Question

Assume the Earth to be a sphere of radius R. What is the radius of the circle of latitude ${\rm{4}}{{\rm{0}}^{\rm{o}}}$S?
A) R cos ${\rm{4}}{{\rm{0}}^{\rm{o}}}$
B) R sin${\rm{8}}{{\rm{0}}^{\rm{o}}}$
C) R sin ${\rm{4}}{{\rm{0}}^{\rm{o}}}$
D) R tan${\rm{4}}{{\rm{0}}^{\rm{o}}}$

Answer 1

Hint: In this solution, first, we have to draw the diagram from the given data of the question. We draw a triangle from the centre of the circle. Then we find the side’s using the trigonometric entity formula of a triangle.

Complete step by step answer:
Here, It is given that Earth to be a sphere of radius R.
Latitude ${\rm{4}}{{\rm{0}}^{\rm{o}}}$S
To find the radius of the circle.
Here, Let us draw the diagram from the given data.

In the diagram above,
OR $\left( { = {\rm{OQ}}} \right)$ is the radius of the earth at the equator.
PQ is the radius of the earth at ${\rm{4}}{{\rm{0}}^{\rm{o}}}$ latitude.
OP is the distance on the axis of earth between the two points.
Here, In the triangle OPQ,
Angle Q is ${\rm{4}}{{\rm{0}}^{\rm{o}}}$ (Since alternate angles in a set of parallel lines)
Therefore,
We can write,
Cos ${\rm{4}}{{\rm{0}}^{\rm{o}}}$$ = $Adjacent side/hypotenuse
$ = \,\dfrac{{{\rm{PQ}}}}{{{\rm{OQ}}}}$
$\therefore \,{\rm{PQ}} = {\rm{OQ}}\,{\rm{cos4}}{{\rm{0}}^{\rm{o}}}$
Or,
${\rm{PQ}} = {\rm{OR}}\,{\rm{cos4}}{{\rm{0}}^{\rm{o}}}$ (Since OR is equal to OQ)
Therefore,
Radius of the earth at ${\rm{4}}{{\rm{0}}^{\rm{o}}}$ latitude $ = $ Radius at equator $ \times $${\rm{cos4}}{{\rm{0}}^{\rm{o}}}$
Thus, the radius of the circle of latitude ${\rm{4}}{{\rm{0}}^{\rm{o}}}$S is R cos ${\rm{4}}{{\rm{0}}^{\rm{o}}}$

Hence the correct option is A.

Note: Radius is the distance between the centre and any points on the surface area. First, we have to draw the diagram from the given data of the question. We draw a triangle from the centre of the circle which is triangle OPQ where angle O and angle Q are equal since alternate angles in a set of parallel lines that is${\rm{4}}{{\rm{0}}^{\rm{o}}}$. Then we find the side PQ using the formula of the triangle.