A cylindrical metal can has volume of 600cm^3.

Question

what dimention will allow for the minimum amount of metal to produce the can.

Answer 1

Volume of cylender V = 600 cm³

Formula for Volume of cylender V = πr²h

600 = πr²h

h = 600/(πr²)

 

Area of cylender S = 2πr² + 2πrh = 2πr² + 2πr[ 600/(πr²) ] = 2πr² + [ 1200πr/(πr*r) ]

S = 2πr² + 1200/r

Apply derivative , for the minimum amount of metal.

dS/dr = 2π(2r) + 1200(-1/r²) = 4πr - 1200/r²

Area s is constant.So dS/dr = 0

4πr - 1200/r² = 0

4πr =1200/r²

r³ = 1200/4π = 300/π

r = 4.571 cm

 

h = 600/(πr²) = 600/(π(4.571)²) = 9.14 cm

The dimensions of volume for minimum amount of metal is r = 4.571 cm , h = 9.14 cm.