Question

Drag and drop an answer to each box to correctly complete the explanation for deriving the formula for the volume of a sphere

Answer 1

The answers that would fill in the blanks are

• 2r
• a circle
• an annulus
• 1/3πr³
• 4/3πr³

What is the Cavalier's principle?

This principle states that if two solids are of equal altitude then the sections that the planes would make would have to be parallel and also be at the same distances from their bases which are equal such that the volumes of the solids would be equal.

Now we have to fill in the blanks with the solution.

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 base of cylinder, is 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 2πr³, and the volume of each cone with radius r and height r is 1/3πr³. So the volume of the cylinder minus the two cones is 4/3πr³. Therefore, the volume of the sphere is by Cavalieri's principle

Read more on Cavalieri's principle here

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