34. From The Top Of A Building 50 M High, The Angles Of Depression Of The Top And Bottom Of A Tower Are Observed To Be 30° And 60°. Find The Height Of The Tower And Distance Between The Building And The Tower. (Take √√3 = 1.73)

Mar 1, 2025 by ADMIN 229 views

Introduction

In trigonometry, the concept of angles of depression is used to find the height of an object or the distance between two objects. The angles of depression are the angles formed by the line of sight from the observer's eye to the object and the horizontal line. In this problem, we are given the height of a building and the angles of depression of the top and bottom of a tower. We need to find the height of the tower and the distance between the building and the tower.

Given Information

Height of the building = 50 m
Angle of depression of the top of the tower = 30°
Angle of depression of the bottom of the tower = 60°

Step 1: Draw a Diagram

To solve this problem, we need to draw a diagram. Let's draw a diagram with the building, the tower, and the angles of depression.

          +---------------+
          |               |
          |  Building    |
          |  (50 m)      |
          +---------------+
                  |
                  |
                  v
+---------------+---------------+
|                         |
|  Tower          |          |
|  (h)          |          |
+---------------+---------------+

Step 2: Identify the Triangles

From the diagram, we can see that there are two right-angled triangles: ∆ABC and ∆ABD.

          +---------------+
          |               |
          |  Building    |
          |  (50 m)      |
          +---------------+
                  |
                  |
                  v
+---------------+---------------+
|                         |
|  ∆ABC          |          |
|  (30°, 50 m)  |          |
+---------------+---------------+
                  |
                  |
                  v
+---------------+---------------+
|                         |
|  ∆ABD          |          |
|  (60°, h)     |          |
+---------------+---------------+

Step 3: Use Trigonometry to Find the Height of the Tower

We can use the tangent function to find the height of the tower.

tan(30°) = 50 / x
x = 50 / tan(30°)
x = 50 / 0.57735
x = 86.6 m

Step 4: Use Trigonometry to Find the Distance Between the Building and the Tower

We can use the tangent function to find the distance between the building and the tower.

tan(60°) = h / x
x = h / tan(60°)
x = h / 1.73205
x = 50 / tan(30°) + h / tan(60°)
x = 86.6 + h / 1.73205

Step 5: Solve for h

We can solve for h by substituting the value of x into the equation.

x = 86.6 + h / 1.73205
86.6 = 86.6 + h / 1.73205
0 = h / 1.73205
h = 0

However, this is not possible since the height of the tower cannot be zero. Let's try again.

tan(60°) = h / x
x = h / tan(60°)
x = h / 1.73205
x = 50 / tan(30°) + h / tan(60°)
x = 86.6 + h / 1.73205
h = 50 * tan(60°) - 50 * tan(30°)
h = 50 * 1.73205 - 50 * 0.57735
h = 86.6 m

Step 6: Find the Distance Between the Building and the Tower

Now that we have the height of the tower, we can find the distance between the building and the tower.

x = 50 / tan(30°) + h / tan(60°)
x = 86.6 + 86.6 / 1.73205
x = 86.6 + 50
x = 136.6 m

Conclusion

In this problem, we used trigonometry to find the height of a tower and the distance between a building and the tower. We drew a diagram, identified the triangles, and used the tangent function to find the height of the tower and the distance between the building and the tower. The height of the tower is 86.6 m, and the distance between the building and the tower is 136.6 m.

Final Answer

The final answer is:

Height of the tower = 86.6 m
Distance between the building and the tower = 136.6 m

Q&A

Q: What is the concept of angles of depression in trigonometry?

A: The concept of angles of depression in trigonometry is used to find the height of an object or the distance between two objects. The angles of depression are the angles formed by the line of sight from the observer's eye to the object and the horizontal line.

Q: How do we use trigonometry to find the height of the tower?

A: We use the tangent function to find the height of the tower. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.

Q: What is the formula to find the height of the tower?

A: The formula to find the height of the tower is:

h = 50 * tan(60°) - 50 * tan(30°)

Q: How do we find the distance between the building and the tower?

A: We use the tangent function to find the distance between the building and the tower. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.

Q: What is the formula to find the distance between the building and the tower?

A: The formula to find the distance between the building and the tower is:

x = 50 / tan(30°) + h / tan(60°)

Q: What is the final answer to the problem?

A: The final answer to the problem is:

Height of the tower = 86.6 m
Distance between the building and the tower = 136.6 m

Q: What is the significance of the angles of depression in this problem?

A: The angles of depression in this problem are 30° and 60°. These angles are used to find the height of the tower and the distance between the building and the tower.

Q: How do we use the tangent function to find the height of the tower and the distance between the building and the tower?

A: We use the tangent function to find the height of the tower and the distance between the building and the tower. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.

Q: What is the relationship between the height of the tower and the distance between the building and the tower?

A: The height of the tower and the distance between the building and the tower are related by the tangent function. The tangent function is used to find the height of the tower and the distance between the building and the tower.

Q: How do we solve for the height of the tower and the distance between the building and the tower?

A: We solve for the height of the tower and the distance between the building and the tower by using the tangent function and the given information.

Q: What is the final answer to the problem in terms of the height of the tower and the distance between the building and the tower?

A: The final answer to the problem in terms of the height of the tower and the distance between the building and the tower is:

Height of the tower = 86.6 m
Distance between the building and the tower = 136.6 m

Conclusion

In this Q&A article, we have discussed the concept of angles of depression in trigonometry, how to use trigonometry to find the height of the tower and the distance between the building and the tower, and the final answer to the problem. We have also discussed the significance of the angles of depression in this problem and how to use the tangent function to find the height of the tower and the distance between the building and the tower.