Solving the Height of the Hill Problem using Trigonometry

This problem involves understanding angles of elevation and applying trigonometric ratios to find the height of a hill relative to the height of a building.

Let's define the variables:

• Let H be the height of the hill.
• Let h be the height of the building.
• Let x be the horizontal distance between the building and the hill.

We are given two angles of elevation to the top of the hill:

1. From the bottom of the building, the angle of elevation is $\pi/3$ (which is $60^{\circ}$).
2. From the top of the building, the angle of elevation is $\pi/6$ (which is $30^{\circ}$).

Consider the right-angled triangle formed by the bottom of the building, the base of the hill, and the top of the hill. Using the angle of elevation from the bottom of the building:

$\tan(\pi/3) = \frac{\text{Height of the hill}}{\text{Horizontal distance}}$

$\tan(\pi/3) = \frac{H}{x}$

Since $\tan(\pi/3) = \sqrt{3}$, we have:

$\sqrt{3} = \frac{H}{x} \quad (Equation\ 1)$

Now, consider the observation from the top of the building. The observer is at a height h above the ground. The height of the hill above the observer's level is H - h. The horizontal distance remains x. Using the angle of elevation from the top of the building:

$\tan(\pi/6) = \frac{\text{Height of the hill above the observer}}{\text{Horizontal distance}}$

$\tan(\pi/6) = \frac{H - h}{x}$

Since $\tan(\pi/6) = \frac{1}{\sqrt{3}}$, we have:

$\frac{1}{\sqrt{3}} = \frac{H - h}{x} \quad (Equation\ 2)$

Now we have a system of two equations with two variables (H and x):

1. $\sqrt{3} = \frac{H}{x}$
2. $\frac{1}{\sqrt{3}} = \frac{H - h}{x}$

From Equation 1, we can express x in terms of H:

$x = \frac{H}{\sqrt{3}}$

Substitute this expression for x into Equation 2:

$\frac{1}{\sqrt{3}} = \frac{H - h}{\frac{H}{\sqrt{3}}}$

Simplify the right side:

$\frac{1}{\sqrt{3}} = \frac{\sqrt{3}(H - h)}{H}$

Multiply both sides by $\sqrt{3}$:

$1 = \frac{3(H - h)}{H}$

Multiply both sides by H:

$H = 3(H - h)$

Distribute the 3 on the right side:

$H = 3H - 3h$

Rearrange the terms to solve for H:

$3h = 3H - H$

$3h = 2H$

$H = \frac{3h}{2}$

Thus, the height of the hill is $\frac{3h}{2}$.

Let's summarise the steps:

Revision Table: Key Trigonometric Values

Additional Information: Angles of Elevation

An angle of elevation is the angle formed by a horizontal line and the line of sight to an object above the horizontal line. It is measured upwards from the horizontal line.

In problems involving heights and distances, the angle of elevation is crucial for setting up trigonometric equations using sine, cosine, or tangent, depending on the known and unknown sides of the right-angled triangle formed.

Understanding the basic trigonometric ratios (SOH CAH TOA) is fundamental:

• $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

In this problem, we used the tangent function because we related the vertical height (opposite side) to the horizontal distance (adjacent side).