From the top of a `7` m high building, the angle of elevation of the top of a cable tower is `60^@`and the angle of depression of its foot is `45^@`. Determine the height of the tower.

Options

A. `7(sqrt2+1)m`
B. `7(sqrt3+1)m`
C. `5(sqrt3+1)m`
D. `13(sqrt3+1)m`

Explanation

To solve the problem step by step, we will use trigonometric ratios and properties of right triangles.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have a building of height \( AB = 7 \) m.
- From the top of the building (point A), the angle of elevation to the top of the cable tower (point C) is \( 60^\circ \).
- The angle of depression to the foot of the cable tower (point D) is \( 45^\circ \).
- We need to find the total height of the tower \( CD \).

2. **Setting Up the Diagram:**
- Draw a vertical line for the building (AB) and another vertical line for the cable tower (CD).
- Mark point A at the top of the building, point B at the bottom of the building, point C at the top of the tower, and point D at the bottom of the tower.

3. **Finding the Distance from A to D (horizontal distance):**
- From point A, the angle of depression to point D is \( 45^\circ \).
- In triangle \( ABD \), we can use the tangent function:
\[ \tan(45^\circ) = \frac{AB}{AD} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{7}{AD} \implies AD = 7 \text{ m} \]

4. **Finding the Height from A to C (height of the tower):**
- Now, we consider triangle \( AEC \) where \( AE = AD = 7 \) m and the angle of elevation \( \angle AEC = 60^\circ \).
- We can again use the tangent function:
\[ \tan(60^\circ) = \frac{CE}{AE} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{CE}{7} \implies CE = 7\sqrt{3} \text{ m} \]

5. **Finding the Total Height of the Tower (CD):**
- The total height of the tower \( CD \) can be calculated as:
\[ CD = CE + ED \]
Here, \( ED = AB = 7 \) m (the height of the building).
\[ CD = 7\sqrt{3} + 7 \]

6. **Final Calculation:**
- Therefore, the height of the tower \( CD \) is:
\[ CD = 7(\sqrt{3} + 1) \text{ m} \]

### Conclusion:
The height of the cable tower is \( 7(\sqrt{3} + 1) \) meters.