Question

The ratio of curved surface area of two spheres 4:9 find the ratio of radii and volumes​

Answer 1

Answer:

ʜᴇʀᴇ ɪs ʏᴏᴜʀ ᴀɴsᴡᴇʀ⬇️

Step-by-step explanation:

sᴜʀғᴀᴄᴇ ᴀʀᴇᴀ ᴏғ ᴀ sᴘʜᴇʀᴇ ɪs ᴅɪʀᴇᴄᴛʟʏ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟ ᴛᴏ sǫᴜᴀʀᴇ ᴏғ ʀᴀᴅɪᴜs,ʜᴇɴᴄᴇ ʀᴀᴛɪᴏ ᴏғ ʀᴀᴅɪɪ ᴏғ ᴛᴡᴏ sᴘʜᴇʀᴇs ɪs 2:3.

Answer 2

Given :

• Ratio of Curved surface area of two spheres = 4:9

To find :

• Ratio of there radius
• Ratio of volume

Formula used :

• Curved surface area of sphere = 4πr²
• Volume of sphere = (4/3)πr³

Solution :

Let -

• Radius of 1st sphere = r
• Radius of 2nd sphere = R

Part 1- CSA

$\sf \: CSA \: of \: First \: sphere \: : \: CSA \: of \: second \: sphere \: = \: 4 \: : \: 9 \:$

$\sf \implies \: \dfrac{CSA \: of \: First \: sphere \: }{ \: CSA \: of \: second \: sphere \: } = \: \dfrac{4}{9}$

$\sf \implies \: \dfrac{4\pi \: {r}^{2} }{4\pi \: {R}^{2} } \: = \: \dfrac{4}{9}$

$\sf \implies \: \dfrac{ {r}^{2} }{{R}^{2} } \: = \: \dfrac{4}{9}$

$\sf \implies \: \bigg( \dfrac{ {r} }{{R}} \bigg)^{2} \: = \: \dfrac{ {2}^{2} }{ {3}^{2} }$

$\sf \implies \: \bigg( \dfrac{ {r} }{{R}} \bigg)^{2} \: = \: \bigg( \dfrac{ 2}{3} \bigg)^{2}$

$\sf \implies \: \bigg( \dfrac{ {r} }{{R}} \bigg) \: = \: \dfrac{2}{3}$

$\sf \implies \: radius \: of \: First \: sphere \: \: radius \: of \: second \: sphere \: = \: \bold{ 2 \: : \: 3}$

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$\sf \: \: volume \: of \: first \: sphere \: : \: volume \: of \: second \: sphere \: = \: \dfrac{volume \: of \: first \: sphere \: : \: }{volume \: of \: second \: sphere \: }$

$\sf \: \implies \:\dfrac{volume \: of \: first \: sphere \: : \: }{volume \: of \: second \: sphere \: } \: = \dfrac{ \frac{4}{3} \pi \: {r}^{3} }{ \frac{4}{3}\pi \: { R }^{3} } \\ \\ \sf \: \implies \:\dfrac{volume \: of \: first \: sphere \: : \: }{volume \: of \: second \: sphere \: } \: = \dfrac{ {r}^{3} }{ { R }^{3} } \\ \\ \sf \: \implies \:\dfrac{volume \: of \: first \: sphere \: : \: }{volume \: of \: second \: sphere \: } \: = \: \bigg(\dfrac{ {r} }{ \: { R }} \bigg)^{3} \\ \\ \sf \: \implies \:\dfrac{volume \: of \: first \: sphere \: : \: }{volume \: of \: second \: sphere \: } \: = \: \bigg(\dfrac{ {2} }{ { 3}} \bigg)^{3} \\ \\ \sf \: \implies \:\dfrac{volume \: of \: first \: sphere \: : \: }{volume \: of \: second \: sphere \: } \: = \: \dfrac{ {2} ^{3} }{ { 3} ^{3} } \\ \\ \sf \: \implies \:\dfrac{volume \: of \: first \: sphere \: : \: }{volume \: of \: second \: sphere \: } \: = \: \dfrac{ 8}{ 27}\\ \\ \sf \: \: volume \: of \: first \: sphere \: : \: volume \: of \: second \: sphere \: = \: \bold{8 \: : \: 27}$

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ANSWER :

• Ratio of radius = 2 : 3
• Ratio of volume = 8 : 27