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# The moon subtends an angle of 57 minute at the base-line equal to the radius of the earth. What is the distance of the moon from the earth? [Radius of earth=$6.4\times {{10}^{6}}m$]A. $11.22\times {{10}^{8}}m$B. $3.86\times {{10}^{8}}m$C. $3.68\times {{10}^{-3}}cm$D. $3.68\times {{10}^{8}}cm$

Last updated date: 25th Jun 2024
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Hint: To solve this problem first we will convert the given angle in degree than in radians to calculate the required distance, because for small angles , angle can be written as arc length divided by radius. In the above problem arc is radius of earth and radius is distance of the moon from earth.

Formula used:
$\theta =\dfrac{arc}{radius}$

Complete step by step solution:
Given, moon subtends an angle of 57 minute at the base-line equal to radius of earth so, by converting it into degrees we have,
$\theta =57'$
$\Rightarrow \theta =\dfrac{57}{60}=\dfrac{19}{20}={{0.95}^{\circ }}$
$\Rightarrow \theta ={{0.95}^{\circ }}$

Now, to change angle theta into radian form we need to multiply the above value of theta with $\dfrac{\pi }{180}$so,
$\Rightarrow \theta ={{0.95}^{\circ }}\times \dfrac{\pi }{180}$
Now, we know that the above angle is very small so we can write angle theta by,
$\Rightarrow \theta =\dfrac{arc}{radius}$
Here, arc (Radius of earth R) and radius (Distance of the moon from earth D).
$\Rightarrow \theta =\dfrac{R}{D}$
$\Rightarrow D=\dfrac{R}{\theta }$
By putting the value of R from the problem and of theta calculated above we have,
$\Rightarrow D=\dfrac{6.4\times {{10}^{6}}}{{{0.95}^{\circ }}\times \dfrac{\pi }{180}}m$
$\Rightarrow D=\dfrac{11.52\times {{10}^{8}}}{{{0.95}^{\circ }}\times \pi }m$
$\Rightarrow D=3.86\times {{10}^{8}}m$

$\therefore$ The distance of the moon from the earth is $3.86\times {{10}^{8}}m$ so option (B) is correct.

In mathematics both radians and degrees represent angles in different notations, in which radian value can be defined using arc of circle and we know that $\pi$ radians is equal to ${{180}^{\circ }}$ so we can easily calculate the value of 1 degree which is given by,
$\Rightarrow {{1}^{\circ }}=\dfrac{\pi }{180}$