A cylindrical can, open at the top, is to hold 500 cm3 of liquid. Find the height and

radius that minimizes the amount of material needed to manufacture the can.

Find h with respect to r:

V = πr²h = 500

h = 500/πr²

Plug this into the surface area equation:

SA = πr² + 2πrh

= πr² + 2πr (500/πr²)

= πr² + 1000/r

Differentiate and set to 0, solve for r:

dSA/dr = 2πr - 1000/r² = 0

2πr = 1000/r²

r³ = 500/π

r = (500/π) ^1/3

≈ 5.42 cm

find h:

h = 500/πr²

= 500/[π (5.42) ²]

= 5.42cm