Answered

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.

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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