Question

Suppose we need to construct a tin can with a fixed volume V cm3 in the shape of a cylinder with radius r cm and height h cm. (Here V should be regarded as a constant. In some sense, your answers should be independent of the exact value of V .) The can is made from 3 pieces of metal: a rectangle for the side and two circles for the top and bottom. Suppose that these must be cut out of a rectangular sheet of metal. Our goal is to find the values of r and h, and the dimensions of this rectangular sheet that minimize its area.

1. Draw a picture of how the rectangle and two circles could be cut out of a larger rectangle. There are multiple ways to do this (I can think of at least 3). Draw as many as you can, solve the problems below for each arrangement and then compare your answers.

2. Label the sides of the rectangle in terms of r and h. Express the rectangle’s area in terms of r and h. Also, note whether there are any assumptions about r and h that you need to make in order for your picture to make sense. (For example, if you draw a circle with diameter 2r inside of a rectangle with side l, then you must have 2r ≤ l.)

3. Use the fact that the can’s volume is V = πr2h to express h in terms of r, and write the rectangle’s area as a function of r. (Or else, you may alternatively solve for r and write the area as a function of h.)

4. Find the value of r (or h) that minimizes the rectangle’s area. What is the correspond- ing value of h (or r), and the dimensions of the rectangle? Your answers will most likely be in terms of V , but the ratio h/r might be a number. What is the minimum area of the rectangle in terms of V ?

5. As mentioned above, you should complete (1)-(4) for as many different arrangements as you can think of. (The math for some might be very simple.) Then compare your answers to find the best way of arranging the 2 circles and rectangle inside the larger rectangle, and the minimum possible area of the rectangle.

6. What if you need to make 2 (or more) cans in the same way. Can you find an arrangement of all the necessary pieces inside a single rectangle that is even more efficient?

Answer 1