   Question Answers

# Two cones have their heights in the ratio 1:3 and their radii of their bases in the ratio 3:1. Find the ratio of their volumes.  Hint- In order to solve to this problem, we must choose formula of volume of cone which is equal to $\dfrac{1}{3}(\pi \times {{\text{r}}^2} \times {\text{h)}}$ along with the proper understanding of the figure that is being provided in the question. Here in this question it is given the ratios of heights and bases radii of two different cones and we have to find the volume ratio.
We know that, the volume of cone is $\dfrac{1}{3}(\pi \times {{\text{r}}^2} \times {\text{h)}}$ ……………………………... (1)
Let the radius, height and Volume of 1st cone be ${{\text{r}}_{1{\text{ }}}}{\text{, }}{{\text{h}}_1}{\text{ and }}{{\text{v}}_1}$ respectively
And
The radius, height and Volume of 2nd cone be ${{\text{r}}_{{\text{2 }}}}{\text{, }}{{\text{h}}_2}{\text{ and }}{{\text{v}}_2}$ respectively
So Using expression (1)
Volume of 1st cone ${{\text{v}}_1}{\text{ will be }}\dfrac{1}{3}(\pi \times {{\text{r}}_1}^2 \times {{\text{h}}_1}{\text{)}}$ …………………………………...(2)
And volume of 2nd cone ${{\text{v}}_2}{\text{ will be }}\dfrac{1}{3}(\pi \times {{\text{r}}_2}^2 \times {{\text{h}}_2}{\text{)}}$ …………………………………….(3)
Now on dividing equation (2) by (3),
${\text{Hence }}\dfrac{{{{\text{v}}_1}}}{{{{\text{v}}_2}}} = \dfrac{{\dfrac{1}{3}(\pi \times {{\text{r}}_1}^2 \times {{\text{h}}_1}{\text{)}}}}{{\dfrac{1}{3}(\pi \times {{\text{r}}_2}^2 \times {{\text{h}}_2}{\text{)}}}}$

On cancelling $\dfrac{1}{3}$ and $\pi$ from numerator and denominator,
$\therefore \dfrac{{{{\text{v}}_1}}}{{{{\text{v}}_2}}} = \dfrac{{({{\text{r}}_1}^2 \times {{\text{h}}_1}{\text{)}}}}{{({{\text{r}}_2}^2 \times {{\text{h}}_2}{\text{)}}}}$
$\therefore \dfrac{{{{\text{v}}_1}}}{{{{\text{v}}_2}}} = {\left( {\dfrac{{{{\text{r}}_1}}}{{{{\text{r}}_2}}}} \right)^2} \times \left( {\dfrac{{{{\text{h}}_1}}}{{{{\text{h}}_2}}}} \right)$ ………………………………….(4)
In the given question it is given that
$\therefore \left( {\dfrac{{{{\text{r}}_1}}}{{{{\text{r}}_2}}}} \right) =$ $\dfrac{3}{1}$ And $\left( {\dfrac{{{{\text{h}}_1}}}{{{{\text{h}}_2}}}} \right) =$$\dfrac{1}{3}$
Now putting the value of $\left( {\dfrac{{{{\text{r}}_1}}}{{{{\text{r}}_2}}}} \right)$ and $\left( {\dfrac{{{{\text{h}}_1}}}{{{{\text{h}}_2}}}} \right)$ in equation (4)
$\therefore \dfrac{{{{\text{v}}_1}}}{{{{\text{v}}_2}}}$ $= {\left( {\dfrac{3}{1}} \right)^2} \times \left( {\dfrac{1}{3}} \right)$
On solving
$\therefore \dfrac{{{{\text{v}}_1}}}{{{{\text{v}}_2}}} =$ $3$
Hence, the volume ratio of given cones will be $\dfrac{3}{1}$

Note: -Whenever we face such types of problems the key concept we have to remember is that always remember the formula of volume of cone which is stated above, then using this formula calculate the ratio of volume of two cones and simplify, we will get the required answer.
View Notes
Why do we Have Two Eyes?  Physical Properties of Alkanes and Their Variations  Excretory Products and Their Elimination  CBSE Class 9 Maths Chapter 13 - Surface Areas and Volumes Formulas  CBSE Class 6 Maths Chapter 12 - Ratio and Proportion Formulas  Ratio and Proportion Formula  Table of 13 - Multiplication Table of 13  CBSE Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables Formula  CBSE Class 9 Maths Chapter 6 - Lines and Angles Formulas  CBSE Class 8 Maths Chapter 13 - Direct and Inverse Proportions Formulas  Important Questions for CBSE Class 9 Maths Chapter 13 - Surface Areas and Volumes  NCERT Books for Class 9 Maths Chapter 13  Important Questions for CBSE Class 9 Science Chapter 13 - Why Do We Fall ill  Important Questions for CBSE Class 6 Science Chapter 9 - The Living Organisms and Their Surroundings  NCERT Books for Class 9 Science Chapter 13  CBSE Class 6 Science The Living Organisms and Their Surroundings Worksheets  Important Questions for CBSE Class 9 Maths Chapter 4 - Linear Equations in Two Variables  Important Questions for CBSE Class 6 Maths Chapter 12 - Ratio and Proportion  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 1 - A Tale of Two Birds  Important Questions for CBSE Class 9 Maths Chapter 9 - Areas of Parallelograms and Triangles  CBSE Class 12 Maths Question Paper 2020  CBSE Class 10 Hindi A Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2009  Hindi A Class 10 CBSE Question Paper 2015  Hindi A Class 10 CBSE Question Paper 2007  Hindi A Class 10 CBSE Question Paper 2013  CBSE Class 10 Maths Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2016  Hindi A Class 10 CBSE Question Paper 2012  Hindi A Class 10 CBSE Question Paper 2010  RD Sharma Solutions for Class 9 Maths Chapter 20 - Surface Area and Volume of A Right Circular Cone  RD Sharma Solutions for Class 9 Maths Chapter 13 - Linear Equations in Two Variables  NCERT Solutions for Class 9 Maths Chapter 13 Surface Areas and Volumes (Ex 13.9) Exercise 13.9  RD Sharma Solutions for Class 9 Maths Chapter 19 - Surface Area and Volume of A Right Circular Cylinder  RD Sharma Class 9 Solutions Chapter 13 - Linear Equation In Two Variables (Ex 13.3) Exercise 13.3  RD Sharma Class 9 Solutions Chapter 13 - Linear Equation In Two Variables (Ex 13.2) Exercise 13.2  NCERT Solutions for Class 9 Maths Chapter 13 Surface Areas And Volumes (Ex 13.7) Exercise 13.7  NCERT Solutions for Class 9 Maths Chapter 13 Surface Area and Volumes In Hindi  RD Sharma Class 9 Solutions Chapter 13 - Linear Equation In Two Variables (Ex 13.1) Exercise 13.1  RD Sharma Class 9 Solutions Chapter 13 - Linear Equation In Two Variables (Ex 13.4) Exercise 13.4  