Question

The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

Answer 1

Hint: Here we need to find what fraction of the volume of earth is the volume of the moon by considering the formula of volume as\[\dfrac{4}{3}\pi {r^3}\].

Complete step-by-step answer:
It is given to us that the diameter of the moon is approximately one-fourth of the diameter of earth and with the help of that we have to find out what fraction of the volume of earth is the volume of the moon.
Let’s say Diameter of the earth be $ {d_e} $ and radius of the earth as $ \dfrac{{{d_e}}}{2} $ and diameter of the moon be \[\dfrac{{{d_e}}}{4}\] and radius of the moon as $ \dfrac{{{d_e}}}{8} $
Since, Volume of moon is \[\dfrac{4}{3}\pi {r^3}\]and hence putting the value of radius of moon i.e $ \dfrac{{{d_e}}}{8} $ we get
Volume of moon=\[\dfrac{4}{3}\pi {\left( {\dfrac{{{d_e}}}{8}} \right)^3}\].
Similarly, Volume of earth=\[\dfrac{4}{3}\pi {\left( {\dfrac{{{d_e}}}{2}} \right)^3}\].
Therefore Ratio $ {\text{ = }}\dfrac{{{\text{Volume of the Moon}}}}{{{\text{Volume of the Earth}}}} = \dfrac{{\dfrac{4}{3}\pi {{\left( {\dfrac{{{d_e}}}{8}} \right)}^3}}}{{\dfrac{4}{3}\pi {{\left( {\dfrac{{{d_e}}}{2}} \right)}^3}}} $
Hence on Solving we have
Ratio $ {\text{ = }}\dfrac{{{\text{Volume of the Moon}}}}{{{\text{Volume of the Earth}}}} = \dfrac{{{{\left( {\dfrac{{{d_e}}}{8}} \right)}^3}}}{{{{\left( {\dfrac{{{d_e}}}{2}} \right)}^3}}} = \dfrac{8}{{64 \times 8}} = \dfrac{1}{{64}} $ .
Therefore, Volume of Moon = $ \dfrac{1}{{64}} $ (Volume of Earth)

Note: So in this type of question first of all we have to find the volume of the moon and volume of the earth and then with the help of that we can find the ratio.
Here instead of diameter we can take the radius also. We will get the same answer.