Question

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

Answer 1

Answer:

The ratio of the Heights of two cones is 1 : 4 .

Step-by-step explanation:

Let the volumes of two cones V1 & V2 and radii be r1 & r2 and height be h1 & h2.

Given :

Two cones have their radii in the ratio 2 : 1. Volumes of two cones are equal i.e V1 = V2

Let  Radii of two cones be  r1 = 2x ,  r2 = 1x

Volume of first cone (V1) / Volume of second cone(V2) = ⅓ πr1²h1/ ⅓ πr2²h2

V1/V2 = ⅓ πr1²h1/ ⅓ πr2²h2

V1/V2 = r1²h1/r2²h2

V1/V2 = r1²/ r2² × h1/h2

V1/V1 = (2x/1x)² × h1/h2

[V1 = V2]

1  = 4/1 × h1/h2

¼ = h1/h2

h1 : h2 = 1 : 4  

Hence, the ratio of the Heights of two cones is 1 : 4 .

HOPE THIS ANSWER WILL HELP YOU…..

Answer 2

$\huge \boxed{ \underline{ \underline{ \bf{Answer}}}}$

GIVEN :

Ratio of radii of two cones are(R : r) = 2 : 1.

Volume of the two cones are equal.

TO FIND :

Ratio of their height (H : h)

PROCESS :

Let the two cones be ABC and PQR.

Now,

V. of Cone ABC = V. of Cone PQR

$\implies \: \sf{\frac{1}{3} \pi {R }^{2}H \: = \: \frac{1}{3} \pi {r}^{2} h}$

$\implies \: \sf{ {R }^{2}H \: = {r}^{2} h}$

$\implies \: \sf { \frac{ {R}^{2} }{ {r}^{2} } = \: \frac{h}{H}}$

$\implies \: \sf{{ (\frac{R}{r})}^{2} = \: \frac{h}{H}}$

$\implies \: \sf{{ (\frac{2}{1})}^{2} = \: \frac{h}{H}}$

$\implies \: \sf{4 = \: \frac{h}{H}}$

$\implies \: \sf { \frac{H}{h} = \frac{1}{4} }$

$\implies \: \boxed{ \sf{H : h \: = \: 1 : 4}}$

Hence, the ratio of their height is 1 : 4