Identify the given information and the formula for distance between two points on the same latitude.

The two points are P($27^\circ$N; $16^\circ$W) and Q($27^\circ$N; $97^\circ$W).
The latitude ($\phi$) is $27^\circ$N.
The radius of the Earth ($r$) is given as $6400 \text{ km}$.
The value for $\pi$ is given as 3.142.

Since both points are on the same latitude, the distance between them along that parallel of latitude can be calculated using the formula:

$$\text{Distance} = \frac{\theta}{360^\circ} \times 2 \pi r \cos(\phi)$$

where $\theta$ is the difference in longitude.

Both longitudes are West, so we find the absolute difference.

$$\theta = |97^\circ \text{W} - 16^\circ \text{W}| = |97^\circ - 16^\circ| = 81^\circ$$

$$\text{Distance} = \frac{81^\circ}{360^\circ} \times 2 \times 3.142 \times 6400 \text{ km} \times \cos(27^\circ)$$

First, calculate $\frac{81}{360}$:

$$\frac{81}{360} = 0.225$$

Next, calculate $\cos(27^\circ)$:

$$\cos(27^\circ) \approx 0.8910065$$

Now, substitute these values back into the distance formula:

$$\text{Distance} = 0.225 \times 2 \times 3.142 \times 6400 \text{ km} \times 0.8910065$$ $$\text{Distance} = 0.225 \times 40217.6 \text{ km} \times 0.8910065$$ $$\text{Distance} = 9048.96 \text{ km} \times 0.8910065$$ $$\text{Distance} \approx 8062.57 \text{ km}$$

The distance between the two points is approximately $8062.57 \text{ km}$.