Two circular broadcast zones, P and Q, have radii of 8 units and 5 units, respectively. If the distance between the centers of these two zones is 13 units, how many common tangent lines can be drawn to both circles P and Q simultaneously?

The correct answer is 3.

Step 1: Identify the given dimensions of the two circles.

Let the radii of circle P and circle Q be $r_1$ and $r_2$, respectively, and let $d$ be the distance between their centers.

From the problem statement, we have:

$r_1=8\text{units}$

$r_2=5\text{units}$

$d=13\text{units}$

Step 2: Determine the relative position of the two circles.

Calculate the sum of the radii of the two circles:

$r_1+r_2=8+5=13\text{units}$

Since the distance between their centers ($d=13$) is equal to the sum of their radii ($r_1+r_2=13$), the two circles touch each other externally at exactly one point.

Step 3: Count the total number of common tangents.

For two circles that touch externally:

1. Direct (outer) common tangents: Exactly 2 direct common tangents can be drawn touching both circles on opposite sides without passing between them.
2. Transverse (inner) common tangent: Exactly 1 transverse common tangent can be drawn passing directly through the single point of contact between the two circles.

Therefore, the total number of common tangent lines is:

$2+1=3$

Thus, exactly 3 common tangent lines can be drawn to both circles P and Q simultaneously.