Question

The volume of a cone is the same as that of the volume of a cylinder whose height is \[9cm\] and diameter \[40cm\]. Find the radius of the base of the cone if its height is \[108cm\].

Answer 1

Hint: In the given question, we have been given that the volume of two shapes is the same. Then, we have been given the required parameters of one of the figures. We have to find one parameter of the other shape whose one parameter is given. To solve that, we equate the formula of volume of the two quantities, put in the values and simplify their values.

Formula used:
We are going to use the formula of volume of cone and cylinder:
\[{V_{cone}} = \dfrac{1}{3}\pi {r^2}h\] and \[{V_{cyl}} = \pi {r^2}h\]

Complete step by step solution:

Volume of cone, \[{V_{cone}} = \dfrac{1}{3}\pi {r^2} \times 108 = \pi {r^2} \times 36 = \pi {r^2} \times {\left( 6 \right)^2}\]
Volume of cylinder, \[{V_{cyl}} = \pi \times {\left( {20} \right)^2} \times 9 = \pi \times {\left( {60} \right)^2}\]
Since the volumes are equal, let us equate them,
\[\pi {r^2} \times {\left( 6 \right)^2} = \pi \times {\left( {60} \right)^2}\]
Cancelling \[\pi \] from both sides and taking \[{\left( 6 \right)^2}\] to the other side,
\[{r^2} = \dfrac{{{{\left( {60} \right)}^2}}}{{{{\left( 6 \right)}^2}}} = {\left( {10} \right)^2}\]
Hence, \[r = 10cm\]

Note: In the given question, we were given that the volume of two shapes was the same. We had to find the value of one parameter of one shape where we gave the value of the other parameter and all the parameters of the other shape. To solve that, we just equated the formula of volume of the two shapes and calculated the answer.