The angle of elevation of the top of a tower which is incomplete at a point 120 ft. from its base is `45^(@)`. If the elevation at the same point of the top is desired to be `60^(@)` then the tower should be raised by `120(sqrt(3)-1)` ft.

Answer

To solve the problem, we need to determine how much the height of the incomplete tower should be raised in order to change the angle of elevation from 45 degrees to 60 degrees at a point 120 feet away from its base.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Distance from the base of the tower (D) = 120 ft
- Initial angle of elevation (θ1) = 45 degrees
- Desired angle of elevation (θ2) = 60 degrees

2. **Calculate the Height of the Tower at 45 degrees:**
- In triangle BPA (where A is the top of the tower, B is the point on the ground directly below A, and P is the point from which the angle of elevation is measured):
- Using the tangent function:
\[ \tan(45^\circ) = \frac{H}{D} \]
- Since \(\tan(45^\circ) = 1\):
\[ 1 = \frac{H}{120} \]
- Therefore, the height of the tower (H) is:
\[ H = 120 \text{ ft} \]

3. **Calculate the Height of the Tower at 60 degrees:**
- In triangle BPC (where C is the new top of the tower):
- Using the tangent function:
\[ \tan(60^\circ) = \frac{H + X}{D} \]
- Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{H + X}{120} \]
- Rearranging gives:
\[ H + X = 120\sqrt{3} \]

4. **Substituting the Height from Step 2:**
- We already found that \(H = 120\):
\[ 120 + X = 120\sqrt{3} \]
- Solving for X:
\[ X = 120\sqrt{3} - 120 \]
- Factoring out 120 gives:
\[ X = 120(\sqrt{3} - 1) \]

5. **Conclusion:**
- The height of the tower should be raised by \(120(\sqrt{3} - 1)\) ft.