Question

The radii of two cones are in the ratio 2:3 and their volumes are in the ratio 1:3. What is the ratio of their heights?​

Answer 1

Given :

• Ratio of Radii of 2 cones = 2:3
• Ratio of Volume of 2 cones = 1:3

To Find :

• Ratio of their height.

Solution :

✰ As we know that, Volume of Cone is given by ⅓ πr²h . So Simply we will put the given values in the formula to find the ratio of their height.

⠀

According to the Question :

${:} \longrightarrow \sf{\sf{Volume \: of \: 2 \: Cones \: = \: \dfrac{ \dfrac{1}{3} \: \pi \: {r}^{2} \: h_{1} }{\dfrac{1}{3} \: \pi \: {r}^{2} \: h_{2}} }}$

${:} \longrightarrow \sf{\sf{\dfrac{1}{3} \: = \: \dfrac{ \dfrac{1}{3} \: \pi \: {(2)}^{2} \: h_{1} }{\dfrac{1}{3} \: \pi \: {(3)}^{2} \: h_{2}} }}$

${:} \longrightarrow \sf{\sf{ \dfrac{1}{3} \: = \: \dfrac{ \cancel{ \dfrac{1}{3} \: \pi }\: {(2)}^{2} \: h_{1} }{ \cancel{\dfrac{1}{3} \: \pi} \: {(3)}^{2} \: h_{2}} }}$

${:} \longrightarrow \sf{\sf{ \dfrac{1}{3} \: = \: \dfrac{ {(2)}^{2} \: h_{1} }{ {(3)}^{2} \: h_{2}} }}$

${:} \longrightarrow \sf{\sf{ \dfrac{1}{3} \: = \: \dfrac{ 4 \: \times \: h_{1} }{ 9 \: \times \: h_{2}} }}$

${:} \longrightarrow \sf{\sf{ \dfrac{1}{3} \: \times \: \dfrac{9}{4} \: = \: \dfrac{ h_{1} }{h_{2}} }}$

${:} \longrightarrow \sf \red{\boxed{\bf\green{ \dfrac{3}{4} \: = \: \dfrac{ h_{1} }{h_{2}} }}}$

Thus Ratio of their height is 3:4

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