Question

Volumes of two spheres are in the ra-
tio of 8:27, the ratio of their surface​

Answer 1

Answer:

Volume of sphere is

3

4

πr

3

V

2

V

1

=

3

4

πr

2

3

3

4

πr

1

3

=

r

2

3

r

1

3

=

27

8

⇒

r

2

r

1

=

3

2

⇒

S

2

S

1

=

4πr

2

2

4πr

1

2

=

r

2

2

r

1

2

=(

3

2

)

2

=

9

4

Answer 2

GivEn:

• Ratio of volume of two spheres = 8:27

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To find:

• Ratio of their surface area

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SoluTion:

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${\underline{\sf{\bigstar\;As\; per\;given\; Question\;:}}}$

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Ratio of volume of two spheres = 8:27

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As we know that,

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$\star\;{\boxed{\sf{\purple{Volume\;of\;sphere = \dfrac{4}{3} \pi r^3}}}}$

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• Let radius of 1st sphere be r

• Let radius of 2nd sphere be R

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$:\implies\sf \dfrac{Volume\;of\;1st\;sphere}{Volume\;of\;2nd\;sphere} = \dfrac{8}{27}$

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$:\implies\sf \dfrac{ \frac{4}{3} \pi r^3}{ \frac{4}{3} \pi R^3} = \dfrac{8}{27}$

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$:\implies\sf \dfrac{ \cancel{ \frac{4}{3} \pi} r^3}{ \cancel{ \frac{4}{3}} \pi R^3} = \dfrac{8}{27}$

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$:\implies\sf \dfrac{r^3}{R^3} = \dfrac{8}{27}$

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$\small\sf\;\;\;\;\;\underline{Taking\;cube\;root\;both\; sides\;:}$

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$:\implies\sf \sqrt[3]{ \dfrac{r^3}{R^3}} = \sqrt[3]{ \dfrac{2^3}{3^3}}$

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

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Therefore,

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• Radius of 1st sphere = 2
• Radius of 2nd sphere = 3

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As we know that,

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$\star\;{\boxed{\sf{\pink{Surface\;area\;of\;sphere = 4 \pi r^2}}}}$

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Therefore,

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${\underline{\sf{\bigstar\;Ratio\;of\;there\;surface\;area\;:}}}$

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$:\implies\sf \dfrac{Surface\;area\;of\;1st\;sphere}{Surface\;area\;of\;2nd\;sphere}$

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

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$:\implies\sf \dfrac{ \cancel{4 \times \pi} \times 2^2}{ \cancel{4 \times \pi} \times 3^2}$

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$:\implies\bf \dfrac{4}{27}$

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$\therefore$ Ratio of the curved surface area of spheres is 4:27.