Question

What length of a solid cylinder \[2cm\] in diameter must be taken to recast into a hollow cylinder of length \[16cm\], external diameter \[20cm\], and thickness \[2.5mm\]?

Answer 1

Hint: We have to find the length of the solid cylinder when converted into a solid cylinder. So we have to keep the volumes of both the cylinders equal to each other and then the height of the solid cylinder will be obtained. Also, the thickness of the solid cylinder will be obtained by subtracting the inner radius from the outer radius.

Complete step-by-step solution:
A cylinder is a kind of polygon that has three faces and vertices are not present. A cylinder has a curved surface. A cylinder can be both a hollow cylinder or a solid cylinder.
This is the question related to surface area and volume. So in the above question, we have to find the length of the solid cylinder that is converted into a hollow cylinder.
So in the question, it is given that.
Diameter of the solid cylinder, \[d=2cm\].
So the radius of the solid cylinder will be, \[{{r}_{1}}=1cm\].
Also, it is given that.
Length of a hollow cylinder, \[{{h}_{2}}=16cm\].
External diameter of the cylinder, \[D=20cm\].
So the external radius will be, \[{{R}_{2}}=10cm\].
The thickness of the cylinder will be given as, \[t=2.5mm\].
On converting it into centimeters, we get \[t=0.25cm\].
Now we have to calculate the inner radius, which will be calculated as follows.
\[t={{R}_{2}}-{{r}_{2}}\]
\[\begin{align}
  & \Rightarrow 0.25=10-{{r}_{2}} \\
 & \Rightarrow {{r}_{2}}=10-0.25 \\
 & \Rightarrow {{r}_{2}}=9.75cm \\
\end{align}\]
So the inner radius of the hollow cylinder comes out to be \[{{r}_{2}}=9.75cm\].
As the solid cylinder is converted into a hollow cylinder so we will keep their volumes equal to each other and then we will solve our question further.
\[volume\text{ }of\text{ }solid\text{ }cylinder=volume\text{ }of\text{ }hollow\text{ }cylinder\]
We know that the volume of a cylinder is given by the formula as shown below.
\[volume\text{ }of\text{ }cylinder=\pi {{r}^{2}}h\]
Where ‘r’ is known as the radius of the cylinder and ‘h’ is known as the height of the cylinder.
On keeping both the volumes equal, we get
\[\pi {{r}_{1}}^{2}{{h}_{1}}=\pi {{h}_{2}}(({{R}_{2}}^{2}-{{r}_{2}}^{2}))\]
On putting the given values in the above equation, we get
\[\pi {{(1)}^{2}}{{h}_{1}}=\pi {{h}_{2}}({{(10)}^{2}}-{{(9.75)}^{2}})\]
\[\Rightarrow \pi {{(1)}^{2}}{{h}_{1}}=\pi (16)({{(10)}^{2}}-{{(9.75)}^{2}})\]
The value of \[\pi \] will cancel out each other on both sides and the following results will be obtained
\[\begin{align}
  & {{h}_{1}}=16\times (10-9.75)(10+9.75) \\
 & \Rightarrow {{h}_{1}}=16\times 19.75\times 0.25 \\
\end{align}\]
\[\Rightarrow {{h}_{1}}=79cm\]
So the height of the solid cylinder comes out to be \[79cm\].

Note: There are two kinds of shapes in mathematics. The first one is a two-dimensional shape and the second one is a three-dimensional shape. We can find the area of the three-dimensional shapes by calculating the area and the area of two-dimensional shapes can find out by simply calculating the area of the figure.