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?

Question

111697 views around the world

You can reuse this answer
Creative Commons License

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#