Question

The surface areas of two spheres are in the ratio 1: 4. Find the ratio of
their volumes.

Answer 1

Given :-

The surface areas of two spheres are in the ratio 1:4

To find :-

The ratio of their volumes.

Solution :-

Given that

The ratio of the surface areas of two spheres

= 1:4

Let the radius of the first sphere be r units

The Surface Area of the first sphere

= 4πr² sq.units

Let the radius of the second sphere be R units

The Surface Area of the second sphere

= 4πR² sq. units

The ratio of the surface areas of the two spheres

= 4πr² : 4πR²

= (4πr²)/(4πR²)

= r²/ R²

= r²:R²

Therefore, r² : R² = 1 : 4

=> r²/R² = 1/4

=> r²/R² = 1²/2²

=> (r/R)² = (1/2)²

=> r/R = 1/2 -------(1)

The ratio of their radii = 1:2

We know that

The Volume of the first sphere = (4/3)πr³ cubic units

The Volume of the second sphere = (4/3)πR³ cubic units

Ratio of their volumes

= (4/3)πr³ : (4/3)πR³

= (4/3)πr³ / (4/3)πR³

= r³/R³

= (r/R)³

= (1/2)³ (from (1))

= 1/8

= 1:8

Answer :-

The ratio of the volumes of the two spheres is 1:8

Used formulae:-

♦ The Surface Area of the first sphere = 4πr² sq.units

♦ The Volume of the first sphere = (4/3)πr³ cubic units

Answer 2

Question :-

• The surface areas of two spheres are in the ratio 1: 4. Find the ratio oftheir volumes.

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Given :-

• Surface areas of two spheres are in ratio 1:2.

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To find :-

• The ration in there volumes.

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Solution :-

Let the first radius be r and the second radius be R .

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$\implies \frac{4\pi r^{2} }{4\pi R^{2} }=\frac{1}{4}\\ \\ \implies\frac{r}{R} = \frac{1}{2}$

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Volume :-

$\implies\frac{\frac{4}{3}\pi r^{3} }{\frac{4}{3}\pi R^{3} } = \frac{(1)^{3} }{(2)^{3} } = \frac{1}{8}$

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Therefore, the ratio of their volumes are 1:8.

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Joyful learning!!! ⛅