Question

The ratio of surface area of a sphere and curved surface area of a hemisphere is 9:2. Then find the ratio of their volumes​

Answer 1

Step-by-step explanation:

Let the radius of sphere be r₁

And radius of hemisphere be r₂

The surface area of a sphere = 4πr₁²

The curved surface area of hemisphere = 2π(r₂)²

The ratio of them is 9:2

$\rm \dashrightarrow \dfrac{4\pi { r_{1} }^{2} }{2\pi { r_{2} }^{2} } = \dfrac{9}{2}$

$\rm \dashrightarrow \dfrac{2 { r_{1} }^{2} }{ { r_{2} }^{2} } = \dfrac{9}{2}$

$\rm \dashrightarrow \bigg(\dfrac{ { r_{1} } }{ { r_{2} } } \bigg) ^{2} = \dfrac{9}{4}$

$\rm \dashrightarrow \dfrac{ r_{1} }{ r_{2} } = \sqrt{ \dfrac{9}{4}}$

$\rm \dashrightarrow \dfrac{ r_{1} }{ r_{2} } = \dfrac{3}{2}$

As we know that:-

Volume of sphere = $\rm\dfrac{4}{3}\pi { r_{1}}^{3}$

Volume of hemisphere =$\dfrac{2}{3}\pi { r_{2}}^{3}$

$\rm \dashrightarrow \dfrac{ \: \: \: \: \dfrac{4}{3}\pi { r_{1}}^{3} \: \: \: \: }{ \dfrac{2}{3}\pi { r_{2}}^{3}}$

$\rm \dashrightarrow 2 \times \bigg( \dfrac{ { r_{1}} }{ r_{2} } \bigg)^{3}$

$\rm \dashrightarrow 2 \times \bigg( \dfrac{ 3 }{ 2 } \bigg)^{3}$

$\rm \dashrightarrow 2 \times \dfrac{27}{8} = \dfrac{27}{4}$

$\underline{ \boxed{ \therefore \rm The \: ratio \: of \: the \: volumes \: of \: sphere = 27 : 4}}$

Answer 2

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