Question

Cavalieri’s principle states that if two solids of equal height have equal cross-sectional areas at every level parallel to the respective bases, then the two solids have equal volume.
The two shaded solids both have a height of 2r units. At every level, the areas of the cross sections of both solids equal π(r2 – b2).

Therefore, the formula for the volume of the sphere can be derived by writing an expression that represents the volume of
one cone within the cylinder.
the two cones within the cylinder.
the solid between the two cones and the cylinder.
the cylinder.

Answer 1

Answer:

one cone within the cylinder.

Step-by-step explanation:

First of all we have to start with half of a sphere (also known as a hemisphere), whose radius is r, and calculate its volume. Then we can double our answer at the end to find the volume of the complete sphere. But first to apply the Cavalieri's Principle, we start with a cone with the same radius r and suppose the height is also r, so the volume of this cone is:

$V_{cone}=\frac{1}{3}\pi r^{3}$

Next, let's look at a cylinder and suppose this cylinder also has a radius r and height r, so the volume of this cylinder is:

$V_{cylinder}=\pi r^{3}$

Next, we place the the cone inside the cylinder and the volume that's inside the cylinder, but outside the cone is:

$V_{cylinder}-V_{cone}=\pi r^{3}-\frac{1}{3}\pi r^{3}=\frac{2}{3}\pi r^{3}$

And this is the volume of a hemisphere. Finally, if we double this value we get the volume of a complete sphere, which is:

$V_{sphere}=\frac{4}{3}\pi r^{3}$

Answer 2

Answer:

C. The solid between the two cones and the cylinder

Step-by-step explanation: