# Math Help - Optimization Involving A Cylinder

1. ## Optimization Involving A Cylinder

I am not sure how to go about doing this question:

A cylindrical can needs to hold 500cm^3 of apple juice. The height of the can must be between 6cm and 15cm, inclusive. How should the can be constructed so that a minimum amount of material will be used in the construction? (Assume that there will be no waste.)

Any help is appreciated.

2. Originally Posted by Shapeshift
A cylindrical can needs to hold 500cm^3 of apple juice. The height of the can must be between 6cm and 15cm, inclusive. How should the can be constructed so that a minimum amount of material will be used in the construction? (Assume that there will be no waste.)
A good place to start might be with the formulas for the volume V and surface area SA of a cylinder with height h and radius r. Also, it would probably be useful to note that you have been given that V = 500, 6 < h < 15, and that you are trying to minimize SA. So:

i) What is the formula for the volume V of a cylinder with height h and radius r?

ii) Noting that V = 500, solve for r in terms of h. (You have bounds on h, so you know you need to be working with h.)

iii) What is the formula for the surface area SA of a cylinder with height h and radius r?

iv) Into the formula in (iii), substitute for r using the results of (ii). You should now have SA in terms only of h.

v) Differentiate with respect to h, and minimize. Remember to check the endpoints of the interval for absolute minima.

vi) Back-solve for the radius, and clearly state the cylinder's dimensions.