Qasim spots an airplane on radar that is currently approaching in a straight line, and that will fly directly overhead. The plane maintains a constant altitude of 7200 feet. Qasim initially measures an angle of elevation of $16^{\circ}$ to the plane at point $A$. At some later time, he measures an angle of elevation of $38^{\circ}$ to the plane at point $B$. Find the distance the plane traveled from point $A$ to point $B$. Round your answer to the nearest foot if necessary.

Step 1 ：Form two right triangles, one with the angle of elevation 16 degrees and the other with the angle of elevation 38 degrees. The altitude of the plane forms the opposite side of the triangle and the distance the plane traveled forms the adjacent side of the triangle.

Step 2 ：Use the tangent of the angle of elevation which is equal to the ratio of the opposite side to the adjacent side to find the distance the plane traveled.

Step 3 ：Calculate the distance at point A using the formula \(\text{distance}_A = \frac{\text{altitude}}{\tan(\text{angle}_A)}\), where altitude = 7200 feet and angle_A = 16 degrees. This gives \(\text{distance}_A = 25109.383995654545\) feet.

Step 4 ：Calculate the distance at point B using the formula \(\text{distance}_B = \frac{\text{altitude}}{\tan(\text{angle}_B)}\), where altitude = 7200 feet and angle_B = 38 degrees. This gives \(\text{distance}_B = 9215.579751790166\) feet.

Step 5 ：Subtract the distance at point A from the distance at point B to find the distance the plane traveled from point A to point B. This gives \(\text{distance}_{AB} = \text{distance}_B - \text{distance}_A = 15894\) feet.

Step 6 ：Final Answer: The distance the plane traveled from point $A$ to point $B$ is \(\boxed{15894}\) feet.