A) Find the longitudinal difference between town P (49^ exto extE) and town Q (131^ exto extW). b) If town R is 840 km due east to town Q, find the position of town R, given Q is at (36^ exto extN, 13

a) Find the longitudinal difference between the two towns.

Step 1: Identify the longitudes of town P and town Q.
Town P is at $49^\circ \text{E}$.
Town Q is at $131^\circ \text{W}$.

Step 2: Calculate the longitudinal difference.
Since one town is East and the other is West of the Prime Meridian, the longitudinal difference is the sum of their longitudes.

$$\text{Longitudinal difference} = 49^\circ + 131^\circ = 180^\circ$$

The longitudinal difference between the two towns is $\boxed{180^\circ}$.

b) If another town R is 840 km due east to town Q, find the position of town R.

Step 1: Identify the position of town Q and the given information.
Town Q is at $(36^\circ \text{N}, 131^\circ \text{W})$.
Distance $D = 840 \text{ km}$ due east of Q.
Radius of the Earth $R = 6370 \text{ km}$.
$\pi = \frac{22}{7}$.
The latitude of town Q is $\alpha = 36^\circ \text{N}$. Since town R is due east of Q, it will be on the same parallel of latitude, so its latitude will also be $36^\circ \text{N}$.

Step 2: Calculate the change in longitude ($\Delta \lambda$) corresponding to the distance $D$ along the parallel of latitude.
The formula for the distance along a parallel of latitude is:

$$D = \frac{\Delta \lambda}{360^\circ} \times 2\pi R \cos \alpha$$

Rearrange the formula to solve for $\Delta \lambda$:

$$\Delta \lambda = \frac{D \times 360^\circ}{2\pi R \cos \alpha}$$

Step 3: Substitute the given values into the formula.
First, calculate $2\pi R$:

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