Ex.13.8 Q4 Surface Areas and Volumes Solution - NCERT Maths Class 9

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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?

Text Solution

Reasoning:

Volume of the sphere \(\begin{align} = \frac{4}{3} \end{align}\) \(\pi r^{3}\). So, the fraction of the volume of the earth is the volume of the moon is the ratio of the volume of the earth to the volume of the moon

What is known?

Ratio between the diameters of moon and earth.

What is unknown?

Fraction of the volume of the earth is the volume of the moon

Steps:

Diameter of the Earth \(=2r\)

Radius of the Earth \(\begin{align} = \frac{1}{2}(2r) = r \end{align}\)

Diameter of moon \(\begin{align} = \frac{1}{4}(2r) = \frac{1}{2}r \end{align}\)

Radius of the moon \(\begin{align} = \frac{1}{2}(\frac{r}{2}) = \frac{r}{4} \end{align}\)

Volume of the earth \(\begin{align}({v_1}) = \frac{4}{3} \end{align}\)

Volume of the moon \(\begin{align}({v_2}) = \frac{4}{3}\pi {(\frac{r}{4})^3} \end{align}\)

\[ = \frac{1}{{64}}\left( {\frac{4}{3}\pi {r^3}} \right)\]

\(\begin{align} = \frac{1}{{64}} \end{align}\) Volume of earth

Answer:

Hence the volume of the moon is \(\begin{align}\frac{1}{{64}} \end{align}\)th fraction of the volume of the earth.