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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.

Verified Answer :

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

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