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Two solid spheres made of the same metal have weights 5920 g and 740 g, respectively. Determine the radius of the larger sphere, if the diameter of the smaller one is 5 cm.

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Answer : $5\; cm$

Density (D) of an object is mass (m) per unit volume (v).

$D=\frac{m}{v}$

So mass is directly proportional to volume for same metal (as density remains same)

For Solid 1

Let Mass = $M_{1}$

Volume = $V_{1}$

For solid 2

Mass = $M_{2}$

Volume = $V_{2}$

Now, $\frac{M_{1}}{M_{2}}=\frac{V_{1}}{V_{2}}$

In the question, the objects are spheres

Volume of sphere $=\frac{4}{3}\pi R^{3}$ (where R is radius)

So, volume is directly proportional to $R^{3}$

Hence, $\frac{M_{1}}{M_{2}}=\frac{V_{1}}{V_{2}}=\frac{R_{1}^{3}}{R_{2}^{3}}$

Putting the given values $\left ( 5920\; g \; and\; 740\; g \right )$

$\Rightarrow \frac{5920}{740}=\frac{R_{1}^{3}}{R_{2}^{3}}$

$\Rightarrow \frac{R_{1}^{3}}{R_{2}^{3}}=8$

$\Rightarrow \frac{R_{1}}{R_{2}}=\sqrt[3]{8}$

$\Rightarrow \frac{R_{1}}{R_{2}}=2$

So, $R_{1}=R_{2}\times 2$

Diameter of the smaller one is 5 cm

So, $R_{2}=\frac{5}{2}=2.5\; cm$

$R_{1}=2.5 \times 2$

Hence, $R_{1}=5\; cm$

Therefore radius of larger sphere is 5 cm.

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infoexpert23

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