From a point on the ground 40 m away from the foot of a tower, the angle of elevation of the top of the tower is `30^(@)`. The angle of elevation of the top of a water tank(on the top of tower) is `45^@`.find the height of the tower and depth of tank

Explanation

To solve the problem, we will use trigonometric ratios, specifically the tangent function, which relates the angle of elevation to the opposite side (height) and the adjacent side (distance from the tower).

### Step-by-Step Solution:

1. **Identify the given data:**
- Distance from the point on the ground to the foot of the tower (adjacent side) = 40 m
- Angle of elevation to the top of the tower = 30°
- Angle of elevation to the top of the water tank = 45°

2. **Calculate the height of the tower (h):**
- Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
- For the angle of elevation to the top of the tower (30°):
\[ \tan(30°) = \frac{h}{40} \]
- We know that \(\tan(30°) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{h}{40} \]
- Cross-multiplying gives:
\[ h = 40 \cdot \frac{1}{\sqrt{3}} = \frac{40}{\sqrt{3}} \approx 23.09 \text{ m} \]

3. **Calculate the total height to the top of the water tank (H):**
- For the angle of elevation to the top of the water tank (45°):
\[ \tan(45°) = \frac{H}{40} \]
- We know that \(\tan(45°) = 1\):
\[ 1 = \frac{H}{40} \]
- Thus, we find:
\[ H = 40 \text{ m} \]

4. **Calculate the depth of the water tank (d):**
- The depth of the water tank can be found by subtracting the height of the tower from the total height:
\[ d = H - h = 40 - \frac{40}{\sqrt{3}} = 40 - 23.09 \approx 16.91 \text{ m} \]

### Final Results:
- Height of the tower (h) ≈ 23.09 m
- Depth of the water tank (d) ≈ 16.91 m