Question

The base radii of two right circular cones of the same height are in the ratio \[3:5\]. Find the ratio of their volumes.

Answer 1

Hint:
We can see in the given question the cones have the same height and different radii. That has radii as r for cone 1 and R for cone 2. Now we will go to use formula for volume of cone and which is to both the cones and then put the cone’s values by dividing them to each other that will give you the answer.

Complete step by step solution:
Let there be cone 1 and cone 2 respectively.

Let the r and R be the radii of the two right circular cones respectively.

Ratio of base radii \[ = 3:5\]
Volume of cone \[ = \left( {\dfrac{1}{3} \times \pi \times {r^2} \times h} \right)\]
Volume of cone 1 \[{V_1} = \left( {\dfrac{1}{3} \times \pi \times {r^2} \times h} \right)\]
Volume of cone 2 \[{V_2} = \left( {\dfrac{1}{3} \times \pi \times {R^2} \times h} \right)\]
Now, we will divide the both volumes and find the ratio

Volume of cone 1 divided by Volume of cone 2 \[ = \dfrac{{{V_1}}}{{{V_2}}}\]
We will put the ratio given in the question and then we will calculate the numbers
 \[ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{\left( {\dfrac{1}{3} \times \pi \times {{(3x)}^2} \times h} \right)}}{{\left( {\dfrac{1}{3} \times \pi \times {{(5x)}^2} \times h} \right)}}\]
After cancelling the similarity from numerator and denominator we get

 \[ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{{3^2}}}{{{5^2}}}\]
Squaring the values, we get

 \[ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{9}{{25}}\]
Hence, we have,

 \[ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = 9:25\]


So, the ratio of their volumes is \[9:25\].

Note:
Let \[{r_1}\] ​ and \[{r_2}\] be the radii of two cones and \[{V_1}\] and \[{V_2}\] be their volumes. Let h be the height of the two cones. Then,
 \[{V_1} = \left( {\dfrac{1}{3} \times \pi \times {r_1}^2 \times h} \right)\]

​ \[{V_2} = \left( {\dfrac{1}{3} \times \pi \times {r_2}^2 \times h} \right)\]
Divide the given volumes and put the values of the radii
 \[ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{\left( {\dfrac{1}{3} \times \pi \times {3^2} \times h} \right)}}{{\left( {\dfrac{1}{3} \times \pi \times {5^2} \times h} \right)}}\]
By cancelling similar terms, we can get the answer directly.
Hence, the ratio of the volumes of two cones is \[9:25\] .