A cylindrical can, open at the top, is to hold 640 cm3 of liquid. Find the height and radius that minimize the amount of material needed to manufacture the can.

Question

Enter your answer with rational exponents, and use pi to represent .

Radius=

Height=

Answer 1

Find h with respect to r:

V = πr²h = 640

h = 640/πr²

Plug this into the surface area equation:

SA = πr² + 2πrh

= πr² + 2πr(640/πr²)

= πr² + 1280/r

Differentiate and set to 0, solve for r:

dSA/dr = 2πr - 1280/r² = 0

2πr = 1280/r²

r³ = 640/π

r = (640/π)^1/3

≈ 5.88 cm

Now find h:

h = 640/πr²

= 640/[π(5.88)²]

≈ 5.89 cm