The ratio of surface area of a sphere and curved surface area of a hem

1.	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» 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 = $\dfrac{2}{3}\pi { r_{2}}^{3}$ $\rm \dashrightarrow \dfrac{ \: \: \: \: \dfrac{4}{3}\pi { r_{1}}^{3} \: \: \: \: }{ \dfrac{2}{3}\pi { r_{2}}^{3}}$ $\rm \dashrightarrow 2 \times \bigg( \dfrac{ { r_{1}} }{ r_{2} } \bigg)^{3}$ $\rm \dashrightarrow 2 \times \bigg( \dfrac{ 3 }{ 2 } \bigg)^{3}$ $\rm \dashrightarrow 2 \times \dfrac{27}{8} = \dfrac{27}{4}$ $\underline{ \boxed{ \therefore \rm The \: ratio \: of \: the \: volumes \: of \: sphere = 27 : 4}}$

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»

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 = $\dfrac{2}{3}\pi { r_{2}}^{3}$

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

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

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

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