Question

Assuming the distance of Earth from the moon to be 38,400 km and the angle subtended by the moon at the eye of a person on earth to be $31\prime $, find the diameter of the moon.

Answer 1

Hint:At first, convert the given angle in minutes to degree by dividing it by 60, then change it to radian by multiplying by $\dfrac{\pi }{180}$. Consider the diameter of the moon as an arc of length and distance between moon and earth as radius,Then use formula $s=r\theta $ where r is distance of earth from Moon and $\theta $ is angle in radian and s be the arc of length i.e diameter of moon.

Complete step-by-step answer:
In the question we are given the distance of Earth from the arc to be 38,400 Km and the angle subtended by the moon at the eye of a person on Earth to be $31\prime $.
Let it be represented as in a diagram as shown below,

Here E be the point of person, AB can be considered the diameter of Moon as the distance between Moon and Earth is very large.
Here In the question the diameter of the Moon will be considered as an arc of a circle with radius 38400 Km and angle as $31\prime $.
At first, we will convert angle to degree by using fact ${{1}^{\circ }}=60\prime $ so we can write it as $60\prime ={{1}^{\circ }}$. Hence $1\prime ={{\left( \dfrac{1}{60} \right)}^{\circ }}$ . So, $31\prime ={{\left( \dfrac{31}{60} \right)}^{\circ }}$. Then we will convert the angle in degree to radians.
Before proceeding we will first briefly say something about radian.
The radian is an S.I. unit for measuring angles and is the standard unit of angular measure used in areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radius of the angle that it subtends; one radian is just under 57.3 degrees. The unit is formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered as a SI derived unit.
Radian describes the plain angle subtended by a circular arc as the length of arc divided by radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of such a subtend angle is equal to the ratio of the arc length to the radius of circle; that is $\theta \ =\ \dfrac{s}{r}$, where $\theta $ is the subtended angle in radius, s is arc length and r is radius.
Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radius that is $s=r\theta $.
So, we will convert ${{\left( \dfrac{31}{60} \right)}^{\circ }}$ it into radian by multiplying the angle degree by $\dfrac{\pi }{180}$ so we get,
$\theta =\dfrac{31}{60}\times \dfrac{\pi }{180}$.
Now to find the length of the arc we will use formula $s=r\theta $, where r is radius, s is the length of arc and $\theta $ is angle in radian.
Hence, on using formula we get,
$s=r\theta \ =\ 384000\times \dfrac{31}{60}\times \dfrac{\pi }{180}$
Which on calculation after substituting $\pi =3.14$ we get,
$s\ =\ 3460.9\ km$
Hence the length of the arc or the diameter of the moon is 346.09 km.

Note: Considering the diameter of the moon as an arc of length and distance between moon and earth as radius is the key point for solving the question.Drawing a proper diagram helps us to approach the solution Students generally misunderstand the quantity of $\theta $. Generally, most students have confusion that ‘$\theta $’ in the question is in degree or in radian. So, they should clearly know that the value of $\theta $ is in radian. If $\theta $ is in degree then we can convert it into radian by multiplying with $\dfrac{\pi }{180}$.If $\theta $ is in radian then we can convert it into degree by multiplying with $\dfrac{180 }{\pi}$.