How do you solve this? I know that you have to use the Surface area equation but I got lost while doing the work. Please help :(

Question

The material for the cans is cut from sheets of metal. The sylindrical sides are formed by bending rectangles. These rectangles can be cut without very much waste. But if the tops and bottoms are cut from squares of side 2r then some amount of the metal goes to waste. Show that if this method is used, then the amount of metal used is minimized when h/r = 8/π ≈ 2.55

( I think you need to somehow get 8/π as an answer, but like I'm not getting that )

Answer 1

The surface area formula now changes. It is now 2πrh + 2(2r)² = 2πrh + 8r²

Then, as V = πr²h, h = V/(πr²)

Thus, we minimize A=2πrh + 8r² = 2πrV/πr² + 8r² = 2V/r + 8r²

dA/dr = 0

-2V/r² + 16r = 0

2V/r² = 16r

2V = 16r³

r = 1/2 V^1/3

h = V/πr² = V/(π(1/2 V^1/3)²)= V/(πV^2/3/4) = 4V^1/3/π

Then, h/r = 4V^1/3/π/(1/2 V^1/3) = 8/π