What are the dimensions of the closed cylindrical can that has surface area 381 cm^2 and contains the maximum volume?

Question

radius= ________cm

height= ________cm

Answer 1

The formula for volume is:

V = πr²h

And the formula for the surface area is:

A = 2πr² + 2πrh

Solve the value of h in terms A and r:

-2πrh = 2πr² - A

h = (2πr² - A)/(-2πr)

h = A /(2πr) - r

Given A = 381 cm²

h = 381/(2πr) - r

Now plugin the value of h in the volume formula:

V = πr² (381/(2πr) - r)

V = (381/2)r - πr³

Find dV/dr:

dV/dr = (381/2) - 3πr²

The zero of the derivative is the local minimum and/or maximum:

0 = (381/2) - 3πr²

3πr² = 381/2

r² = 381/(6π)

r² = 127/(2π)

r = √(127/(2π))

The radius will be:

r ≈ 4.496 cm

The height will be:

h = 381/(2π(4.496...)) - 4.496...

h ≈ 8.992 cm

The maximum volume will be:

V = (381/2)(4.496...) - π(4.496...)³

V ≈ 570.973 cm³