A pole of length 7 m is fixed vertically on the top of a tower. The angle of elevation of the top of the pole observed from a point on the ground is `60^(@)` and the angle of depression of the same point on the ground from the top of the tower is 45 The height (in m) of the tower is:

Options

A. `7(2sqrt3-1)`
B. `""_(2)^(7)(sqrt3+2)`
C. `7sqrt3`
D. `""_(2)^(7)(sqrt3+1)`

Answer

To solve the problem, we need to determine the height of the tower given the conditions about the pole and the angles of elevation and depression. Let's break it down step by step.

### Step 1: Understand the Problem
We have a pole of height 7 m fixed on top of a tower. From a point on the ground:
- The angle of elevation to the top of the pole is \(60^\circ\).
- The angle of depression from the top of the tower to the point on the ground is \(45^\circ\).

### Step 2: Define Variables
Let:
- \(h\) = height of the tower (in meters).
- The total height from the ground to the top of the pole = \(h + 7\) m.

### Step 3: Set Up the Right Triangle for Angle of Elevation
From the point on the ground, we can form a right triangle with:
- The opposite side = height from the ground to the top of the pole = \(h + 7\).
- The adjacent side = distance from the point on the ground to the base of the tower = \(d\).

Using the tangent of the angle of elevation:
\[ \tan(60^\circ) = \frac{h + 7}{d} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{h + 7}{d} \quad \text{(1)} \]

### Step 4: Set Up the Right Triangle for Angle of Depression
From the top of the tower, we can form another right triangle:
- The opposite side = height of the tower = \(h\).
- The adjacent side = same distance \(d\).

Using the tangent of the angle of depression:
\[ \tan(45^\circ) = \frac{h}{d} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ 1 = \frac{h}{d} \quad \text{(2)} \]

### Step 5: Solve for \(d\) from Equation (2)
From equation (2):
\[ d = h \quad \text{(3)} \]

### Step 6: Substitute \(d\) in Equation (1)
Substituting \(d\) from equation (3) into equation (1):
\[ \sqrt{3} = \frac{h + 7}{h} \]

### Step 7: Cross-Multiply and Solve for \(h\)
Cross-multiplying gives:
\[ \sqrt{3}h = h + 7 \]
Rearranging the equation:
\[ \sqrt{3}h - h = 7 \]
\[ (\sqrt{3} - 1)h = 7 \]
Now, solve for \(h\):
\[ h = \frac{7}{\sqrt{3} - 1} \]

### Step 8: Rationalize the Denominator
To rationalize the denominator:
\[ h = \frac{7(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{7(\sqrt{3} + 1)}{3 - 1} = \frac{7(\sqrt{3} + 1)}{2} \]

### Step 9: Final Calculation
Thus, the height of the tower is:
\[ h = \frac{7(\sqrt{3} + 1)}{2} \approx 7 \times 1.366 = 9.562 \text{ m} \text{ (approximately)} \]

### Conclusion
The height of the tower is approximately \(9.56\) m.