Understanding the Problem: Height of Hill Calculation

This problem involves trigonometry, specifically using angles of elevation to find unknown heights. We have a tower and a hill, and we are given the heights of one (the tower) and angles of elevation between the top of one structure and the foot of the other.

Defining Angles of Elevation

The angle of elevation is the angle between the horizontal line from an observer's eye to an object, when the object is above the horizontal line. In this case, the horizontal line is the ground level between the tower and the hill.

• The angle of elevation of the top of the hill at the foot of the tower is 60°. This means if you stand at the base of the tower and look up at the top of the hill, the angle your line of sight makes with the horizontal ground is 60°.
• The angle of elevation of the top of the tower from the foot of the hill is 30°. This means if you stand at the base of the hill and look up at the top of the tower, the angle your line of sight makes with the horizontal ground is 30°.

We are given the height of the tower is 50 m. We need to find the height of the hill.

Setting up the Triangles

Let's imagine the situation forms two right-angled triangles. Let:

• \(H\) be the height of the hill.
• \(T\) be the height of the tower, so \(T = 50\) m.
• \(D\) be the horizontal distance between the foot of the tower and the foot of the hill.

From the problem description, we can form two right-angled triangles:

1. Triangle formed by the tower's height, the horizontal distance, and the line of sight from the foot of the hill to the top of the tower. In this triangle, the angle of elevation is 30°, the opposite side is \(T = 50\) m, and the adjacent side is \(D\).
2. Triangle formed by the hill's height, the horizontal distance, and the line of sight from the foot of the tower to the top of the hill. In this triangle, the angle of elevation is 60°, the opposite side is \(H\), and the adjacent side is \(D\).

Using Trigonometry to Solve

We can use the tangent function, which relates the opposite side and the adjacent side in a right-angled triangle: \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\).

Step 1: Find the Horizontal Distance (D)

From Triangle 1 (using the tower):

\(\tan(30^\circ) = \frac{\text{Height of Tower}}{\text{Distance Between Bases}}\)

\(\tan(30^\circ) = \frac{50}{D}\)

We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\). So,

\(\frac{1}{\sqrt{3}} = \frac{50}{D}\)

Solving for \(D\):

\(D = 50 \times \sqrt{3}\) m

Step 2: Find the Height of the Hill (H)

From Triangle 2 (using the hill):

\(\tan(60^\circ) = \frac{\text{Height of Hill}}{\text{Distance Between Bases}}\)

\(\tan(60^\circ) = \frac{H}{D}\)

We know that \(\tan(60^\circ) = \sqrt{3}\). So,

\(\sqrt{3} = \frac{H}{D}\)

Now, substitute the value of \(D\) we found in Step 1:

\(\sqrt{3} = \frac{H}{50\sqrt{3}}\)

Solving for \(H\):

\(H = \sqrt{3} \times (50\sqrt{3})\)

\(H = 50 \times (\sqrt{3} \times \sqrt{3})\)

\(H = 50 \times 3\)

\(H = 150\) m

So, the height of the hill is 150 m.

Summary of Calculation

We used the given angles of elevation and the height of the tower to first find the horizontal distance between the two structures, and then used that distance along with the other angle of elevation to find the height of the hill.

Revision Table: Trigonometry Basics

Additional Information: Applications of Elevation Angles

Angles of elevation and depression are widely used in various fields:

• Surveying: To determine the height of buildings, trees, or geographical features without physically measuring them.
• Navigation: In aerial and maritime navigation to determine positions and distances.
• Astronomy: To measure the altitude of celestial bodies.
• Engineering: In construction and design, such as planning ramps or bridges.

These angles are crucial tools for indirect measurement using the principles of trigonometry.