A straight highway leads to the foot of a tower of height 50m. From the top of the tower, the angles of depression of two cars standing on the highway are 30 degree and 60 degree. What is the distance between the two cars and how far is each car from the tower ?

To solve this problem, we can use basic trigonometry principles.

Let's start by drawing a diagram of the situation. We have a tower of height 50m and two cars on the highway, which we can call Car A and Car B.

From the top of the tower, we have two angles of depression:
- The angle of depression to Car A is 30 degrees.
- The angle of depression to Car B is 60 degrees.

Now, let's label the distances we need to find:
- Distance between Car A and Car B: we'll call this distance "d".
- Distance from the tower to Car A: we'll call this distance "x".
- Distance from the tower to Car B: we'll call this distance "y".

Now, let's use trigonometry to solve for these distances.

For Car A:
In a right triangle with the tower as the vertical side and the distance "x" as the horizontal side, we have the angle of depression of 30 degrees.

Using trigonometry, we know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is 50m (the height of the tower) and the adjacent side is "x" (the distance from the tower to Car A). So, we have the equation:

tan(30 degrees) = 50 / x

To solve for "x", we can rearrange the equation:

x = 50 / tan(30 degrees)

Now, we can calculate the value of "x":

x ≈ 86.60m

Similarly, for Car B:
In a right triangle with the tower as the vertical side and the distance "y" as the horizontal side, we have the angle of depression of 60 degrees.

Using the same trigonometric relationship, we can set up the equation:

tan(60 degrees) = 50 / y

Rearranging the equation to solve for "y":

y = 50 / tan(60 degrees)

Calculating the value of "y":

y ≈ 28.87m

Now, let's find the distance between Car A and Car B, which we labeled as "d". We can use the Pythagorean theorem to find "d":

d^2 = (x - y)^2 + 50^2

Substituting the calculated values:

d^2 = (86.60 - 28.87)^2 + 50^2

Calculating "d":

d ≈ 92.61m

Therefore, the distance between the two cars is approximately 92.61 meters. Car A is about 86.60 meters away from the tower, and Car B is approximately 28.87 meters away from the tower.

The 1st(closest) car:

X1 = Distance from tower.
A1 = 60o = Angle between X1 and hyp.

The 2nd car:
X2 = Dist. bet. the 2 cars.
X1+X2 = Dist. from 2nd car to tower.
A2 = 30o = Angle bet. (X1+X2) and hyp.

tan 60 = 50/X1
X1 = 50/tan 60 = 28.87 m.

tan 30 = 50/(X1+X2)
X1+X2 = 50/tan 30 = 86.60 m.
28.87 + X2 = 86.60
X2 = 57.73 m.