Question

The inner and outer radii of the hollow cylinder are 15cm and 20cm respectively. The cylinder is melted and recast into a solid cylinder of the same length. Find the radius of the base of the new cylinder.

Answer 1

Hint: The volumes of the new solid and the original hollow cylinder are the same as the new solid is made out of the original cast of the hollow cylinder.
The volume of a cylinder is
\[Volume=\pi {{r}^{2}}h\]\[\left( Where\text{ }\!\!'\!\!\text{ }r'=radius\text{ }of\text{ }the\text{ }cylinder\text{ }and\text{ }\!\!'\!\!\text{ }h'=height\text{ }of\text{ }the\text{ }cylinder\text{ }and\text{ }value\text{ }of~\pi =\dfrac{22}{7} \right)\]
The volume of a hollow cylinder is
\[Volume=\left[ \pi {{(outer\text{ }radius)}^{2}}height-\pi {{(inner\text{ }radius)}^{2}}height \right]\] (This can be seen as the smaller solid cylinder is being taken out of the bigger solid cylinder)



Complete step-by-step answer:
As mentioned in the question, both the inner and the outer radii are given as 15cm and 20cm respectively.
Now , a hollow cylinder can be imagined as a solid cylinder with radius equal to the outer radius(as mentioned above) with a solid cylinder of radius equal to the inner radius(as mentioned above) is taken out from it.
Now, using the formula for the volume of a hollow cylinder, we get \[\]
\[Volume=\pi \left( {{(outer\text{ }radius)}^{2}}-{{(inner\text{ }radius)}^{2}} \right)height\]
\[Volume=\pi \left( {{(20)}^{2}}-{{(15)}^{2}} \right)height\]
\[Volume=\pi \left( (400)-(225) \right)h\] (Where h is the height of cylinders as both of the cylinders are having same height)
\[Volume=\pi \left( 175 \right)h\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...(a)\]
Now, this volume of the hollow cylinder is equal to the new solid cylinder which is formed by melting the hollow cylinder as given in the question.
So,
Volume of new cylinder = \[Volume\] (from (a))
Volume of new cylinder = \[\pi \left( {{r}^{2}} \right)height\] (Where height of the new cylinder is equal to the height of the hollow cylinder)
Therefore, \[\pi \left( 175 \right)h=\pi ({{r}^{2}})h\] (Using the above two equations)
On cancelling h and \[\pi \] from both the sides, we get
\[{{r}^{2}}=175\]
On taking the square root of both sides, we get
\[r=\sqrt{175}\]
\[r=5\sqrt{7}\]
\[r=13.2cm\]
Hence, the radius of the new solid cylinder formed from the cast of the hollow cylinder is \[r=13.2cm\] .

NOTE: -
There is no actual need of knowing the value of \[\pi \] in the question and the height of the hollow as well as the new solid cylinder that is formed later from the cast of the original hollow cylinder. The volume of the new solid cylinder is equal to the volume of the originally given hollow cylinder because of the fact that the new solid cylinder is made from the same cast which we got on melting the hollow cylinder.