from the top of a cliff 50 m high the angle of depression of the top and bottom of tower are observed to be 30 and 45 degree resp. find h of tower ?

well, you know the distance to the base of the tower is 50 m (a 45 deg angle).

Then, you can compute the length of the hypotenuse of that triangle on the common side (ie, 50sqrt2).

now you have one angle (15 deg, one side, and another angle (the 45 deg angle from the tower to the long side 50sqrt2)

Now use the law of sines:

sin15/h=50sqrt2/sin125

To solve this problem, we will use trigonometric relations. Let's first label the relevant parts of the diagram:

```
B
/ |
/ |
h / |
/ | 50 m
/ |
/ |
/________\
A C
```

Here, A represents the top of the cliff, B represents the top of the tower, and C represents the bottom of the tower. We are given that the angle of depression at point A is 30 degrees and at point C is 45 degrees.

Using trigonometric relations, we can determine the height of the tower.

Step 1: Find the distance AC:
Since AC is the horizontal distance, it can be calculated from tangent of angle of depression.
tan(30 degrees) = AC / 50 m
AC = 50 m * tan(30 degrees)
≈ 28.87 m

Step 2: Find the distance BC:
Similarly, we can use tangent of angle of depression to find BC.
tan(45 degrees) = BC / 50 m
BC = 50 m * tan(45 degrees)
≈ 50 m

Step 3: Calculate the height of the tower:
The height of the tower is the difference between AB and BC.
AB = AC + BC
≈ 28.87 m + 50 m
≈ 78.87 m

Therefore, the height of the tower is approximately 78.87 meters.

To find the height of the tower, we can use the concept of trigonometry. Let's break down the problem and understand the given information:

1. The height of the cliff: 50 meters.
2. The angles of depression: 30 degrees and 45 degrees.

Now, let's solve the problem step by step:

Step 1: Draw a diagram:
Draw a diagram representing the given situation. Start by drawing a straight vertical line to represent the cliff, which is 50 meters high. At the bottom of the cliff, draw another line perpendicular to the ground. This line represents the tower.

|
|
-----
/ \
/ \
/_____\
Cliff

Step 2: Identify the trigonometric relationships:
The angles of depression help us form right-angled triangles. From the diagram, we can see that we have two right-angled triangles: one with a 30-degree angle of depression and another with a 45-degree angle of depression.

Step 3: Determine the height of the tower using trigonometry:
Let's start with the triangle formed by the 30-degree angle of depression.

In the triangle:

Height of the cliff (opposite side) = 50 meters
Length along the ground (adjacent side) = h meters (height of the tower)
Angle of depression (angle) = 30 degrees

Using the trigonometric ratio "tangent," we can write:

tan(angle) = opposite / adjacent
tan(30) = 50 / h

Now, solve for h:

h = 50 / tan(30)

Calculating the value of h:

h ≈ 86.60 meters

Step 4: Repeat the process for the triangle formed by the 45-degree angle of depression:
In this triangle, the opposite side (height of the cliff) remains the same (50 meters). The adjacent side represents the total height (h) of the tower plus the height of the cliff.

Using the trigonometric ratio "tangent":

tan(angle) = opposite / adjacent
tan(45) = 50 / (h + 50)

Now, solve for h:

h + 50 = 50 / tan(45)
h + 50 = 50

Subtracting 50 from both sides:

h = 0

This means that the height of the tower is zero meters, which is not logical.

Step 5: Final answer:
Since h = 0 for the triangle formed by the 45-degree angle of depression, it means that there is an error in the given information. It is possible that the angle or observation was not accurately measured.

Therefore, based on the valid information (30-degree angle of depression), the height of the tower is approximately 86.60 meters.