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 ?

See previous post.

To find the distance between the two cars and the distance of each car from the tower, we can use trigonometry. Let's label the points:

- Car A is the car with the angle of depression of 30 degrees.
- Car B is the car with the angle of depression of 60 degrees.
- The foot of the tower is point T.

First, let's find the distance between the two cars. We can consider the line connecting Car A and Car B as a base line. Let's label the midpoint of this line as point M.

Now, we need to find the height of the tower above point M. We know that the height of the tower is 50m, and the angle of depression from the top of the tower to Car A is 30 degrees.

Using trigonometry, we can use the tangent function since we have the opposite side (height of the tower) and the adjacent side (distance from the top of the tower to Car A):
tan(30) = opposite/adjacent
tan(30) = 50/x
x = 50 / tan(30)

Next, we need to find the height of the tower above point M using the angle of depression from the top of the tower to Car B, which is 60 degrees.

Again, using the tangent function:
tan(60) = 50/y
y = 50 / tan(60)

Now, we can find the distance between the two cars by subtracting the distances from the tower to each car:
Distance between the two cars (M) = x - y

To find the distances of each car from the tower, we simply add or subtract their respective distances from M:
Distance from Car A to the tower = x
Distance from Car B to the tower = y

Now, let's calculate the values:

x = 50 / tan(30) ≈ 86.60 m
y = 50 / tan(60) ≈ 28.87 m

Distance between the two cars (M) = x - y ≈ 86.60 - 28.87 ≈ 57.73 m

So, the distance between the two cars is approximately 57.73 meters. Car A is approximately 86.60 meters away from the tower, and Car B is approximately 28.87 meters away from the tower.