Problem:
From the top of a vertical tower, the angles of depression of two cars in the same straight with the base of a tower at an instant are found to be 45 degrees and 60 degrees. If the cars are 100 meters apart and on the same side of the tower, find the height of the tower. (use root 3 equal to 1.73)

Solution:

Let's consider the given information and use it to solve the problem step by step:

Step 1: Identify the Given Information
- Angle of depression of the first car = 45 degrees
- Angle of depression of the second car = 60 degrees
- Distance between the two cars = 100 meters

Step 2: Understanding the Problem
We need to find the height of the tower based on the given angles of depression and the distance between the two cars.

Step 3: Draw a Diagram
Let's draw a diagram to represent the situation:

T
|
|
|
|
|-----------------
A B

In the diagram, T represents the top of the tower, A and B represent the positions of the two cars, and the line segment AB represents the distance between the two cars.

Step 4: Analyzing the Diagram
From the diagram, we can observe the following:
- Angle TAC = 45 degrees (angle of depression of the first car)
- Angle TBC = 60 degrees (angle of depression of the second car)
- Distance AC = Distance BC = 100 meters (given distance between the two cars)

Step 5: Applying Trigonometry
We can use the tangent function to find the height of the tower. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.

Let's consider the right-angled triangles TAC and TBC:

In triangle TAC:
- tan(45 degrees) = height of the tower / AC

In triangle TBC:
- tan(60 degrees) = height of the tower / BC

Step 6: Calculating the Height of the Tower
Using the given values and equations, we can solve for the height of the tower:

- tan(45 degrees) = height of the tower / AC
- tan(45 degrees) = height of the tower / 100

Simplifying the equation, we get:
- 1 = height of the tower / 100
- height of the tower = 100 meters

Similarly, for triangle TBC:
- tan(60 degrees) = height of the tower / BC
- tan(60 degrees) = height of the tower / 100

Simplifying the equation, we get:
- √3 = height of the tower / 100
- height of the tower = 100 * √3
- height of the tower = 100 * 1.73
- height of the tower = 173 meters

Step 7: Final Answer
Therefore, the height of the tower is 173 meters.