# Calculus

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A metal cylindrical container with an open top is to hold 1 cubic foot. If there is no waste in construction, find the dimensions which will require the least amount of material.

• Calculus -

V = πr^2 h
h = 1/(πr^2)

Surface Area (SA)
= bottom + collar of cylinder
= πr^2 + 2πrh
=πr^2 + 2πr(1/(πr^2)
= πr^2 + 2/r
dSA/dr = 2πr - 2/r^2 = 0 for a min of SA
2πr = 2/r^2
r^3 = 1/π
r = 1/π^(1/3) = appr .693 ft
then h = 1/(π(.693^2)) = .683

• Calculus -

V = π r ² h = 1ft³
h = 1/π r ²

SA = π r ² + 2 π r h
= π r ² + 2 π r * 1/π r ²
= π r ² + 2 / r

SA ' = 2π r - 2 / r² = 0

2π r = 2 / r²
r³ = 1/π

r = 1/ ³√π ft

h = 1/(π r ²)
= 1/(π (1/ ³√π) ²)
= ³√π ² / π
= 1/ ³√π ft

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