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A rectangular piece of sheet metal with a perimeter of cm is rolled into a cylinder with two open ends.

Horizontal rectangle, bottom edge labeled, x.Cylinder, circumference of base labeled, x.

  1. Write equations for the radius and height of the cylinder in terms of .

    Look at the diagram of the cylinder. The circumference of the base is equivalent to .

    Solve for .

  2. Express the volume of the cylinder as a function of .

    . Substitute.

  3. Determine the value of that will maximize the volume and calculate the maximum value.

    The volume will be at a maximum when its derivative is .