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Based on my results from Part A, I can conclude that if cones and pyramids have the same height and radius, their volumes are the same.

Part C:
The volume of the pyramid is one-third of the area of its base multiplied by its height, which is [tex]$\frac{1}{3} \pi r^2 h$[/tex] cubic units. Using this measurement and your answer from Part B, derive a formula for the volume of a cone.

The volume of a pyramid is [tex]$=\frac{1}{3} \pi r^2 h$[/tex]. Using this measurement and my answer from Part B, the formula for the volume of a cone is [tex]$=\frac{1}{3} \pi r^2 h$[/tex].


Sagot :

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.
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