Understanding the Angle of Elevation Problem

This problem involves trigonometry, specifically the concept of the angle of elevation. We are given the heights of a platform and a tower, and the angle of elevation from the top of the platform to the top of the tower. We need to find the horizontal distance between the platform and the tower.

Setting up the Geometry

Let's visualize the scenario. We have two vertical lines representing the platform and the tower, standing on a horizontal ground. The observer is at the top of the platform. The line of sight goes from the top of the platform to the top of the tower. The angle of elevation is formed between the horizontal line from the top of the platform and the line of sight upwards to the top of the tower.

• Height of the platform = 5 m
• Height of the tower = 45 m
• Angle of elevation from the top of the platform to the top of the tower = 30°

We are looking for the horizontal distance between the platform and the tower.

Forming a Right-Angled Triangle

Draw a horizontal line from the top of the platform parallel to the ground, extending towards the tower. This line, the vertical segment from this line up to the top of the tower, and the line of sight form a right-angled triangle. The right angle is where the horizontal line meets the vertical line representing the tower.

• The angle of elevation (30°) is one of the acute angles in this triangle.
• The horizontal distance we want to find is the base of this triangle (the side adjacent to the 30° angle).
• The vertical side of this triangle is the difference in height between the top of the tower and the horizontal line originating from the top of the platform.

Calculating the Height Difference

The top of the platform is 5 m above the ground. The top of the tower is 45 m above the ground. The vertical height difference relevant to the triangle is the height of the tower above the level of the top of the platform.

Vertical height difference = Height of tower - Height of platform

Vertical height difference = \(45 \text{ m} - 5 \text{ m} = 40 \text{ m}\)

This 40 m is the length of the side opposite to the 30° angle in our right-angled triangle.

Using Trigonometry to Find the Distance

In the right-angled triangle, we know:

• Angle = 30°
• Opposite side = 40 m
• Adjacent side = Horizontal distance (what we need to find)

The trigonometric function that relates the opposite side and the adjacent side to an angle is the tangent function:

\(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

Substituting the values:

\(\tan(30^\circ) = \frac{40 \text{ m}}{\text{Horizontal Distance}}\)

We know the value of \(\tan(30^\circ)\):

\(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)

So, the equation becomes:

\(\frac{1}{\sqrt{3}} = \frac{40}{\text{Horizontal Distance}}\)

Solving for the Horizontal Distance

To find the Horizontal Distance, we can rearrange the equation:

\(\text{Horizontal Distance} = 40 \times \sqrt{3}\)

\(\text{Horizontal Distance} = 40\sqrt{3} \text{ m}\)

The horizontal distance away from the tower that the platform was positioned is \(40\sqrt{3}\) m.

Summary of Calculation Steps

• Identified the heights of the platform and tower.
• Calculated the vertical height difference relevant to the triangle.
• Set up the trigonometric equation using the tangent function.
• Used the known value of \(\tan(30^\circ)\).
• Solved the equation for the horizontal distance.

Revision Table: Key Trigonometry Concepts

Additional Information: Angles of Elevation and Depression

Angles of elevation and depression are crucial concepts in trigonometry used to solve problems involving heights and distances.

• Angle of Elevation: This is the angle measured upwards from a horizontal line to the line of sight to an object above the observer. Imagine looking straight ahead (horizontal) and then tilting your eyes up to see something; the angle you tilted is the angle of elevation.
• Angle of Depression: This is the angle measured downwards from a horizontal line to the line of sight to an object below the observer. Imagine looking straight ahead (horizontal) and then tilting your eyes down to see something; the angle you tilted is the angle of depression.
• In geometric problems, the angle of elevation from point A to point B is equal to the angle of depression from point B to point A, assuming A and B are at different heights and horizontally separated. This is because they are alternate interior angles when a transversal line (the line of sight) intersects two parallel horizontal lines (one at A's level, one at B's level).
• These angles, along with measured distances or heights, allow us to use sine, cosine, or tangent ratios in right-angled triangles to find unknown distances or heights.