Optimization problem, Please Help!!

Question

I am asked to design a fuel tank for a fighter jet that will hold 300 liters of jet fuel. The shape of the tube is a cylinder with two hemispheres attached to each end. The side wall (the cylinder part) costs $0.1 per cm^2 to make while the rounded tops (hemispheres) cost $0.3 per cm^2 to make. What is the optimal radius that will minimize the manufacturing cost?

Answer 1

Hi Caleb,

 

I will try to get you started on this by giving you a strategy. You will need formulas for volume of a sphere and cylinder and lateral surface area for a cylinder and surface area of a sphere.

 

The Total volume of the fuel tank can be represented by V = 4/3πr³ + πr²h = 300. We need this equation in order to establish a relationship between r and h. Solve for h to get something like h = (900 - 4πr³)/3πr². 

 

Now we need a total cost function. The surface area can be represented by S = 4πr² + 2πrh (sphere + cylinder).

 

The total cost is therefore C = 30(4πr²) + 10(2πrh) where cost is shown in cents (instead of $). 

 

Replace the value of h in the C function with its equivalent value determine by the volume relationship. 

 

Then find C', set that equal to zero and solve for r. Convince yourself that this value for r generates a minimum value for the cost function where r>0.