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A rectangle is bounded by the function and the -axis as shown below.

Downward parabola, vertex at the point (0, comma 5), passing through the points (negative 2, comma 2), & (2, comma 2), shaded rectangle, bottom edge on x axis, top edge at, y = 2, with top left & top right vertices on the curve.

  1. If the base of the rectangle is , what is the height?

    Notice that the rectangle is symmetric across the -axis.

    Because of the symmetry, the endpoints of the rectangle can be found at and after all, the distance between and is , the length of the base.

    Consequently, the height of the rectangle can be evaluated at :

  2. What is the maximum area that the rectangle can enclose?

    Area = (base)(height)
    Use the base and height you found in part (a).Then optimize the Area function.

    To find the maximum Area:
    1. Let .
    2. Solve for .
    3. Use the and your expressions for base & height to calculate the maximum area.

  3. If you make the rectangle from part (b) into a flag and spin it around the -axis, what is the resulting volume?

    A cylinder with height __________ and radius __________.

  4. Determine the value of so that the volume of the rotated rectangle is a maximum.

    Volume of a cylinder:
    If you need guidance optimizing the volume, refer to steps in part (b).