Question

Find the radius of a globe which is such that the distance between two places on the same meridian whose latitude differs by \[{1^ \circ }10'\] may be half-an-inch.

Answer 1

Hint: Here we will find the radius of the globe by using Arc length formula. First, we will convert the unit of our latitude from degree to radian by using the formula. Then we will change the unit of Arc length from inch to meter. Finally, we will substitute all the values in the Arc length formula and solve it further to get the radius.

Formula used:
We will use the following formulas:
1.Arc length \[ = \] Radius \[ \times \theta \] (in radians), where, \[\theta \] is the value of latitude between the two points
2.Degree \[ = \] (Degree \[ + \] (Minutes or Seconds) \[ \div \] 60) \[ \div \] 60
3.Radian \[ = \] Decimal Degree \[ \times \dfrac{\pi }{{180}}\]

Complete step-by-step answer:
It is given to us that the latitude between the two points is \[{1^ \circ }10'\]. That means,
Difference in latitude \[ = {1^ \circ }10'\]
Also, Arc length is given to be half an inch.
Arc length \[ = 0.5{\rm{inch}}\]
Now, we will convert the inch unit to a meter unit (\[1{\rm{inch}} = 0.025{\rm{m}}\]). Therefore, we get
\[ \Rightarrow \] Arc length \[ = 0.5 \times 0.0254\]
Multiplying the terms, we get
\[ \Rightarrow \] Arc length \[ = 0.0127m\]
We know that Arc length \[ = \] Radius \[ \times \theta \] (in radians).
Therefore using formula Degree \[ = \] (Degree \[ + \] (Minutes or Seconds) \[ \div \] 60) \[ \div \] 60 and Radian \[ = \] Decimal Degree \[ \times \dfrac{\pi }{{180}}\], we get
\[0.0127 = R \times \left( {\left( {{1^ \circ } + {{\dfrac{{10}}{{60}}}^ \circ }} \right) \times \dfrac{\pi }{{180}}} \right)rad\]
Taking LCM inside the bracket, we get
\[ \Rightarrow 0.0127 = R \times \left( {{{\dfrac{7}{6}}^ \circ } \times \dfrac{\pi }{{180}}} \right)rad\]
Simplifying the expression, we get
\[\begin{array}{l} \Rightarrow 0.0127 = R \times 0.02025\\ \Rightarrow R = \dfrac{{0.0127}}{{0.02025}}\end{array}\]
Dividing the terms, we get
\[ \Rightarrow R = 0.627\]
So, Radius is \[0.627\] metres.

Note: We know that the arc length is the distance along a section of curve between two points. We know that a globe is circular in shape; therefore, we used arc length formula to find the radius between the two points. Here, it is important to convert the angle given in radian so that the radius value comes in meter. In addition, latitude is the degree between the two points.