A fuel tank is being designed to contain 200 m^3 of gasoline; however, the maximum length tank that can be safely transported to clients is 16 m long. The design of the tank calls for a cylindrical part in the middle with hemispheres at the end. If the hemispheres cost $2/unit and the cylindrical wall costs $1/unit, find the radius and height of the cylinder part so that the cost of manufacturing the tank will be minimal. Give the answer correct to the nearest centimeter.
Here's what I have so far:
V= πr^2h + 4/3πr^3
200= πr^2h + 4/3πr^3
(200 - 4/3πr^3)/(πr^2)= h
SA= 4πr^2 + 2πrh
= 4πr^2 + 2πr[(200 - 4/3πr^3)/(πr^2)]
= 4πr^2 + (400πr - 8/3π^2r^4)/r^2
C= 8πr^2 + (400πr - 8/3π^2r^4)/(πr^2)
C'= 16πr - [16(r^3π + 75)]/3r^2 ??
I'm pretty lost and I'm sure I made several large errors because this is just not making sense. Any help is very much appreciated. Thanks!