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5-40.

A rectangle is bounded by the function y = –x2 + 5 and the x-axis as shown below. Homework Help ✎

y= – x^2 + 5 with a shaded rectangle inscribed from y = 2 to the x-axis

  1. If the base of the rectangle is 2x, what is the height?

    Notice that the rectangle is symmetric across the y-axis.

    Because of the symmetry, the endpoints of the rectangle can be found at −x and +x... after all, the distance between −x and x is 2x, the length of the base.

    Consequently, the height of the rectangle can be evaluated at f(x):
    height = y = −x2 + 5

  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 A'(x) = 0.
    2. Solve for x.
    3. Use the x 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 x-axis, what is the resulting volume?

    A cylinder with height __________ and radius __________.

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

    Volume of a cylinder: V = πr2h
    If you need guidance optimizing the volume, refer to steps in part (b).