At the foot of a mountain the elevation of its summit is `45 ^∘` , after ascending 2km towards the mountain up a slope of `30^∘` inclination, the elevation is found to be `60^∘` .Find the height of the mountain.

Answer

To solve the problem, we will break it down step by step.

### Step 1: Understand the problem and create a diagram
We have a mountain with a summit that has an elevation angle of \(45^\circ\) from point A (the foot of the mountain). After ascending 2 km towards the mountain at a slope of \(30^\circ\), the elevation angle from point P (the new position after ascending) is \(60^\circ\). We need to find the height of the mountain (OB).

### Step 2: Set up the diagram
Let:
- \(O\) be the summit of the mountain.
- \(A\) be the foot of the mountain.
- \(B\) be the point directly below the summit on the ground.
- \(P\) be the point after ascending 2 km.

### Step 3: Use triangle AOB
In triangle \(AOB\):
- The angle \(AOB = 45^\circ\).
- Using the tangent function:
\[ \tan(45^\circ) = \frac{OB}{AO} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ OB = AO \]
Let \(OB = h\). Thus, \(AO = h\).

### Step 4: Use triangle AKP
In triangle \(AKP\):
- The angle \(KAP = 30^\circ\).
- The length \(AK = 2\) km.
- Using the sine function:
\[ \sin(30^\circ) = \frac{KP}{2} \]
Since \(\sin(30^\circ) = \frac{1}{2}\), we have:
\[ \frac{1}{2} = \frac{KP}{2} \implies KP = 1 \text{ km} \]

### Step 5: Find the height of point O from point P
From the diagram, we can see that:
- \(OD = KP = 1\) km.
- Therefore, \(AP = AO - OP\).
- Let \(AP = h - \sqrt{3}\) (as derived from the cosine function).

### Step 6: Use triangle BKD
In triangle \(BKD\):
- The angle \(KBD = 60^\circ\).
- Using the tangent function:
\[ \tan(60^\circ) = \frac{h - 1}{h - \sqrt{3}} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{h - 1}{h - \sqrt{3}} \]

### Step 7: Cross-multiply and solve for h
Cross-multiplying gives:
\[ \sqrt{3}(h - \sqrt{3}) = h - 1 \]
Expanding:
\[ \sqrt{3}h - 3 = h - 1 \]
Rearranging:
\[ \sqrt{3}h - h = 2 \implies h(\sqrt{3} - 1) = 2 \]
Thus,
\[ h = \frac{2}{\sqrt{3} - 1} \]

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

### Step 9: Calculate the numerical value
Using \(\sqrt{3} \approx 1.732\):
\[ h \approx 1.732 + 1 = 2.732 \text{ km} \]

### Final Answer
The height of the mountain is approximately \(2.732\) km.

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