Question

$\sf{Question :-}$
The diameter of a moon is approximately one-fourth of the diameter of the earth. What fraction of the earth is the volume of moon.
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Answer 1

Answer:

Let d1 and d2 be the diameters of the moon and the earth respectively.

Then, d1 = 1/4 d2

⇒r1/r2 =1/4

We know that volume of sphere = 3/4 πr^3

⟹

Volume of moon/Volume of earth= (4/3πr1³)/(4/3πr2³)

= (¼r2/r2)³

= (¼)³

= 1/64

Answer 2

Let's assume that ;

• Diameter of Moon = d units
• Diameter of Earth = d' units

Also , given that ;

Diameter of moon is ¼th of diameter of Earth which implies that ;

• d'/4 = d

Now , continuing further ;

• Radius of Moon = d/2 units
• Radius of Earth = d'/2 = 4d/2 = 2d units

Now , we have to Find V/V' , where V' represents volume of Earth while V represents volume of Moon

Continuing further ;

${: \implies \quad \sf \dfrac{V}{V^{\prime}} = \dfrac{\dfrac{4}{3}\pi {\left(\dfrac{d}{2}\right)}^{3}}{\dfrac{4}{3}\pi (2d)^{3}}}$

After cancelling 4/3 and π from both numerator and denominator we have ;

${: \implies \quad \sf \dfrac{V}{V^{\prime}} = \dfrac{\dfrac{d^3}{8}}{8d^{3}}}$

${: \implies \quad \sf \dfrac{V}{V^{\prime}} = \dfrac{d^3}{64d^{3}}}$

After cancelling d³ we are left with ;

${: \implies \quad \sf \dfrac{V}{V^{\prime}} =\dfrac{1}{64}}$

Hence , 1/64 of volume of Earth is volume of Moon

Used Concepts :-

• ${\bf Volume \: of \: Sphere \: =\: \dfrac{4}{3}\pi {r^3}}$

where r is the radius of Sphere

Additional Information :-

${\begin{array}{|c|c|c|}\cline{1-3}\bf Shape&\bf Volume\ formula&\bf Surface\ area formula\\\cline{1-3}\sf Cube&\tt l^3}&\tt 6l^2\\\cline{1-3}\sf Cuboid&\tt lbh&\tt 2(lb+bh+lh)\\\cline{1-3}\sf Cylinder&\tt {\pi}r^2h&\tt 2\pi{r}(r+h)\\\cline{1-3}\sf Hollow\ cylinder&\tt \pi{h}(R^2-r^2)&\tt 2\pi{rh}+2\pi{Rh}+2\pi(R^2-r^2)\\\cline{1-3}\sf Cone&\tt 1/3\ \pi{r^2}h&\tt \pi{r}(r+s)\\\cline{1-3}\sf Sphere&\tt 4/3\ \pi{r}^3&\tt 4\pi{r}^2\\\cline{1-3}\sf Hemisphere&\tt 2/3\ \pi{r^3}&\tt 3\pi{r}^2\\\cline{1-3}\end{array}}$

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