Question

*Volume of 2 cones are in the ratio 1:4 and their diameter are in ratio 4:5, the ratio of their heights is 64:25* ...True or false...with solution​

Answer 1

Answer:

heights=h1,h2

radius, ri,r2

volume VI, V2

Vol. of Cone=1/3πr 2h

h=3V/πr 2h

hi: h2 = 3V1/πr 1^2: 3 V2/πr 2^2

V1:V2 = 1:4

ri: r2=4:5

hi: h2 = 3x1/πx16:.3x4/πx25

= 3/16:12/25

= 25:64. Ans

The given answer 64:25 is false

Answer 2

Answer:

$\large\bold\red{FALSE}$

Explanation:

Let,

For the first cone ,

• Volume = v
• Diameter = d
• Radius = r = d/2
• Height = h

And,

For the second cone,

• Volume = V
• Diameter = D
• Radius = R = D/2
• Height = H

Now,

It is given that,

• $\bold{\frac{v}{V}=\frac{1}{4}}$

And,

• $\bold{\frac{d}{D}=\frac{4}{5}}$

But,

We know that,

• Volume of a cone is = $\bold{\frac{1}{3}\pi{r}^{2}h}$

Therefore,

We get,

$= > \frac{ \frac{1}{3} \pi{r}^{2} h}{ \frac{1}{3} \pi{R}^{2}H} = \frac{1}{4} \\ \\ = > \frac{ {r}^{2} h}{ {R}^{2} H} = \frac{1}{4} \\ \\ = > { (\frac{r}{R} )}^{2} \frac{h}{H} = \frac{1}{4}$

But,

We know that,

• Radius = Half of diameter

Therefore,

Substituting the terms,

We get,

$= > {( \frac{ \frac{d}{2} }{ \frac{D}{2} } )}^{2} \frac{h}{H} = \frac{1}{4} \\ \\ = > { (\frac{d}{D} )}^{2} \frac{h}{H} = \frac{1}{4} \:$

But,

It is Given that,

• $\bold{\frac{d}{D}=\frac{4}{5}}$

Therefore,

Putting the values,

We get,

$= > {( \frac{4}{5}) }^{2} \frac{h}{H} = \frac{1}{4} \\ \\ = > \frac{16h}{25H} = \frac{1}{4} \\ \\ = > \frac{h}{H} = \frac{1}{4} \times \frac{25}{16} \\ \\ = > \frac{h}{H} = \frac{25}{64}$

Therefore,

The ratio of their heights is $\large\bold{\frac{25}{64}}$

Hence,

It's $\large\bold{FALSE }$