Two circular cylinders of equal volume have heights in the ratio of 1:2 ratio of their radii is

Two circular cylinders of equal volumes have their heights in the ratio 1 : 2.

Question:

Two circular cylinders of equal volumes have their heights in the ratio 1 : 2. Find the ratio of their radii.

Solution:

Here, V1 = Volume of cylinder 1

V2 = Volume of cylinder 2

 r1 = Radius of cylinder 1

r2 = Radius of cylinder 2

h1 = Height of cylinder 1

h2 = Height of cylinder 2

Volumes of cylinder 1 and 2 are equal.

Height of cylinder 1 is half the height of cylinder 2.

​∴ V1 = V2

(πr12h1) = (πr22h2) 

(πr12h) = (πr222h) 

$\frac{r_{1}{ }^{2}}{r_{2}{ }^{2}}=\frac{2}{1}$

$\frac{r_{1}}{r_{2}}=\sqrt{\frac{2}{1}}$

Thus, the ratio of their radii is $\sqrt{2}: 1$.

Two circular cylinders of equal volumes have their heights in the ratio 1 : 2. Find the ratio of their radii.

Given data is as follows:

`h_1/h_2 = 1/2`

Volume of cylinder1 = Volume of cylinder2

We have to find the ratio of their radii

Since the volumes of the two cylinders are equal,

`"Volume of cylinder"_1/"  Volume of cylinder"_2=1`

`(pir_1^2h_1)/(pir_2^2 h_2 )= 1`

`(r_1/r_2)^2(h_1/h_2)=1`

But it is given that,

`h_1/h_2=1/2`

Therefore,

`(r_1/r_2)^2 xx 1/2 = 1`

          ` (r_1/r_2)^2= 2`

         ` (r_1/r_2)^2 = 2/1`

                `r_1/r_2 = sqrt(2)/1`

Therefore, the ratio of the radii of the two cylinders is `sqrt(2) : 1`

Concept: Surface Area of Cylinder

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Text Solution

Solution : Volumes of two cylinders are equal.<br> Ratio of heights, `h/H=1/2` <br> h=height of one cylinder and H=Height of other cylinder<br> Ratio of their volumes<br> v=volume of one cylinder and V=volume of other cylinder<br> Radius is same to both of cylinders i.e. r<br> `v/V=`(pir^2h)/(piR^2H)`<br> `1= `(r/R)^2(1/2)`<br> `(r/R)^2=2/1`<br> `therefore` `r/R=sqrt2/1`<br> Hence, ratio of their radii is `2:1`.