Question

If the heights of two cylinders are in the ratio of 5 : 3 and their radii are in the ratio of 3 : 5 then what is the ratio of their volumes?

Answer 1

Answer:

The ratio of the volumes of the given Cylinders is 3:5

Step-by-step explanation:

Given:-

the heights of two cylinders are in the ratio of 5 : 3 and their radii are in the ratio of 3 : 5

To find:-

what is the ratio of their volumes?

Solution:-

The ratio of the heights of the two cylinders = 5:3

Let they be 5X units and 3X units

Let Height of the first Cylinder (h1) = 5X units

Height of the second cylinder (h2)= 3X units

The ratio of the radii of the two cylinders = 3:5

Let they be 3Y units and 5Y units

Let radius of the first Cylinder ( r1 )= 3Y units

Radius of the second cylinder (r2 )= 5Y units.

Volume of a cylinder = πr^2h cubic units

Volume of the first Cylinder =

V1 = π(3Y)^2 (5X) cubic units

=>V1 = π(9Y^2)(5X)

=>V1 = 45πXY^2 ------------(1)

Volume of the second cylinder

V2 = π(5Y)^2(3X) cubic units

=>V2= π(25Y^2)(3X)

=>V2 = 75πXY^2 cubic units -----(2)

Ratio of their volumes =V1:V2

From (1)&(2)

=>45πXY^2:75πXY^2

=>(45πXY^2)/(75πXY^2)

On cancelling πXY^2

=>45/75

=>(3×15)/(5×15)

On cancelling 15

=>3/5

=>3:5

the ratio of their volumes = 3:5

Answer:-

The ratio of the volumes of the given Cylinders is 3:5

Used formulae:-

• If r is the radius and h is the height of the cylinder then the Volume of a cylinder is πr^2h cubic units

Answer 2

$\Large{\underbrace{\underline{\sf Understanding\: the\; Question}}}}$

Here, this is a question about finding ratio of volume of two cylinders whose ratios of radius and height is given in the question. To find the ratio of volume of cylinder, we will first assume the given parameters and put them in formula.

So let's do it!!

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★ Given:

• Ratio of heights=5:3

• Ratio of radii =3:5

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★ To find:

• Ratio of volume of both cylinders

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★ Formula used:

• Volume of cylinder=πr²h

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★ Solution:

~Assumption

Let us assume the ratio of heights of cylinder be $\sf 5x:3x$.

Now let us assume the ratio of radii of cylinder be $\sf 3z:5z$

~We get:

• The height $\sf (h_1)$ and radius $\sf (r_1)$ of first cylinder will be $\sf 5x \:and \:3z$ respectively.

• The height $\sf (h_2)$ and radius  $\sf (r_2)$ of second cylinder will be $\sf 3x\;and\;5z$ respectively.

~Now finding ratios of volume of cylinders:

$\longmapsto\sf \dfrac{Volume\;of\; Cylinder_{\bf 1}}{Volume\;of\; Cylinder_{\bf 2}}$

~Now putting formula for volume of cylinder:

$\longmapsto\sf \dfrac{\pi( r_{1})^2( h_1)}{\pi (r_2)^2(h_2)}$

~Now putting values of radius and height:

$\longmapsto\sf \dfrac{\pi( 3z)^2( 5x)}{\pi (5z)^2(3x)}$

~Now solving it:

$\longmapsto\sf \dfrac{\pi\times 9z^2\times 5x}{\pi\times 25z^2\times 3x}$

~Now cancelling π, x² and z from both numerator and denominator:

$\longmapsto\sf \dfrac{\not\pi\times 9\not z^2\times 5 \not x}{\not\pi\times 25\not z^2\times 3\not x}$

$\longmapsto\sf \dfrac{9\times 5 }{25 \times 3}$

$\longmapsto\sf \dfrac{45}{75}$

~Now divide 15 from both numerator and denominator:

$\longmapsto\sf \dfrac{45\div 15}{75\div 15}$

$\longmapsto\sf \dfrac{3}{5}$

So the ratios of volume of both cylinders is 3:5.

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