Much needed optimization help!

[sylas7306]

The makers of a soda are constructing a container that has a shape of a right circular cylinder. The container must hold 10π cubic inches of soda. It costs $0.02 per square inch to construct the top and bottom of the can and $0.01 per square inch to construct the side of the can. Find the dimensions that will minimize the cost.

I've spent hours on this, as well as looking up tutorials/explanations and cannot find anything. Please help. Much appreciated.

 

[HallsofIvy]

Since you spent hours on this, surely you have some work you could show? We cannot tell you what you did wrong if you do not tell us what you did!

The problem is to minimize the cost of a cylinder that will hold \(\displaystyle 10\pi\) cubic inches. The "dimensions" of a "right circular cylinder" are its height and radius, h and r. What is the formula for volume of a cyinder? set that equal to \(\displaystyle 10\pi\) to get a formula relating r and h. What is the total area of both top and bottom (they are circles of radius r, of course)? Multiply that by $0.02 to get the cost. What is the area of the sides (it is a rectangle with height h and width the circumference of the circular top and bottom, \(\displaystyle 2\pi r\). Multiply that by $0.01 to get the cost. Add those two costs together to get the total cost. Use the fact that the volume is \(\displaystyle 10]pi\) to replace h with a function of r.

Since this is posted in Calculus, I presume you know you can find a max or min of a function by setting its derivative equal to 0. But I also see that the cost function is quadratic in r and so you could find its minimum by "completing the square".