40 if the volume of two spheres is in the ratio 27 64 then the ratio of their radii is

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Correct Answer:

Description for Correct answer:

\( \Large \frac{(a_{1})^{3}}{(a_{2})^{3}}=\frac{27}{64} \)\( \Large \frac{a_{1}}{a_{2}}=\frac{3}{4} \)Ratio of their total surface area\( \Large =\frac{6a_{1}^{2}}{6a_{2}^{2}}= \left(\frac{a_{1}}{a_{2}}\right)^{2} \)

\( \Large = \left(\frac{3}{4}\right)^{2}=\frac{9}{16}=9:16 \)

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The volumes of two spheres are in the ratio 27: 64 . What is the ratio of their radii?

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The volumes of the two spheres are in the ratio 64:27. Find the ratio of their surface areas.

Text Solution

Solution : Let the radius of two spheres be `r_(1)` and `r_(2)` <br> Given, the ratio of the volume of two spheres = 64: 27 <br> `(V_(1))/(V_(2)) =(64)/(27) rArr ((4)/(3)pir_(1)^(3))/((4)/(3)pir_(2)^(3)) = (64)/(27)` <br> `rArr" "((r_(1))/(r_(2)))^(3) = ((4)/(3))^(3) " "[because "volume of sphere" =(4)/(3) pir^(3)]` <br> `rArr " "(r_(1))/(r_(2)) =(4)/(3)` <br> Let the surface areas of the two spheres `S_(1)` and `S_(2)` <br> `therefore" "(S_(1))/(S_(2)) = (4pir_(1)^(2))/( 4pir_(2)^(2)) = ((r_(1))/(r_(2)))^(2) rArr S_(1),S_(2) = ((4)/(3))^(2) = (16)/(9)` <br> `rArr" "S_(1),S_(2) = 16:9` <br> Hence, the ratio of the their surface areas is 16: 9.