Understanding the Problem: Finding Hill Height

This problem involves using trigonometry to find the height of a hill. We are given the height of a building and the angles of elevation to the top of the hill observed from both the top and the bottom of that building. We need to express the hill's height in terms of the building's height and the given angles of elevation.

Setting up the Geometry

Let's visualize the scenario. Imagine a building standing on horizontal ground, and a hill some distance away. The top of the hill is observed from two points on the building: its base and its top.

• Let the height of the building be \(h\).
• Let the height of the hill be \(H\). This is what we need to find.
• Let the horizontal distance between the building and the hill be \(x\).

We have two angles of elevation:

• The angle of elevation from the bottom of the building to the top of the hill is \(q\).
• The angle of elevation from the top of the building to the top of the hill is \(p\).

Since the observation from the top of the building is from a higher point, the angle of elevation \(p\) must be less than the angle of elevation \(q\) (assuming the hill is taller than the building, which is implied by having positive elevation angles from the top of the building).

Applying Trigonometry

Let's set up right-angled triangles based on the given information.

Consider the observation from the bottom of the building:

• We have a right-angled triangle formed by the bottom of the building, the base of the hill, and the top of the hill.
• The opposite side is the height of the hill, \(H\).
• The adjacent side is the horizontal distance between the building and the hill, \(x\).
• The angle of elevation is \(q\).

Using the tangent ratio:

\(\tan q = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H}{x}\)

From this, we can express the horizontal distance \(x\) in terms of \(H\) and \(\cot q\):

\(x = \frac{H}{\tan q} = H \cot q \quad \text{(Equation 1)}\)

Now, consider the observation from the top of the building:

• We draw a horizontal line from the top of the building towards the hill.
• This forms a right-angled triangle where the vertex at the top of the building is the observation point.
• The vertical side opposite the angle \(p\) is the height of the hill above the top of the building. This height is \(H - h\).
• The horizontal side adjacent to the angle \(p\) is the same horizontal distance \(x\) between the building and the hill.
• The angle of elevation is \(p\).

Using the tangent ratio in this triangle:

\(\tan p = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H - h}{x}\)

From this, we can express the horizontal distance \(x\) in terms of \(H - h\) and \(\cot p\):

\(x = \frac{H - h}{\tan p} = (H - h) \cot p \quad \text{(Equation 2)}\)

Solving for the Height of the Hill

We now have two expressions for the horizontal distance \(x\). We can equate them:

\(H \cot q = (H - h) \cot p\)

Now, we need to solve this equation for \(H\). Let's expand the right side:

\(H \cot q = H \cot p - h \cot p\)

Collect the terms involving \(H\) on one side:

\(h \cot p = H \cot p - H \cot q\)

\(h \cot p = H (\cot p - \cot q)\)

Finally, isolate \(H\) by dividing by \((\cot p - \cot q)\):

\(H = \frac{h \cot p}{\cot p - \cot q}\)

This formula gives the height of the hill \(H\) in terms of the height of the building \(h\) and the angles of elevation \(p\) and \(q\).

Checking the Options

Let's compare our derived formula with the given options:

• Option 1: \(\frac{{h\cot q}}{{\cot q - \cot p}}\)
• Option 2: \(\frac{{h\cot p}}{{\cot p - \cot q}}\)
• Option 3: \(\frac{{2h\tan p}}{{\tan p - \tan q}}\)
• Option 4: \(\frac{{2h\tan q\;}}{{\tan q - \tan p}}\)

Our derived formula is \(\frac{{h\cot p}}{{\cot p - \cot q}}\), which matches Option 2.

Summary of the Solution Process

Revision Table: Key Trigonometry Concepts

To solve problems like finding the height of a hill or distances using angles of elevation and depression, it is crucial to understand basic trigonometric ratios (SOH CAH TOA) and how they apply to right-angled triangles. The cotangent function is the reciprocal of the tangent function, i.e., \(\cot \theta = \frac{1}{\tan \theta}\) or \(\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}}\).

Additional Information: Angles of Elevation and Depression

Angle of Elevation: This is the angle between the horizontal line from the observer's eye to an object and the line of sight to the object, when the object is above the horizontal line. It's always measured upwards from the horizontal.

Angle of Depression: This is the angle between the horizontal line from the observer's eye to an object and the line of sight to the object, when the object is below the horizontal line. It's always measured downwards from the horizontal.

In this problem, both angles \(p\) and \(q\) are angles of elevation, observed from horizontal lines extending from the top and bottom of the building towards the hill.