The angular depression on the top and the bottom of a 9 M high building for the top of the tower are 30° find the height of Tower and distance between the building in the tower

To solve this problem, we can use trigonometry. Let's denote:

- \( h \) as the height of the tower.
- \( d \) as the horizontal distance between the building and the tower.

### Given:
- The height of the building \( = 9 \) meters.
- The angle of depression to the top of the tower \( = 30^\circ \).
- The angle of depression to the bottom of the tower \( = 30^\circ \).

### Step 1: Set up the problem

Since both the angles of depression to the top and the bottom of the tower are \( 30^\circ \), this implies that the building and the tower are at the same horizontal level, with the tower's base at the same height as the building's base.

Let's denote:

- \( h_1 = h - 9 \), the height from the top of the building to the top of the tower.
- \( d \) as the horizontal distance between the building and the tower.

### Step 2: Apply Trigonometry

For the angle of depression \( \theta = 30^\circ \):

Using the tangent function for the height from the top of the building to the top of the tower:

\[
\tan(30^\circ) = \frac{h_1}{d}
\]

Using the fact that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):

\[
\frac{1}{\sqrt{3}} = \frac{h - 9}{d}
\]

So, we have:

\[
h - 9 = \frac{d}{\sqrt{3}}
\]

### Step 3: Solve for \( h \)

Now, use the tangent function for the height of the building to the bottom of the tower (which is the same angle):

\[
\tan(30^\circ) = \frac{9}{d}
\]

\[
\frac{1}{\sqrt{3}} = \frac{9}{d}
\]

Solving for \( d \):

\[
d = 9\sqrt{3}
\]

### Step 4: Substitute \( d \) back into the first equation:

\[
h - 9 = \frac{9\sqrt{3}}{\sqrt{3}}
\]

\[
h - 9 = 9
\]

\[
h = 18 \text{ meters}
\]

### Final Answer:

- The height of the tower \( h \) is **18 meters**.
- The horizontal distance between the building and the tower \( d \) is **9√3 meters** or approximately **15.59 meters**.

Would you like more details or have any questions?

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**Relative Questions:**

1. What is the difference between angle of depression and angle of elevation?
2. How does the height of the building affect the distance between the building and the tower?
3. What trigonometric identities are commonly used to solve problems involving angles of depression?
4. Can you find the height of the building if only the angle of elevation to the top of the tower is given?
5. How would the problem change if the angles of depression were different for the top and bottom of the tower?

**Tip:** When working with angles of depression, remember that they are measured from the horizontal line downwards, often requiring the use of complementary angles in trigonometry.

Learn how to calculate the height of a tower and the distance from a building using trigonometry. Solve a problem where the angular depression to the top and bottom of a 9-meter high building is given as 30°, and find the tower's height and horizontal distance from the building. Detailed step-by-step solution provided.

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