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Question
A solid sphere and a solid hemisphere have an equal total surface area. Prove that the ratio of their volume is `3sqrt(3):4`
Solution
Total surface area of a sphere = `4pi"r"_1^2` sq.units
Total surface area of a hemisphere = `3pi"r"_2^2` sq.units
Ratio of Total surface area = `4pi"r"_1^2 : 3pi"r"_2^2`
1 = `(4pi"r"_1^2)/(3pi"r"_2^2)` ...(Same Surface Area)
1 = `(4"r"_1^2)/(3"r"_2^2)`
∴ `("r"_1^2)/("r"_2^2) = 3/4`
r12 : r22 = 3 : 4
r1 : r2 = `sqrt(3) : 2`
Ratio of their volume
= `4/3 pi"r"_1^3 : 2/3 pi"r"_2^3`
= `2"r"_1^3 : "r"_2^3`
= `2 xx (sqrt(3))^3` : 23
= `2 xx 3 sqrt(3)` : 8 ...(÷ 2)
= `3sqrt(3) : 4`
Ratio of their volumes = `3sqrt(3) : 4`
Hence it is proved.
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