An airplane is flying between two towns at an altitude of 8km. One town is to the west of the plane and the other is to the east. The angle of depression to one town in de34grees and to the other town is 49degrees. What is the distance between the two towns

If the point directly below the plane is P, and the two towns are W and E, then their separation is

8cot34° + 8cot49°

To determine the distance between the two towns, we can use the concept of trigonometry. Specifically, we will use the tangent function, as the angles of depression are given.

Let's assume that the distance between the two towns is represented by "d."

If we draw a right triangle, the height (altitude of the airplane) can be considered as the opposite side, and the distance between the two towns (d) as the adjacent side.

For the angle of depression of 34 degrees, the tangent of the angle is defined as the ratio of the opposite side (8 km) to the adjacent side (d):

tan(34°) = opposite side / adjacent side
tan(34°) = 8 km / d

Similarly, for the angle of depression of 49 degrees:

tan(49°) = opposite side / adjacent side
tan(49°) = 8 km / d

We now have a system of two equations with two variables. We can solve this system of equations to find the value of "d," which represents the distance between the two towns.

To solve the system of equations, we can take the ratio of the two equations:

[tan(34°) / tan(49°)] = (8 km / d) / (8 km / d)
[tan(34°) / tan(49°)] = 1

Now, we can rearrange this equation to solve for "d":

d = (8 km) * [tan(49°) / tan(34°)]

Using a scientific calculator, we can find the values of tan(49°) and tan(34°), and then substitute them into the equation to get the distance, "d."