Question

Can someone help me with this geometry question? It’s part C I will provide you with more information about the question if needed.

Answer 1

$V=\frac{1}{3}*\pi *r^2*h$

C)

1) For cones and pyramids, we're going to use this general formula:

$V=\frac{1}{3}*A_{base}*h$

2) And since the pyramid and the cone have the same height, we can find the common heigh they share by plugging into the Volume of pyramid formula the given height:

$\begin{gathered} V=\frac{1}{3}*Ab*h \\ \frac{144}{3}\pi h=\frac{1}{3}*Ab.h \\ \frac{144}{3}\pi h=\frac{1}{3}Abh \\ \end{gathered}$

Note that the base of a cone is a circumference, so we can replace that with the formula for the area of the circle:

$\begin{gathered} \frac{144\pi}{3}h=\frac{1}{3}*(\pi *r^2)*h \\ \frac{144}{3}\pi h=\frac{\pi *r^2}{3}*h \\ 3*\frac{144}{3}\pi h=\frac{\pi r^{2}}{3}h*3 \\ r^2=144 \\ r=\sqrt{144} \\ r=12 \\ ---- \\ \end{gathered}$

3) Finally, let's test

$\begin{gathered} \frac{144}{3}\pi h=\frac{1}{3}\pi r^2*h \\ \frac{144}{3}\pi h=\frac{1}{3}\pi *12^2*h \\ \frac{144}{3}\pi h=\frac{144}{3}\pi h \\ \end{gathered}$

4) Thus the formula for the volume of a cone is:

$V=\frac{1}{3}*\pi *r^2*h$