Question

of mdii of two spheres is 2:3. Find the ratio of their surface areas and volumes.
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Answer 1

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Answer 2

Correct question:

The ratio of the two spheres is 2 : 3. Find the ratio of their surface areas and volumes.

Answer:

Ratio of surface area = 4 : 9

Ratio of volumes of sphere = 8 : 27

Step-by-step explanation:

Given:

Ratio of radii = 2:3

To Find:

• The ratio of surface area.

• Ratio of volume.

Formula used :

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies \: Surface \: \: \: area \: \: \: of \: \: \: sphere = 4(\pi) {r}^{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies \: \: \: \: volume \: \: \: of \: \: \: sphere = {{\dfrac{4}{3} }\pi {r}^{3} }$

Solution

Let the common ration be 'r'.

So, radius of first sphere is = 2r

Radius of second sphere = 3r

Finding ratio of surface area :

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies Ratio \: of \: surface \: = \: \dfrac{4\pi {(2r)}^{2} }{4\pi {(3r)}^{2} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies Ratio \: \: \: of \: \: \: surface \: \: \: = \: \bigg( \dfrac{2r \times 2r }{3r \times 3r} \bigg)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies Ratio \: of \: surface \: = \: \dfrac{4}{9}$

$\green{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies Ratio \: \: \: of \: \: \: surface \: \: \: = \:4 \: : \: 9}$

Finding ratio of volume :

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies Ratio \: of \: surface \: = \: \bigg(\dfrac{ \frac{4}{3}\pi {(2r)}^{3} }{ \frac{4}{3} \pi {(3r)}^{3} } \bigg)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies Ratio \: \: \: of \: \: \: surface \: \: \: = \: \: \bigg( \dfrac{2r \times 2r \times 2r}{3r \times 3r \times 3r} \bigg)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies Ratio \: of \: surface \: = \: \dfrac{8}{27}$

$\pink{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies Ratio \: \: \: of \: \: \: surface \: \: \: = \: \: 8 \: : \: 27}$