Optimisation Question [Cylinder]

[recoil33]

Q. You have been asked to design a can shaped like a right circular cylinder with height h and radius r. Given that the can must hold exactly 130 cm3, what values of h and r will minimise the total surface area (including the top and bottom faces)?

Volume(Cylinder) = pi(r)²*h
130 = pi(r)²*h

Surface Area(Cylinder) = 2pi(r)*h
______________________________________
Because I'm working with 2 variales, r and h I'm a bit confused. Let alone, working with the two equations (Surface Area & Volume).

Firstly, changing h in the form of r.

h = 130/pi(r)²

therefore:

130 = pi(r)²*(130/pi(r)²)

Now, i should differentiate this.

0 = 2pi(r) * (0*(pi(r)²) - (130*2pi(r)) / ((pi(r)²)²)

I'm a bit confused as of what to do now, I've just finished previous optimisation questions although nothing with a cylinder like this.

Any help would be appreciated, thank you.

 

[I like Serena]

Q. You have been asked to design a can shaped like a right circular cylinder with height h and radius r. Given that the can must hold exactly 130 cm3, what values of h and r will minimise the total surface area (including the top and bottom faces)?

Volume(Cylinder) = pi(r)²*h
130 = pi(r)²*h

Surface Area(Cylinder) = 2pi(r)*h
______________________________________
Because I'm working with 2 variales, r and h I'm a bit confused. Let alone, working with the two equations (Surface Area & Volume).

Firstly, changing h in the form of r.

h = 130/pi(r)²

therefore:

130 = pi(r)²*(130/pi(r)²)

Now, i should differentiate this.

0 = 2pi(r) * (0*(pi(r)²) - (130*2pi(r)) / ((pi(r)²)²)

I'm a bit confused as of what to do now, I've just finished previous optimisation questions although nothing with a cylinder like this.

Any help would be appreciated, thank you.

First of, your area of the cylinder does not take the top and bottom into account.
Can you add those?

Second, it's right to use the volume to find an h expressed in r.
But you need to optimise the area, not the volume.
So you need to insert the expression for h in the area, then take the derivative of the area, and solve it for being equal to zero.

 

[hunt_mat]

This is a method of Lagrange Multipliers problem...