Question

A hollow cylindrical pipe is of length 40 cm. Its internal and external radii are 4 cm and 12 cm respectively. It is melted and casted into a solid cylinder of length 20 cm. Find the radius(in cm) of the new solid.

Answer 1

Hint: First, by using the formula for the volume V of the hollow cylinder as $ V=\pi \left( {{R}^{2}}-{{r}^{2}} \right)h $ . Then, by substituting the value of r as 4cm, R as 12cm and height H as 40cm, we get the volume of the hollow cylinder. Then, we know that if we cast any material to some other material its volume remains the same. Then, by equating both the volume of the figure, we get the desired result.

Complete step-by-step answer:
In this question, we are supposed to find the radius(in cm) of the new solid formed by melting a hollow cylinder given in the question with internal radius r as 4 cm and external radius R as 12 cm and height h as 40 cm.

Now, by using the formula for the volume V of the hollow cylinder as:
 $ V=\pi \left( {{R}^{2}}-{{r}^{2}} \right)h $
Then, by substituting the value of r as 4cm, R as 12cm and height H as 40cm, we get:
 $ \begin{align}
  & V=\pi \left( {{12}^{2}}-{{4}^{4}} \right)\times 40 \\
 & \Rightarrow V=\pi \left( 144-16 \right)\times 40 \\
 & \Rightarrow V=\pi \left( 128 \right)\times 40 \\
 & \Rightarrow V=5120\pi c{{m}^{3}} \\
\end{align} $
So, we get the volume of the hollow cylinder as $ 5120\pi c{{m}^{3}} $ .
Now, we need to melt and cast the hollow cylinder pipe into a solid cylinder which is 20 cm in height and we need to calculate its radius.
So, let the radius of the solid cylinder be r and solid cylinder is as:

Then, we know that if we cast any material to some other material its volume remains the same.
So, from the above fact the volume of the solid cylinder is the same as the volume of the hollow cylinder.
Now, by using the above condition also using the formula of the volume of the solid cylinder as $ \pi {{r}^{2}}h $ , we get:
 $ \begin{align}
  & \pi {{r}^{2}}h=5120\pi \\
 & \Rightarrow {{r}^{2}}\times 20=5120 \\
 & \Rightarrow {{r}^{2}}=\dfrac{5120}{20} \\
 & \Rightarrow {{r}^{2}}=256 \\
 & \Rightarrow r=16 \\
\end{align} $
So, the radius of the new solid cylinder formed is 16cm.
Hence, the radius is found to be 16cm.

Note: Here, in this type of question the most common mistake we occur is in the understanding of the shapes as hollow cylinder and solid cylinder are two different shapes as hollow cylinder is having two radius outer radius R and inner radius r but solid cylinder has single radius r. Also, the formulas for the volume of the two are different as:
Volume of the hollow cylinder is $ V=\pi \left( {{R}^{2}}-{{r}^{2}} \right)h $ .
But, the volume of the solid cylinder with radius and height h is $ V=\pi {{r}^{2}}h $ .