Optimal Calculus problems.

Thanks for your help.

Problem

Cans of soft drinks are made from aluminum. Due to the soaring prices, the beverage company executives are looking for all possible cost-cutting measures. You are being asked to find a way to redesign the traditional can to decrease cost as much as possible. The manufacturer fills the containers with 355 cm³ of liquid and leaves 15 cm³ of open space, so the total volume of 370 cm³ must be maintained.

Now we have the information that the can which has a minimum surface:

• Radius = 3.8905 cm
• Diameter = 7.781 cm
• Height = 7.781 cm
• Volume = 370 cm³
• Surface area = 285.3091cm²

A problem with what we've done in minimizing the surface area of the cylinder is that in reality it takes double the material to construct the top and bottom of the can compared to the side of the can. This is because the top and bottom are layered, folded, and bent to make a stronger can. If we reinforce the top and bottom we need to make some changes to our model.

Q1. Find a new surface area formula to reflect this new information, that is, double the top and bottom surface area and leave the side surface area the same as it was before. You now have a new formula for surface area of a reinforced can. Then eliminate the height variable just as you did before, find the derivative, and use it to find the radius at which the surface area of this new can is a minimum. Show your work.

Q2. For the radius you just found for the reinforced can, calculate the diameter and height. Then, find the surface area. Show your work.