At the foot of a mountain the elevation of its summit is `45^(@)`. After ascending 600 meter towards the mountain, upon an incline of `30^(@)`, the elevation changes to `60^(@)`. Find the height of the mountain.

Answer

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

### Step 1: Understand the scenario
We have a mountain where the angle of elevation to the summit from the foot of the mountain is \(45^\circ\). After ascending 600 meters at an incline of \(30^\circ\), the angle of elevation changes to \(60^\circ\).

### Step 2: Set up the diagram
Let's denote:
- \(A\) as the foot of the mountain,
- \(B\) as the point after ascending 600 meters,
- \(C\) as the summit of the mountain.

### Step 3: Use trigonometric ratios
1. From point \(A\) (foot of the mountain), the height \(h\) of the mountain can be expressed using the tangent of the angle of elevation:
\[ \tan(45^\circ) = \frac{h}{d} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ h = d \]
where \(d\) is the horizontal distance from point \(A\) to the base of the mountain directly below point \(C\).

2. From point \(B\) (after ascending), the height of the mountain can be expressed using the angle of elevation of \(60^\circ\):
\[ \tan(60^\circ) = \frac{h - 600 \sin(30^\circ)}{d - 600 \cos(30^\circ)} \]
Since \(\tan(60^\circ) = \sqrt{3}\) and \(\sin(30^\circ) = \frac{1}{2}\) and \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), we can substitute these values.

### Step 4: Substitute values
1. From point \(B\):
\[ \sqrt{3} = \frac{h - 600 \cdot \frac{1}{2}}{d - 600 \cdot \frac{\sqrt{3}}{2}} \]
Simplifying gives:
\[ \sqrt{3} = \frac{h - 300}{d - 300\sqrt{3}} \]

### Step 5: Solve the equations
1. We have two equations:
- \(h = d\)
- \(\sqrt{3}(d - 300\sqrt{3}) = h - 300\)

2. Substitute \(h = d\) into the second equation:
\[ \sqrt{3}(d - 300\sqrt{3}) = d - 300 \]
Expanding gives:
\[ \sqrt{3}d - 300 \cdot 3 = d - 300 \]
\[ \sqrt{3}d - d = 900 - 300 \]
\[ (\sqrt{3} - 1)d = 600 \]
\[ d = \frac{600}{\sqrt{3} - 1} \]

### Step 6: Calculate height
1. Now, substituting \(d\) back to find \(h\):
\[ h = d = \frac{600}{\sqrt{3} - 1} \]

2. To rationalize the denominator:
\[ h = \frac{600(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{600(\sqrt{3} + 1)}{3 - 1} = 300(\sqrt{3} + 1) \]

### Final Answer
The height of the mountain is:
\[ h = 300(\sqrt{3} + 1) \text{ meters} \]