The radii of two cones are in the ratio 2:1, | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the radii of the two cones be $r_1$ and $r_2$,and their heights be $h_1$ and $h_2$ respectively.Given that the ratio of the radii is $\frac{r_

Vedclass · Accepted Answer

$1:4$

Vedclass · Answer

(B) Let the radii of the two cones be $r_1$ and $r_2$,and their heights be $h_1$ and $h_2$ respectively.Given that the ratio of the radii is $\frac{r_1}{r_2} = \frac{2}{1}$.Since the volumes of the two cones are equal,we have $V_1 = V_2$.The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$.Therefore,$\frac{1}{3} \pi r_1^2 h_1 = \frac{1}{3} \pi r_2^2 h_2$.Canceling $\frac{1}{3} \pi$ from both sides,we get $r_1^2 h_1 = r_2^2 h_2$.Rearranging to find the ratio of heights,we get $\frac{h_1}{h_2} = \frac{r_2^2}{r_1^2} = \left(\frac{r_2}{r_1}\right)^2$.Substituting the given ratio $\frac{r_1}{r_2} = \frac{2}{1}$,we have $\frac{r_2}{r_1} = \frac{1}{2}$.Thus,$\frac{h_1}{h_2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.So,the ratio of their heights is $1:4$.

The radii of two cones are in the ratio $2:1,$ and their volumes are equal. Find the ratio of their heights.

Similar Questions

What is the length (in metre) of the longest rod that can be placed in a room which is $2$ $metre$ long,$2$ $metre$ broad,and $6$ $metre$ high?

The radius and height of a right circular cone are in the ratio $3: 4$. If its volume is $96\pi\, cm^3$,what is its slant height (in $cm$)?

Volume of the cylinder is $1650\, m^3$,whereas the surface area of its base is $78 \frac{4}{7}\, m^2$. Find out the height of the cylinder (in $m$).

The radius of a wheel is $3.5 \, cm$. What is the distance (in $cm$) travelled by the wheel in $20$ revolutions?

$A$ solid cone of height $24 \, cm$ and base radius $8 \, cm$ is melted to form a solid cylinder of radius $6 \, cm$ and height $6 \, cm$. In the whole process,what percent of material is wasted (in $\%$)?

Vedclass Test Series

Exam Paper Generator

Online Exam Module