Question

the ratio of radiii of two spheres is2:3 find the ratio of their surface areas and volumes​

Answer 1

$\large\bf\underline {To \: find:-}$

• We need to find the ratio surface areas and volumes of both spheres .

$\large\bf\underline{Given:-}$

• Ratio of radii of both spheres = 2:3

$\huge\bf\underline{Solution:-}$

Let the of radius of both spheres be 2r and 3r

$\mid\underline{ \boxed{ \textbf{Surface area of sphere} = \bf \: 4 \pi {r}^{2} }} \mid$

Surface area of 1st sphere having Radius 2r

$\hookrightarrow \tt \:4 \times \pi \times {(2r)}^{2} \\ \\ \hookrightarrow \tt \:4 \times \pi \times 4 {r}^{2} \\ \\ \hookrightarrow \tt \:16 \pi {r}^{2} ....(1)$

Surface area of 2nd sphere having Radius 3r

$\hookrightarrow \tt \:4 \times \pi \times {(3r)}^{2} \\ \\ \hookrightarrow \tt \:4 \times \pi \times 9 {r}^{2} \\ \\ \hookrightarrow \tt \:36 \pi {r}^{2} ....(2)$

Now,

✝️Ratio of two Spheres :-

$\dashrightarrow \tt \:ratio = \cancel\frac{16 \pi {r}^{2} }{36 \pi {r}^{2} } \\ \\ \dashrightarrow \tt \:ratio = \frac{4}{9} \\ \\ \dashrightarrow \red{\bf\:ratio = 4 : 9}$

Now,

$\mid\underline{ \boxed{ \textbf{Volume of sphere = } \bf \: \frac{4 \pi {r}^{3} }{3} }} \mid$

Volume of 1st sphere having Radius 2r

$\hookrightarrow \tt \:volume = \: \frac{4 \times \pi \times {(2r)}^{3} }{3} \\ \\ \hookrightarrow \tt \:volume = \: \frac{4 \times \pi \times {8r}^{3} }{3} \\ \\ \hookrightarrow \tt \:volume = \: \frac{32 \pi {r}^{3} }{3}$

Volume of 2nd sphere having Radius 3r

$\hookrightarrow \tt \:volume = \: \frac{4 \times \pi \times {(3r)}^{3} }{3} \\ \\ \hookrightarrow \tt \:volume = \: \frac{4 \times \pi \times {27r}^{3} }{3} \\ \\ \hookrightarrow \tt \:volume = \: \frac{108 \pi {r}^{3} }{3}$

Now,

✝️Ratio of volumes of both spheres :-

$\tt \dashrightarrow \: ratio = \frac{ \frac{ \frac{32 \pi {r}^{3} }{3} }{108 \pi {r}^{3} } }{3} \\ \\ \tt \dashrightarrow \: ratio = \frac{32 \pi {r}^{3} }{ \cancel3} \times \frac{ \cancel3}{108 \pi {r}^{3} } \\ \\ \tt \dashrightarrow \: ratio = \cancel \dfrac{32 \pi {r}^{3} }{108 \pi {r}^{3} } \\ \\ \tt \dashrightarrow \: ratio = \frac{8}{27} \\ \\ \red{\bf \dashrightarrow \: ratio =8 : 27}$

Hence,

❥Ratio of surface area of Spheres = 4:9

❥Ratio of volumes of Spheres = 8:27

$\rule{200}3$

Answer 2

━━━━━━━━━━━━━━━━━━━━

⏭ Required Answer:

✏ GiveN:

• Ratio of the radii of the two spheres = 2:3

✏ To FinD:

• Ratio of their surface areas.
• Ratio of their volumes.

━━━━━━━━━━━━━━━━━━━━

⏭ How to solve?

To find the ratio between the surface areas and volumes respectively of two spheres, we need to know the formula of Surface area and Volume of sphere.

☄ Here are the formulae,

$\large{ \bold{ \underline{ \underline{ \red{Surface \: area \: of \: sphere}}}}} \\ \\ \large{ = \boxed{ \rm{4\pi \: {r}^{2} }}} \\ \\ { \rm{where \: r \: is \: the \: radius \: of \: sphere}}$

And,

$\large{ \bold{ \underline{ \underline{ \red{Volume \: of \: sphere}}}}} \\ \\ \large{ = \boxed{ \rm{ \frac{4}{3} \pi \: {r}^{3} }}} \\ \\ \rm{where \: r \: is \: the \: radius \: of \: sphere}$

So, Using formulae, we can find the respective ratio.

━━━━━━━━━━━━━━━━━━━━

⏭ Solution:

Given, Ratio of radius of two spheres = 2:3

• Let the two radius he 2x and 3x.

So, this will help in finding the ratio of surface area when we use the formula,

Ratio of surface area of spheres,

$\large{ \rm{ \longrightarrow \frac{4\pi (r_1) {}^{2} }{4\pi (r_2) {}^{2} } }} \\ \\ \large{ \rm{ \longrightarrow \frac{(r_1) {}^{2} }{(r_2) {}^{2} } \: \: \: (\rm{\green{4\pi \: cancels) }}}} \\ \\ \sf{ \dag \: { \purple{we \: have \: r_1 = 2x \: and \: r_2 = 3x}}} \\ \\ \large{ \rm{ \longrightarrow \: \frac{ {(2x)}^{2} }{ {(3x)}^{2} } }} \\ \\ \large{ \rm{ \longrightarrow \: \frac{4 {x}^{2} }{9 {x}^{2} } }} \\ \\ \large{ \rm{ \longrightarrow \: \frac{4}{9} }}$

✏ Hence, our required ratio of surface area = 4 : 9

━━━━━━━━━━━━━━━━━━━━

Ratio of volume of the spheres,

$\large{ \rm{ \longrightarrow \: \frac{ \dfrac{4}{3} \pi {r_1}^{3} }{ \frac{4}{3}\pi {r_2}^{3} } }} \\ \\ \large{ \rm{ \longrightarrow \: \frac{ {r_1}^{3} }{ {r_2}^{3} }\: \: \: (\rm{\green{\frac{4}{3}\pi \: cancels}) }}} \\ \\ { \rm{ \dag \: { \purple{we \: have \: radius \: of \: two \: spheres \: 2x \: and \: 3x}}}} \\ \\ \large{ \rm{ \longrightarrow \: \frac{ {(2x)}^{3} }{ {(3x)}^{3} } }} \\ \\ \large{ \rm{ \longrightarrow \: \frac{8 {x}^{3} }{27 {x}^{3} } }} \\ \\ \large{ \rm{ \longrightarrow \: \frac{8}{27} }}$

✏ Hence, our required ratio of volumes = 8 : 27

$\large{ \therefore{ \underline{ \underline{ \pink{ \rm{Hence \: solved \: \dag}}}}}}$

━━━━━━━━━━━━━━━━━━━━