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`

Answer 1

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.