An aeroplane flies from point P (30° N, 40° W) to point Q due east on the same latitude. The distance PQ is 4500 km. Find the longitude of Q.

The aeroplane flies from point P $(30^\circ N, 40^\circ W)$ to point Q due east on the same latitude.
The distance PQ is $4500 \text{ km}$.
We need to find the longitude of Q.
The latitude of both P and Q is $\alpha = 30^\circ N$.
The radius of the Earth is approximately $R = 6371 \text{ km}$.

The formula for the distance ($D$) along a parallel of latitude is given by:

where $R$ is the Earth's radius, $\alpha$ is the latitude, and $\Delta\lambda_{rad}$ is the difference in longitude in radians.
To convert $\Delta\lambda_{rad}$ to degrees, we use $\Delta\lambda_{deg} = \Delta\lambda_{rad} \times \frac{180^\circ}{\pi}$.
So, the distance formula can also be written as:

The radius of the parallel of latitude at $30^\circ N$ is $r = R \cos(\alpha)$.

Using the distance formula $D = \frac{\Delta\lambda_{deg}}{360^\circ} \times 2\pi r$:

Rearrange to solve for $\Delta\lambda_{deg}$:

Point P is at $40^\circ W$. Since the aeroplane flies due east, the longitude increases (moves towards the east). We can represent west longitudes as negative values.
Longitude of Q = Longitude of P + $\Delta\lambda_{deg}$
Longitude of Q = $-40^\circ + 46.70^\circ$
Longitude of Q = $6.70^\circ E$

Rounding $6.70^\circ E$ to the nearest degree gives $7^\circ E$.