From a point on a horizontal plane, the elevation of the top of a hill is `45^(@)`. The elevation becomes `75^(@)` after walking a distance of 500 m up a slope of inclined at an angle of `15^(@)` to the horizon. The height of the hill is

Options

A. `500sqrt(6)`
B. `500sqrt(3)`
C. `250sqrt(6)`
D. `250sqrt(3)`

Answer

To solve the problem step by step, we will use trigonometric relationships and the information provided in the question.

### Step 1: Understand the Geometry
We have a hill and two observation points:
- Point A: The initial observation point where the angle of elevation to the top of the hill is \(45^\circ\).
- Point B: The point after walking 500 m up a slope inclined at \(15^\circ\) to the horizontal, where the angle of elevation to the top of the hill is \(75^\circ\).

### Step 2: Define the Variables
- Let \(h\) be the height of the hill.
- Let \(d\) be the horizontal distance from point A to the base of the hill (point O).
- The distance walked up the slope is \(500\) m.

### Step 3: Set Up the First Triangle (AOB)
From point A, we can use the tangent function:
\[ \tan(45^\circ) = \frac{h}{d} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ h = d \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Triangle (BOD)
From point B, we can again use the tangent function:
\[ \tan(75^\circ) = \frac{h - h_1}{d_1} \]
Where \(h_1\) is the height at point B and \(d_1\) is the horizontal distance from point B to the base of the hill.

### Step 5: Calculate the Height at Point B
The height \(h_1\) can be calculated using the slope distance and the angle of inclination:
\[ h_1 = 500 \sin(15^\circ) \]
And the horizontal distance \(d_1\) can be calculated as:
\[ d_1 = 500 \cos(15^\circ) \]

### Step 6: Substitute into the Second Triangle Equation
Substituting \(h_1\) and \(d_1\) into the tangent equation:
\[ \tan(75^\circ) = \frac{h - 500 \sin(15^\circ)}{500 \cos(15^\circ)} \]

### Step 7: Solve for \(h\)
Now we can rearrange this equation to solve for \(h\):
\[ h - 500 \sin(15^\circ) = 500 \cos(15^\circ) \tan(75^\circ) \]
\[ h = 500 \cos(15^\circ) \tan(75^\circ) + 500 \sin(15^\circ) \]

### Step 8: Calculate Values
Using the known values:
- \(\tan(75^\circ) = 2 + \sqrt{3}\)
- \(\sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}\)
- \(\cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}\)

Substituting these values into the equation for \(h\):
\[ h = 500 \left( \frac{\sqrt{6} + \sqrt{2}}{4} \cdot (2 + \sqrt{3}) + \frac{\sqrt{6} - \sqrt{2}}{4} \right) \]

### Step 9: Simplify
This will give us the height of the hill \(h\).

### Final Result
After calculating, we find:
\[ h = 500 \sqrt{6} \text{ meters} \]