Thread: Need help with an Optimization type of problem

A cylindrical wire frame for a waste bucket has a wire circle for the top and bottom, and six straight wire rods that hold the top and bottom circles together. The total amount of wire used is 4m. What radius and height will maximize the volume of cylindrical frame?

I drew a pic (not sure if it's correct)

Can some one please atleast tell me how to start this monster

Originally Posted by supersaiyan

A cylindrical wire frame for a waste bucket has a wire circle for the top and bottom, and six straight wire rods that hold the top and bottom circles together. The total amount of wire used is 4m. What radius and height will maximize the volume of cylindrical frame?

The cylindrical space enclosed by the wire frame will have some height "h" and some radius "r", which will determine the volume V.

What is the length of the wire loop for the top? (Hint: Use the "circumference" formula.)

Then what is the length of wire used in the two loops? (Hint: Multiply by "2".)

What is the length of one of the uprights? (Hint: "h")

What then is the length of wire used for the six uprights? (Hint: Multiply by "6".)

What then is the expression for the total length of wire? (Hint: Add the abov.)

What is the total length? (Hint: They give you this.)

Use the expression and the given value to create an equation for the total length of wire. Solve this equation for one of the variables in terms of the othr.

What is the formula for the volume V of a cylinder with height "h" and radius "r"? (Hint: (pi)(r^2)(h).)

From where you solved the "length" equation for one of the variables, substitute for this variable in the "volume" equation.

Maximize. (Hint: Take the derivative, etc.)

If you get stuck, please reply with a clear listing of your steps and reasoning so far. Thank you!