The volume of the two spheres are in the ratio 64:27

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.

Volumes of two spheres are in the ratio 64:27. The ratio of their surface areas is 16:9.

Explanation:

Let the radii of the two spheres are r1 and r2, respectively.

∴ Volume of the sphere of radius

r1 = V1 = `43 pir_1^3`   [∵ Volume of sphere = `4/3pi` (radius)3]   ........(i)

And volume of the sphere of radius 

r2 = V2 = `4/3 pi r_2^3`   ......(ii)

Given, ratio of volumes = V1:V2 = 64:27

⇒ `(4/3 pir_1^3)/(4/3 pir_2^3) = 64/27`  ....[Using equations (i) and (ii)]

⇒ `(r_1^3)/(r_2^3) = 64/27` 

⇒ `r_1/r_2 = 4/3`   .....(iii)

Now, ratio of surface area = `(4 pir_1^2)/(4 pir_2^2)`   ......[∵ Surface area of a sphere = 4π (radius)2]

= `r_1^2/r_2^2`

= `(r_1/r_2)^2 = (4/3)^2`   .....[Using equation (iii)]

= 16:9

Hence, the required ratio of their surface area is 16:9.

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