Question

20. A
13. The diameters of two cylinders are in the ratio 2:3. Then the ratio of their heights if their volume
int
the same is
[
a) 3:2
b) 6:5
a)
c) 9:4
d) 9:5​

Answer 1

The ratio of the diameters of the cylinders = 3 : 2

⇒ The ratio of the radius of the cylinders will be = 3 : 2

Therefore, let the radii of the two cylinder be 3x and 2x respectively.

Let the height of the cylinder be h1 and h2.

Now, Volume of first cylinder = Volume of the second cylinder

i.e. π(3x)²h1 = π(2x)²h2

h1/h2 = (π*4x²)/(π*9x²)

h1/h2 = 4/9

h1 : h2 = 4 :9

Answer 2

$\huge{\underline{\underline{\sf{\pink{Given:}}}}}$

The diameter of the two-cylinder are in the ratio 2:3

$\huge{\underline{\red{\rm{To\ find:}}}}$

The ratio of their heights if their volume is the same

So,

The volume of the two-cylinders is the same.

Let diameter of one cylinder be 2x units

And the diameter of the second cylinder be 3x units

Now,

We know that

$\huge{\boxed{\purple{Diameter = 2r}}}$

OR

$\boxed{\purple{\sf{Radius= \frac{diameter}{2} }}}$

Now,

The radius of one cylinder be 2x/2 = 1x units units

And the diameter of the second cylinder be 3x/2 = 2x/3 units

So,

We can also say the ratio of the radius is

$\bf =\frac{1 \not{x}}{\frac{3}{2} \not{x}} = \frac{2}{3}$

= 2:3

Now,

Volumes are the same,

The volume of the one cylinder = volume of the second cylinder

...

Here,

We will find the ratio of the height of the two-cylinders,

$\boxed{\sf{\implies \frac{Volume\ of\ first\ cylinder}{Volume\ of\ second\ cylinder} }}$

$\bf = \frac{ \not{\pi} r^{2}h}{ \not{\pi} RH} = 1$

[π will be canceled out.]

$\bf \implies \frac{r^{2} \times h}{R \times H} = 1$

$\bf \implies ( \frac{r}{R} )^{2} \times \frac{h}{H} = 1$

Now,

Putting the value of the ratio of the radius

$\implies \bf ( \frac{2}{3} )^{2} \times \frac{h}{H}= 1$

$\implies \bf{\frac{h}{H} = \frac{9}{4}}$

$\implies \boxed{ \frac{h}{H} = \frac{9}{4}}$

Hence

The ratio of the heights of the two-cylinders is 9:4

Hence

Option C) 9:4 is the correct answer.