Surface Area Minimization

[anwaname]

I am working on a calculus problem where i am trying to minimize the SA of a cylinder. The problem reads as follows:
a) Calculate the Volume of the cylinder to 3 decimal places
b) Now the V is to be kept the same, but the proportions are to be changed. Write an equation expressing the total surface area of your cylinder as a function of radius and altitude. Then transform the equation so that the volume is in terms of radius alone.
c) Use the derivative to find the radius and height of the cylinder with minimal SA.
D) Calculate the 2 ratios (original and optimized) of diameter to height

So far I have:
d=7.9cm
r=3.95cm
h=12.6cm
a) V=?(r squared)(h)
V=617.610
b)SA=2?(r squared)+2?(r)(h)
SA=337.533

c)

d)r=(12.6/3.95)h

I'm not sure how to do part b.
I know that after I find the the equation and derivative I can set the derivative equal to zero to find r and from there h.

 

[skeeter]

let V = fixed volume of the cylinder

since \(\displaystyle V = \pi r^2 h\), then
\(\displaystyle h = \frac{V}{\pi r^2}\).

surface area is ...

\(\displaystyle A = 2\pi r^2 + 2 \pi r h\)

substitute ...

\(\displaystyle A = 2\pi r^2 + 2 \pi r \cdot \frac{V}{\pi r^2} = 2\pi r^2 + \frac{2V}{r}\)

now find \(\displaystyle \frac{dA}{dr}\) ... remember that V is a constant.