Question

If two cones have their volumes in the ratio 3:2 and their heights are in the ratio 1:3 then find the ratio of thier radius

Answer 1

Answer:

the ratio of their radius is 3 by under root 2

Answer 2

Answer :

= 3 : √2

Step-by-step explanation :

$\underline{\underline{\bf Cone:}}$

             $\setlength{\unitlength}{1.2mm}\begin{picture}(5,5)\thicklines\put(0,0){\qbezier(1,0)(12,3)(24,0)\qbezier(1,0)(-2,-1)(1,-2)\qbezier(24,0)(27,-1)(24,-2)\qbezier(1,-2)(12,-5)(24,-2)}\put(-0.5,-1){\line(1,2){13}}\put(25.5,-1){\line(-1,2){13}}\multiput(12.5,-1)(2,0){7}{\line(1,0){1}}\multiput(12.5,-1)(0,4){7}{\line(0,1){2}}\put(18,1.6){\sf{r}}\put(9.5,10){\sf{h}}\end{picture}$

• Slant height : It is the distance from the apex to any point on the circumference of the base.          

              $\bf l=\sqrt{h^2+r^2}$

• Total Surface Area : The sum of lateral surface area and surface area of the base

             $\bf TSA= \pi r^2+\pi rl \\\\ TSA=\pi r(r+l)$

• Volume :

             $\bf V=\frac{1}{3} \pi r^2h$

_______________________________

Given,

• Volumes of two cones are in the ratio 3 : 2
• Heights are in the ratio 1 : 3

Let V₁ and V₂ be the volumes of two cones

     h₁ and h₂ be the heights of two cones

     r₁ and r₂ be the radii of two cones

Ratio of the Volumes = 3 : 2

          $\bf \frac{V_1}{V_2}=\frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} \\\\\\ \frac{V_1}{V_2}=\frac{r_1^2h_1}{r_2^2h_2} \\\\\\ \frac{3}{2} =\frac{r_1^2}{r_2^2} \times \frac{1}{3} \\\\\\ \frac{r_1^2}{r_2^2} =\frac{3}{2} \times 3 \\\\\\ (\frac{r_1}{r_2})^2 =\frac{9}{2} \\\\\\ \frac{r_1}{r_2}=\sqrt{\frac{9}{2}} \\\\\\ \frac{r_1}{r_2}=\frac{3}{\sqrt{2}}$

Ratio of their radius = 3 : √2