Question

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)

Answer 1

1) r

2) a circle

3) an annulus

4)2pir^3

5)4/3pir^3

Answer 2

1. r

2. a circle

3. an annulus

4. _2pier³_

5. _4/3pier³_

Explanation:

The paragraph representing Cavalieri's principle is filled as follows;

"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 _r_ and height 2r minus the volume of two cones, each with a radius and height of r. A cross section of the sphere is _a circle_, and a cross section of the cylinder minus the cones, taken parallel to the base of cylinder, is _an annulus_. the volume of the cylinder with radius r and height 2r is _2pier³_, 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 _4/3pier³_ Therefore, the volume of the sphere is 4/3pie r^3 by Cavalieri's principle.