The Problem Consider the perimeter of the two rectangles above - 39 ins. Think of making baseless cylinders from all the rectangles with a 39 inch perimeter. What are the dimensions of the rectangle that would produce the cylinder with the greatest volume?
Explore You may want to cut some rectangles with various lengths and widths with the correct perimeter and roll them to form the cylinders. It's fun to predict which of them you think will have the greatest volume.
Formulas you may need:
Solution Choose which method you want to use to solve the problem:
- Circumference = 2(pi)(r)
- Volume = (pi)(r^2)h
Tabular MethodComplete the table below.
Height of Rectangle Width of Rectangle Radius of Cylinder Volume of Cylinder 1 3 5 7 9 11 13 15 17 19
Where is the volume greatest? Try a value on each side of the radius with the greatest volume. Did one of those give you a greater volume? Keep trying to get closer and closer to get the best answer that you can.
Write an expression for the volume of the cylinder as a function of its radius. (Hint: Use the two formulas provided.) Graph the expression, preferably using a graphing utility. Where does the function seem to achieve its maximum value. Can you zoom in to get a more exact answer? Continue to zoom in until you get the best answer that you can.
Using calculus you can find an exact solution. Write an expression for the volume of the cylinder as a function of its radius. (Hint: Use the two formulas provided.) Find the dimensions of the rectangle that give the volume function's maximum value.
Extension: Solve the problem using any P for the perimeter.
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