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pretest: extending to three dimensions
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12 cm
5 cm
5 cm
a rectangular prism has a height of 12 centimeters and a square base with sides measuring 5 centimeters. a pyramid with the same base and
half the height of the prism is placed inside the prism, as shown in the figure.
the volume of the space outside the pyramid but inside the prism is
cubic centimeters,
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For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with radius and height 2r minus the volume of two cones, each with a radius and height of r. a cross section of the sphere is and a cross section of the cylinder minus the cones, taken parallel to the base of cylinder, is the volume of the cylinder with radius r and height 2r is and the volume of each cone with radius r and height r is 1/3 pie r^3. so the volume of the cylinder minus the two cones is therefore, the volume of the cylinder is 4/3pie r^3 by cavalieri's principle. (fill in options are: r/2- r- 2r- an annulus- a circle -1/3pier^3- 2/3pier^3- 4/3pier^3- 5/3pier^3- 2pier^3- 4pier^3)