Question

A cylindrical container is to be made from certain solid material with the following constraints: It has a
fixed inner volume of V mm³, has a 2 mm thick solid wall and is open at the top. The bottom of the
container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the
container.
If the volume of the material used to make the container is minimum when the inner radius of the container
is 10 mm, then the value of V/250???? is

Answer 1

Let inner radius of the container be r and height be h

∴V=πr2h⇒h=πr2V..(1)

Now volume of the material v=π(r+2)2h+π(r+2)2×2−πr2h

⇒v=4πrh+4πh+π(r+2)2×2=r4V+r24V+2π(r+2)2

Now for minimum material required drdv=0

⇒−r24V−r38V+4π(r+2)=0

⇒100V+500V=π(10+2)⇒250πV=4