Question

Sun is about $330$ times heavier and $100$ times bigger in radius than earth. The ratio of mean density of the Sun to that of earth is
A. $3.3 \times {10^{ - 6}}$
B. $3.3 \times {10^{ - 4}}$
C. $3.3 \times {10^{ - 2}}$
D. $3.3$

Answer 1

Hint:Density is defined as the amount of mass of a substance to that of the volume of the same substance. In order to find the mean density here, we have to find the density of the Sun to that of the Earth. Then by comparing the equations we will find the answer.

Complete step by step answer:
In the given question it is mentioned that the mass of the Sun is about $330$ times heavier than the Earth. Let us assume that the mass of the Sun is $M$ and that of Earth is $m$.According to the given question we know that,
$M = 330m - - - - - - \left( 1 \right)$
It is also mentioned that the Sun is about $100$ times bigger in radius than the Earth. Now we take the radius of the Sun as $R$ and the radius of Earth as $r$.According to the given question,
$R = 100r - - - - \left( 2 \right)$
Let the volume of the Sun be \[V\].
$V = \dfrac{4}{3}\pi {R^3} = \dfrac{4}{3}\pi {\left( {100r} \right)^3} - - - - - - \left( 3 \right)$

And the volume of the Earth is $v$.
$v = \dfrac{4}{3}\pi {r^3} - - - - - \left( 4 \right)$
The density of the Sun is $\rho $ and the density of the Earth is $\rho '$.Therefore,
$\rho = \dfrac{M}{V} = \dfrac{{330 \times m}}{{\dfrac{4}{3}\pi {{\left( {100r} \right)}^3}}}$
And
$\rho ' = \dfrac{m}{v} = \dfrac{m}{{\dfrac{4}{3}\pi {r^3}}}$
Thus, the ratio of the mean density of the Sun to the Earth is,
$\dfrac{\rho }{{\rho '}} = \dfrac{{\dfrac{{330 \times m}}{{\dfrac{4}{3}\pi {{\left( {100r} \right)}^3}}}}}{{\dfrac{m}{{\dfrac{4}{3}\pi {r^3}}}}} \\
\Rightarrow \dfrac{\rho }{{\rho '}}= \dfrac{{330}}{{{{10}^6}}} \\
\therefore \dfrac{\rho }{{\rho '}}= 3.3 \times {10^{ - 4}}$

So, the correct option is B.

Note: It must be noted that mean density is also a mathematical form of density.The mass of the Earth is actually $6 \times {10^{24}}{\text{ }}kg$ and the radius of Earth is $6400{\text{ }}km$. In order to calculate the density we need to have the volume, as the shape of the Sun and the Earth are close to the sphere then their volume is calculated as the formula of the volume of the sphere.