A cylindrical can is to be made to hold 1000cm^3 of oil. How do you find the dimensions that will minimize the cost of metal to manufacture the can?

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Answer 1 · 2016-06-29T04:13:40

Height = 10.84 cm, radius of the base = 5.42 cm and material for the surface =37 square cm, nearly

Let height = h and radius of the base = r.

Then, volume $V=pi r^2h=1000$ cc

and surface area of the can

$S=2pir^2+2pirh$

Now, eliminating h, $S=S(r)=2pi(r^2+1000/(pi r^2))$

$S'=2pi(2r-1000/(pir^3))=0$ , when

$r=(1000/(2pi))^(1/3)=5.42$ cm, nearly.

Correspondingly, h= 10.84 cm, nearly

There is no maximum for S. Also,

$S''=2pi(2+3000/(pir^4))>0$ ..

For this r= 5.42 cm, S = 37.1 $cm^2$