Question

What is the volume of a right circular cylinder whose base area is $606c{{m}^{2}}$ and whose height is 2m?

Answer 1

Hint: We here have been given the base area and height of a cylinder and we need to find its volume. For this, we will need its radius. We will calculate that using the base area which is given by the formula $\text{Base area=}\pi {{r}^{2}}$. Hence, we will get our required radius. Then we will put the value of the calculated radius and given height in the formula for volume given as $V=\pi {{r}^{2}}h$ and hence we will get the required volume.

Complete step by step answer:
Here, we have to find the volume of a cylinder whose base area is $606c{{m}^{2}}$ and height is 2m.
To calculate the volume of a cylinder, we need both its radius and height. Since we know its height, we will calculate its radius first.

Now, we know that base area of a right circular cylinder with radius r is given as:
$\text{Base area=}\pi {{r}^{2}}$
Here, we have the base area as $606c{{m}^{2}}$.
Hence, putting it in the formula for base area, we get the radius as:
$\begin{align}
  & \text{Base area=}\pi {{r}^{2}} \\
 & \Rightarrow 606=\pi {{r}^{2}} \\
\end{align}$
We know that $\pi =\dfrac{22}{7}$.
Thus, putting this in the formula we get:
$\begin{align}
  & 606=\pi {{r}^{2}} \\
 & \Rightarrow 606=\dfrac{22}{7}{{r}^{2}} \\
 & \Rightarrow {{r}^{2}}=606\times \dfrac{7}{22} \\
 & \Rightarrow {{r}^{2}}=\dfrac{4242}{22}=\dfrac{2121}{11}c{{m}^{2}} \\
\end{align}$
Now, we know that the volume ‘V’ of a right circular cylinder is given as:
$V=\pi {{r}^{2}}h$
Where, r=radius of the cylinder
h= height of the cylinder
Now, since we already calculated the value of ${{r}^{2}}$, we can directly put it in this formula along with the height which is already given to us in the question (h=2m).
But we can see that the height is in ‘metres’ but the radius square is in ‘square centimetres’. Thus, we need to change them into their SI unit which is metre.
Now, we know that:
$1c{{m}^{2}}={{10}^{-4}}{{m}^{2}}$
Thus, we get the radius square as:
 $\begin{align}
  & {{r}^{2}}=\dfrac{2121}{11}c{{m}^{2}} \\
 & \Rightarrow {{r}^{2}}=\dfrac{2121}{11}\times {{10}^{-4}}{{m}^{2}} \\
\end{align}$
Now, putting these values in the formula for volume, we get:
$\begin{align}
  & V=\pi {{r}^{2}}h \\
 & \Rightarrow V=\dfrac{22}{7}\times \dfrac{2121}{11}\times {{10}^{-4}}\times 2 \\
 & \Rightarrow V=1212\times {{10}^{-4}} \\
 & \therefore V=0.1212{{m}^{3}} \\
\end{align}$

Hence, the volume of the required cylinder is $0.1212{{m}^{3}}$.

Note: We here didn’t calculate the value of r but we left it as ${{r}^{2}}$ instead because in the formula for the volume, the term ${{r}^{2}}$ exists and calculating r and then ${{r}^{2}}$ again would only be long and stupid.
There is also a shorter and more direct way to find the volume which is explained as follows:
The volume is given as:
$V=\pi {{r}^{2}}h$ .....(i)
Now, the base area is given as:
$\text{Base area=}\pi {{r}^{2}}$
The base area is given to us $606c{{m}^{2}}=606\times {{10}^{-4}}{{m}^{2}}$ . Hence, putting its value we get:
$606\times {{10}^{-4}}=\pi {{r}^{2}}$ .....(ii)
Now, dividing equation (i) by equation (ii), we get:
$\begin{align}
  & \dfrac{V}{606\times {{10}^{-4}}}=\dfrac{\pi {{r}^{2}}h}{\pi {{r}^{2}}} \\
 & \Rightarrow \dfrac{V}{606\times {{10}^{-4}}}=h \\
 & \Rightarrow V=2\times 606\times {{10}^{-4}} \\
 & \therefore V=0.1212{{m}^{3}} \\
\end{align}$