Let's break this down step-by-step:
### Part C: Deriving the Volume Formula for a Cone
1. Volume of a Pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\][/tex]
Here, the base area is [tex]\( 144\pi \)[/tex]. Hence, the volume of the pyramid is:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times 144\pi \times h = \frac{144}{3} \pi h = 48\pi h
\][/tex]
2. Volume of a Cone:
The volume [tex]\( V \)[/tex] of a cone is given by:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\][/tex]
Given that the base of the cone is a circle, the base area of the cone is [tex]\( \pi r^2 \)[/tex]. Assuming that the radius [tex]\( r \)[/tex] of the cone is such that the base area equates to [tex]\( 144\pi \)[/tex], we conclude that:
[tex]\[
r^2 = 144
\][/tex]
[tex]\[
r = \sqrt{144} = 12
\][/tex]
So, the area of the base of the cone is:
[tex]\[
\pi r^2 = \pi (12)^2 = 144\pi
\][/tex]
Thus:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \times 144\pi \times h
\][/tex]
### Conclusions:
Through both cases, it becomes evident that the volume formula for the cone is identical to that of the pyramid when considering their bases and heights:
[tex]\[
V_{\text{cone}} = V_{\text{pyramid}} = \frac{144}{3} \pi h = 48\pi h
\][/tex]
### Final Formula:
Therefore, using the given dimensions and relationships, the formula for the volume of the cone can be written as:
[tex]\[
\boxed{V_{\text{cone}} = \frac{1}{3} \times 144\pi \times h = 48\pi h}
\][/tex]
This concludes the derivation with the volume of the cone formula based on the given measurements and comparisons.
