Suppose you are a pharmacist who manufactures pill capsules. The pill is shaped like a cylinder with two hemispheres on either end. The pill must have a volume of 1000mg. The surface area of the hemispheres cost twice as much as the surface area of the cylinder. Find the dimensions which will minimize the cost of the pill.

Cannot figure this one out any help would be greatly appreciated

2. Originally Posted by Parrishguy
Suppose you are a pharmacist who manufactures pill capsules. The pill is shaped like a cylinder with two hemispheres on either end. The pill must have a volume of 1000mg. The surface area of the hemispheres cost twice as much as the surface area of the cylinder. Find the dimensions which will minimize the cost of the pill.

Cannot figure this one out any help would be greatly appreciated
Start by assigning names: Let R be the radius of the cylinder and h the height. The volume of that cylinder is $\displaystyle \pi R^2 h$ and its surface area is $\displaystyle 2\pi Rh$. The two hemispherical caps must have the same radius, R, and so have total volume $\displaystyle (4/3)\pi R^3$ and surface area $\displaystyle 4\pi R^2$.

We can, without loss of generality, take the cost of the cylinder to be 1 times its area and so the cost of two hemispheres is twice their area:

We want to minimize $\displaystyle 4\pi R^2+ 2(2\pi Rh)= 4\pi(R^2+ Rh)$ subject to the constraint that $\displaystyle \pi R^2h+ (4/3)\pi R^3= 1000$.

At this point I don't know how to help you further since there are several ways to do that and I don't know which would be appropriate for you. I would consider "Lagrange multipliers" to be best. Do you know that method.

By the way, the volume can't be "1000 mg". mg is a measure of mass, not volume. Did you mean 1000 ml or 1000 cc?

3. So at this point I would take the derrivative of 4\pi(R^2+ Rh) then plug it back into the constraint to get my measurements?