Question

The height of the frustum cone is 12ft. The area of the top and bottom circle of the cone is $4f{{t}^{2}}$ and $4f{{t}^{2}}$ respectively. Find its volume.

A. $125f{{t}^{3}}$
B. $126f{{t}^{3}}$
C. $127f{{t}^{3}}$
D. $128f{{t}^{3}}$

Answer 1

Hint: We are given the curved surface area of a frustum cone, the larger circle area and the total surface area, as the total surface area of the frustum is curved surface area plus larger circle area plus smaller circle area. So, as we are given all the values, we can find the smaller circle area by subtracting the larger circle area and the curved surface area from the total surface area. And the volume of the frustum cone is also calculated by the formula, $\dfrac{h}{3}\left[ \pi {{R}^{2}}+\pi {{r}^{2}}+\sqrt{\left( \pi {{R}^{2}} \right)\left( \pi {{r}^{2}} \right)} \right]$.

Complete step by step answer:
According to the question it is asked to us to find the volume of a frustum cone whose dimensions are given to us. If we see the diagram of the frustum cone, then it's clear that height is 12ft. And the area of the top and bottom circle of the cone are given as $4f{{t}^{2}}$ and $4f{{t}^{2}}$.

So, the volume of the frustum cone is:
$V=\dfrac{h}{3}\left[ \pi {{R}^{2}}+\pi {{r}^{2}}+\sqrt{\left( \pi {{R}^{2}} \right)\left( \pi {{r}^{2}} \right)} \right]$
The area of the top cone = $4f{{t}^{2}}$
Area of the bottom cone = $4f{{t}^{2}}$
Height = $12ft$
So, the volume is given as:
$\begin{align}
  & V=\dfrac{12}{3}\left[ 4+4\times \sqrt{4\times 4} \right] \\
 & \Rightarrow V=\dfrac{12}{3}\left[ 8\times \sqrt{16} \right] \\
 & \Rightarrow V=\dfrac{12}{3}\left[ 8\times 4 \right] \\
 & \Rightarrow V=4\times 32=128f{{t}^{3}} \\
\end{align}$
So, the correct answer is “Option D”.

Note: You can also find the surface area if asked there by using the relation: $C.S.A=\pi \left( R+r \right)l$. However here you will be required to find the slant height first so it will be a bit of a lengthy process. The relation of slant height is given as $l=\sqrt{{{\left( R-r \right)}^{2}}+{{h}^{2}}}$. And this is the relation between radius and height.