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प्रश्न
Volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is ______.
पर्याय
3 : 4
4 : 3
9 : 16
16 : 9
उत्तर
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 = `4/3 pi"r"_1^3` ...(i) [∵ Volume of sphere = `4/3pi` (radius)3]
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 pi"r"_1^3)/(4/3 pi"r"_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 pi"r"_1^2)/(4 pi"r"_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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