Question

the radius of 2 cylinder are in ratio 5:7 and heights are in ratio 3:5 the ratio of there csa is​

Answer 1

Given :

• Ratio of radius of two cylinders = 5 : 7
• Ratio of the heights of cylinders = 3 : 5

To Find :

• The Ratio of CSA of cylinders

Solution :

Let the ratio constant be "x" . Then the radius of the two cylinders becomes ,

• 5x and 7x

Let the ratio constant (in height's ratio) be "y". Then the heights of two cylinders becomes ,

• 3y and 5y

Curved surface area of cylinder is given by ,

$\\ \star \: {\boxed{\purple{\sf{CSA_{(cylinder)} =2\pi rh }}}}$

Where ,

• r is radius of cylinder
• h is height of cylinder

By the given data ,

• Radius and height of first cylinder is 5x and 3y
• Radius and height of second cylinder is 7x and 5y.

First let us calculate the CSA of first cylinder,

$\\ : \implies \sf \: CSA_{(cylinder_1)} = 2 \pi(5x)(3y) \: ........(1)$

The CSA of second cylinder is ,

$\\ : \implies \sf \:CSA_{(cylinder_2)} = 2\pi(7x)(5y) \: ........(2)$

Now , Taking ratio ;

$\\ : \implies \sf \:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} = 2\pi(5x)(3y) : \: 2\pi(7x)(5y)$

$\\ : \implies \sf \:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} = (5x)(3y) : \: (7x)(5y)$

$\\ : \implies \sf \:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} = 15xy : \:35xy$

$\\ : \implies \sf \:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} = 15 : 35$

$\\ : \implies{\underline{\boxed{\pink{ \sf{\:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} =3 : 7}}}}} \: \bigstar$

Hence ,

• The Ratio of CSA's of two given cylinders is 3 : 7.

Answer 2

Answer:

Given :

Ratio of radius of two cylinders = 5 : 7

Ratio of the heights of cylinders = 3 : 5

To Find :

The Ratio of CSA of cylinders

Solution :

Let the ratio constant be "x" . Then the radius of the two cylinders becomes ,

5x and 7x

Let the ratio constant (in height's ratio) be "y". Then the heights of two cylinders becomes ,

3y and 5y

Curved surface area of cylinder is given by ,

$\begin{gathered} \\ \star \: {\boxed{\purple{\sf{CSA_{(cylinder)} =2\pi rh }}}} \\ \\ \end{gathered} </p><p>⋆ </p><p>CSA </p><p>(cylinder)</p><p> </p><p> =2πrh</p><p> </p><p> </p><p> </p><p> </p><p></p><p>Where ,</p><p></p><p>r is radius of cylinder</p><p>h is height of cylinder</p><p>By the given data ,</p><p></p><p>Radius and height of first cylinder is 5x and 3y</p><p>Radius and height of second cylinder is 7x and 5y.</p><p>First let us calculate the CSA of first cylinder,</p><p></p><p>\begin{gathered} \\ : \implies \sf \: CSA_{(cylinder_1)} = 2 \pi(5x)(3y) \: ........(1) \\ \\ \end{gathered} </p><p>:⟹CSA </p><p>(cylinder </p><p>1</p><p> </p><p> )</p><p> </p><p> =2π(5x)(3y)........(1)</p><p> </p><p> </p><p></p><p>The CSA of second cylinder is ,</p><p></p><p>\begin{gathered} \\ : \implies \sf \:CSA_{(cylinder_2)} = 2\pi(7x)(5y) \: ........(2)\\ \\ \end{gathered} </p><p>:⟹CSA </p><p>(cylinder </p><p>2</p><p> </p><p> )</p><p> </p><p> =2π(7x)(5y)........(2)</p><p> </p><p> </p><p></p><p>Now , Taking ratio ;</p><p></p><p>\begin{gathered} \\ : \implies \sf \:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} = 2\pi(5x)(3y) : \: 2\pi(7x)(5y)\end{gathered} </p><p>:⟹CSA </p><p>(cylinder </p><p>1</p><p> </p><p> )</p><p> </p><p> :CSA </p><p>(cylinder </p><p>2</p><p> </p><p> )</p><p> </p><p> =2π(5x)(3y):2π(7x)(5y)</p><p> </p><p> </p><p></p><p>\begin{gathered} \\ : \implies \sf \:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} = (5x)(3y) : \: (7x)(5y) \\ \\ \end{gathered} </p><p>:⟹CSA </p><p>(cylinder </p><p>1</p><p> </p><p> )</p><p> </p><p> :CSA </p><p>(cylinder </p><p>2</p><p> </p><p> )</p><p> </p><p> =(5x)(3y):(7x)(5y)</p><p> </p><p> </p><p></p><p>\begin{gathered} \\ : \implies \sf \:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} = 15xy : \:35xy\\ \\ \end{gathered} </p><p>:⟹CSA </p><p>(cylinder </p><p>1</p><p> </p><p> )</p><p> </p><p> :CSA </p><p>(cylinder </p><p>2</p><p> </p><p> )</p><p> </p><p> =15xy:35xy</p><p> </p><p> </p><p></p><p>\begin{gathered} \\ : \implies \sf \:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} = 15 : 35 \\ \\ \end{gathered} </p><p>:⟹CSA </p><p>(cylinder </p><p>1</p><p> </p><p> )</p><p> </p><p> :CSA </p><p>(cylinder </p><p>2</p><p> </p><p> )</p><p> </p><p> =15:35</p><p> </p><p> </p><p></p><p>\begin{gathered} \\ : \implies{\underline{\boxed{\pink{ \sf{\:CSA_{(cylinder_1)} : \: CSA_{(cylinder_2)} =3 : 7}}}}} \: \bigstar\\ \\ \end{gathered} </p><p>:⟹ </p><p>CSA </p><p>(cylinder </p><p>1</p><p> </p><p> )</p><p> </p><p> :CSA </p><p>(cylinder </p><p>2</p><p> </p><p> )</p><p> </p><p> =3:7</p><p> </p><p> </p><p> </p><p> ★</p><p> </p><p> </p><p></p><p>Hence ,</p><p></p><p>The Ratio of CSA's of two given cylinders is 3 : 7.</p><p>$