Suppose we need to construct a tin can with a fixed volume V cm3in the shape of a cylinder with radius r cm and height h cm. (HereV should be regarded as a constant. In some sense, your answersshould be independent of the exact value of V .) The can is madefrom 3 pieces of metal: a rectangle for the side and two circlesfor the top and bottom. Suppose that these must be cut out of arectangular sheet of metal. Our goal is to find the values of r andh, and the dimensions of this rectangular sheet that minimize itsarea.

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

2. Label the sides of the rectangle in terms of r and h. Expressthe rectangle’s area in terms of r and h. Also, note whether thereare any assumptions about r and h that you need to make in orderfor your picture to make sense. (For example, if you draw a circlewith diameter 2r inside of a rectangle with side l, then you musthave 2r ≤ l.)

3. Use the fact that the can’s volume is V = πr2h to express h interms of r, and write the rectangle’s area as a function of r. (Orelse, you may alternatively solve for r and write the area as afunction 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 dimensionsof 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 ofthe rectangle in terms of V ?

5. As mentioned above, you should complete (1)-(4) for as manydifferent arrangements as you can think of. (The math for somemight be very simple.) Then compare your answers to find the bestway of arranging the 2 circles and rectangle inside the largerrectangle, and the minimum possible area of the rectangle.

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