Question

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

• A. The diameter of the moon is $3464\frac{8}{63}$ km
• B. The diameter of the moon is $3564\frac{8}{63}$ km
• C. The diameter of the moon is $346\frac{8}{63}$ km
• D. The diameter of the moon is $3664\frac{8}{63}$ km

Answer 1

The correct option is A The diameter of the moon is $3464\frac{8}{63}$ km

Let $PQ$ be the diameter of the moon and $A$ be the observer.
Given, $\angle PAQ={31}^{\prime }=\frac{31}{60}\times \frac{\pi }{180}$ rad
Since the angle subtended by the moon is very small, its diameter will be approximately equal to the small arc of a circle whose center is the eye of the observer and the radius is the distance of the earth from the moon. Also the moon subtends an angle of ${31}^{\prime }$ at the center of this circle.
We know, $\theta =\frac{\ell }{r}$

$⟹\frac{31}{60}\times \frac{\pi }{180}=\frac{PQ}{38,4000}$
$PQ=\frac{31}{60}\times \frac{22}{7\times 180}\times 38,4000=3464\frac{8}{63}$ km