi r^2 + 2\pi r h$

$A = \pi r^2 + 2\pi r \frac{1}{\pi r^2}$

$A = \pi r^2 + \frac{2}{r}$

now find $\frac{dA}{dr}$ and minimize the surface area