Question

The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.

Answer 1

Formula used

$\bold\red{Surface\: area\: of \: sphere = 4\pi r²}$

where, r denotes the Radius of the sphere.

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๑Here the concept of Surface area of sphere has been used. Radius of 2 sphere has been given as 7 cm and 14cm. We have to find the ratio of surface area of the balloons.

$\mathtt{ \to\color{skyblue}Let \: the \: radius \: of \: small \: sphere \: be\: r_1} \\ \\ \mathtt{ \to \color{purple}Let \: the \: radius\: of \: small\:sphere\: be\: r_2}$

Surface area of the sphere = 4πr²

$\sf{ \color{orange}Ratio \: of \: their \: surface\: area = \frac{surface \: area \: of \: smaller\:sphere}{surface\:area\: of \: bigger \: balloon}}$

$\sf{ \implies Ratio \: of \: their \: surface \: area= \frac{4πr²_{1}}{4πr²_{2}}}\\ \\ \sf{ \implies Ratio\:of \: their\: surface \: area= \frac{r²_{1}}{r²_{2}}}\\\\ \sf{ \implies Ratio \: of \: their \: surface \: area= \frac{7×7}{14×14}}\\\\ \sf{ \implies \color{grey}Ratio \: of \: their \: surface \: area= \frac{1}{4}}$

Thus the required answer is 1:4.

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More to know

⚘TSA of cuboid = 2(lb + bh + lh)

⚘CSA of cuboid = 2h(l + b)

⚘TSA of cube =6l²

⚘CSA of cube = 4l²

⚘CSA of a cylinder = 2π × r × h

⚘TSA of a cylinder=2πr(h + r)

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Answer 2

Solution:-

$\sf{Let } \: S_{1} \: and \: S_{2} \: be \: the \: total \: surface \: area \: in \: two \: cases \: of \: r = 7 \: cm \: and \: R = 14 \: cm$

$\sf{Therefore}, \: S_{1} = 4\pi r {}^{2} = 4 \times \pi \times \times 7 \times 7 {cm}^{2}$

$\sf{and} \: S_{2} = 4\pi r {}^{2} = 4 \times \pi \times 14 \times 14 {cm}^{2}$

$\sf{Required \: Ratio} = \frac{S _{1} }{ S_{2} } = \frac{4 \times \pi \times 7 \times 7}{4 \times\pi \times 14 \times 14} = \frac{1}{4} \: i.e \: 1:4$

Required Answer ⇒ is 1:4.