A rectangular piece of sheet metal with perimeter 100 inches is rolled into a cylinder with open ends. Find the maximum volume of the cylinder.

Answer 1

Let x = the width and y = the height

perimeter = 2x+2y = 100 --> y = 50-x

Volume, V= πr²y

Circumference = x = 2πr --> r = x/2π

V(x) =π(x/2π)²(50-x) = (1/(4π))(50x-x²)

To maximize, take the derivative of V(x), set to zero and solve for x

V'(x) = (1/(4π))(50-2x)

50-2x=0

x=25

y=25

Notice that this is a square.

V(25) = (50•25-625)/(4π) = 625/(4π) ≈ 49.74 in³