The angle of elevation of the top of a tower from a point on the ground is `30^(@)`. After walking 45 m towards the tower, the angle of elevation becomes `45^(@)`. Find the height of the tower.

Answer

To solve the problem, we will use trigonometric ratios and the concept of right triangles. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a tower and two points on the ground where the angles of elevation to the top of the tower are given. The first angle of elevation is \(30^\circ\) from point A, and after walking 45 m towards the tower to point B, the angle of elevation becomes \(45^\circ\).

### Step 2: Set Up the Diagram
Let's denote:
- \(h\) = height of the tower
- \(d\) = distance from point A to the base of the tower
- Point A is where the angle of elevation is \(30^\circ\).
- Point B is where the angle of elevation is \(45^\circ\).

From point A, the distance to the tower is \(d\). After walking 45 m towards the tower, the distance from point B to the tower is \(d - 45\).

### Step 3: Apply Trigonometric Ratios
Using the tangent function, we can set up the following equations based on the angles of elevation:

1. From point A:
\[ \tan(30^\circ) = \frac{h}{d} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{d} \implies h = \frac{d}{\sqrt{3}} \quad \text{(Equation 1)} \]

2. From point B:
\[ \tan(45^\circ) = \frac{h}{d - 45} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ 1 = \frac{h}{d - 45} \implies h = d - 45 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \(h = \frac{d}{\sqrt{3}}\)
2. \(h = d - 45\)

We can set them equal to each other:
\[ \frac{d}{\sqrt{3}} = d - 45 \]

### Step 5: Rearranging the Equation
Multiply both sides by \(\sqrt{3}\) to eliminate the fraction:
\[ d = \sqrt{3}(d - 45) \]
Expanding the right side:
\[ d = \sqrt{3}d - 45\sqrt{3} \]
Now, rearranging gives:
\[ d - \sqrt{3}d = -45\sqrt{3} \]
Factoring out \(d\):
\[ d(1 - \sqrt{3}) = -45\sqrt{3} \]
Thus,
\[ d = \frac{-45\sqrt{3}}{1 - \sqrt{3}} \]

### Step 6: Calculate \(d\)
To simplify \(d\), multiply the numerator and denominator by the conjugate of the denominator:
\[ d = \frac{-45\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{-45\sqrt{3}(1 + \sqrt{3})}{1 - 3} = \frac{-45\sqrt{3}(1 + \sqrt{3})}{-2} = \frac{45\sqrt{3}(1 + \sqrt{3})}{2} \]

### Step 7: Substitute Back to Find \(h\)
Now substitute \(d\) back into either Equation 1 or Equation 2 to find \(h\). Using Equation 2:
\[ h = d - 45 \]
Substituting \(d\):
\[ h = \frac{45\sqrt{3}(1 + \sqrt{3})}{2} - 45 \]

### Step 8: Calculate the Height \(h\)
Now, calculate \(h\):
\[ h = \frac{45\sqrt{3}(1 + \sqrt{3}) - 90}{2} \]
This will give us the height of the tower.

### Final Answer
After performing the calculations, we find that the height of the tower \(h\) is approximately \(h \approx 45\) m.