SOLUTION: a closed top cylindrical oil drum made of metal is to be constructed. the oil drum must hold 50 cubic feet of oil. find the dimensions of the oil drum to be constructed that will m

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Question 1057295: a closed top cylindrical oil drum made of metal is to be constructed. the oil drum must hold 50 cubic feet of oil. find the dimensions of the oil drum to be constructed that will minimize the amount of metal used in the construction.
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
The cylinder has two basic measurements. Height and diameter (or "length" and radius).

v for volume,
v=h%2Api%2Ar%5E2

s for surface area,
s=2%2Api%2Ar%5E2%2Bh%2Api%2A2%2Ar

You are given the volume, in cubic feet,
pi%2Ah%2Ar%5E2=50
h=50%2F%28pi%2Ar%5E2%29
Substitute for h in the surface area formula.

s=2pi%2Ar%5E2%2B%2850%2F%28pi%2Ar%5E2%29%292%2Api%2Ar

s=2pi%2Ar%5E2%2B100%2Fr

Minimizing the area s means finding ds%2Fdr and setting equal to 0.
4pi%2Ar%2B100%28-1%29r%5E-2=0
4pi%2Ar-100%2Fr%5E2=0
4pi%2Ar%5E3%2Fr%5E2-100%2Fr%5E2=0
%284pi%2Ar%5E3-100%29%2Fr%5E2=0
From here, you are interested in the NUMERATOR being 0.
4pi%2Ar%5E3-100=0
pi%2Ar%5E3-25=0
pi%2Ar%5E3=25
r%5E3=25%2Fpi
highlight%28r=root%283%2C25%2Fpi%29%29, and from this you can evaluate h.