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 [tex]$2 r$[/tex] units. At every level, the areas of the cross sections of both solids equal [tex]$\pi\left(r^2 - b^2\right)$[/tex].

Cross-section area [tex]=[tex]$\pi r^2 - \pi b^2$[/tex][/tex]

Cross-section area [tex]=$\pi x^2$[/tex]

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



Answer :

It appears that you have included some statements about Cavalieri's principle and the volumes of solids, but the question is somewhat unclear. However, I believe you're seeking to understand how to use Cavalieri's principle to find the volume of a solid. Let's break down the concept using the context given and link it step-by-step with the key points:

1. Understanding Cavalieri's Principle:
Cavalieri's principle states that if two solids have equal heights and equal cross-sectional areas at every level parallel to their bases, then the two solids have equal volumes.

2. Height and Cross-sectional Area:
Given that both solids have a height of [tex]\( 2r \)[/tex] units and at every level, the areas of the cross sections of both solids are equal to [tex]\( \pi (r^2 - b^2) \)[/tex].

3. Calculating Cross-sectional Area:
The cross-sectional area at any level is calculated as:
[tex]\[\text{Cross-sectional Area} = \pi r^2 - \pi b^2\][/tex]
Further simplifying:
[tex]\[\text{Cross-sectional Area} = \pi (r^2 - b^2)\][/tex]

4. Volume Using Cones and Cylinders:
We can derive the volume of complicated structures by considering simpler constituent parts like cones, cylinders, and the space between them.

5. Volume of a Cylinder:
The volume of a cylinder can be described as:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
Since the height is [tex]\( 2r \)[/tex], for the given cylinder,
[tex]\[ V_{\text{cylinder}} = \pi r^2 (2r) = 2 \pi r^3 \][/tex]

6. Cone within the Cylinder:
Suppose we have two cones in the cylinder, and we want to relate them to the given structure. The volume of each cone can be derived using:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
Here the height [tex]\( h = r \)[/tex], so for one cone:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 r = \frac{1}{3} \pi r^3 \][/tex]
For two such cones, the total volume would be:
[tex]\[ V_{2\text{cones}} = 2 \left(\frac{1}{3} \pi r^3 \right) = \frac{2}{3} \pi r^3 \][/tex]

7. Solid Between Two Cones and Cylinder:
The volume of the remaining solid, where the two cones are inside the cylinder:
[tex]\[ V_{\text{solid}} = V_{\text{cylinder}} - V_{2\text{cones}} \][/tex]
Plugging in the known values:
[tex]\[ V_{\text{solid}} = 2 \pi r^3 - \frac{2}{3} \pi r^3 \][/tex]
[tex]\[ V_{\text{solid}} = \pi r^3 \left( 2 - \frac{2}{3} \right) = \pi r^3 \left( \frac{6}{3} - \frac{2}{3} \right) = \pi r^3 \left( \frac{4}{3} \right) = \frac{4}{3} \pi r^3 \][/tex]

So, by applying Cavalieri's principle and understanding the relationships between the cones and cylindrical shapes, we can derive that the volume of the described solid, which could be a part of a sphere or another geometric structure, involves these foundational calculations, tying the volumes of simpler shapes like cones and cylinders to the more complex overall volume.