Understanding the Problem: Finding Tower Height Using Angle of Elevation

This problem involves using trigonometry to find the height of a tower. We are given the height of a platform, the horizontal distance from the platform to the tower, and the angle of elevation from the top of the platform to the top of the tower.

Let's break down the given information:

• Height of the platform = 7 m
• Horizontal distance between the platform and the tower = \(50\sqrt{3}\) m
• Angle of elevation from the top of the platform to the top of the tower = 30°

We need to find the total height of the tower.

Visualizing the Scenario and Setting up the Geometry

Imagine the platform and the tower standing vertically on the ground. The horizontal distance between them forms the base. From the top of the 7m high platform, a line of sight to the top of the tower creates the angle of elevation. If we draw a horizontal line from the top of the platform towards the tower, this line will be parallel to the ground and \(50\sqrt{3}\) m long. This horizontal line and the vertical segment representing the part of the tower above the platform's height form a right-angled triangle.

Let:

• H be the total height of the tower.
• h be the height of the platform (7 m).
• x be the horizontal distance between the platform and the tower (\(50\sqrt{3}\) m).
• y be the height of the tower above the top of the platform.

The total height of the tower is the sum of the platform's height and the additional height 'y', i.e., \(H = h + y\).

The angle of elevation (30°) is formed in the right-angled triangle between the horizontal line from the platform top and the line of sight to the tower top. In this triangle:

• The side opposite the 30° angle is 'y' (the height above the platform top).
• The side adjacent to the 30° angle is 'x' (the horizontal distance, \(50\sqrt{3}\) m).

Using Trigonometry to Find the Additional Height

We can use the tangent function, which relates the opposite side and the adjacent side in a right-angled triangle:

\(\tan(\text{angle of elevation}) = \frac{\text{Opposite side}}{\text{Adjacent side}}\)

In our case:

\(\tan(30^\circ) = \frac{y}{x}\)

Substitute the known values:

\(\tan(30^\circ) = \frac{y}{50\sqrt{3}}\)

We know that the value of \(\tan(30^\circ)\) is \(\frac{1}{\sqrt{3}}\).

So, the equation becomes:

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

Now, we can solve for 'y':

\(y = \frac{1}{\sqrt{3}} \times 50\sqrt{3}\)

\(y = 50\) m

So, the height of the tower above the platform's top is 50 m.

Calculating the Total Height of the Tower

The total height of the tower is the height of the platform plus the height calculated above:

\(H = h + y\)

\(H = 7 \, \text{m} + 50 \, \text{m}\)

\(H = 57 \, \text{m}\)

Thus, the total height of the tower is 57 m.

Summary of Steps

1. Identify the known values: platform height, horizontal distance, angle of elevation.
2. Recognize that the problem forms a right-angled triangle above the platform height.
3. Use the tangent function relating the angle of elevation, the unknown height above the platform (opposite side), and the horizontal distance (adjacent side).
4. Substitute the values and solve for the unknown height.
5. Add the platform height to the calculated height to find the total tower height.

Revision Table: Key Trigonometry Concepts for Height and Distance Problems

Additional Information: Solving Height and Distance Problems

Problems involving heights and distances often require drawing a diagram to represent the situation correctly. Identifying the right-angled triangle is crucial. Based on what is given (angles, sides) and what needs to be found, you choose the appropriate trigonometric ratio (sin, cos, tan).

Remember common trigonometric values for standard angles like 0°, 30°, 45°, 60°, and 90°. In this problem, the value of \(\tan(30^\circ)\) was essential.

• \(\sin(30^\circ) = 1/2\)
• \(\cos(30^\circ) = \sqrt{3}/2\)
• \(\tan(30^\circ) = 1/\sqrt{3}\)
• \(\sin(45^\circ) = 1/\sqrt{2}\)
• \(\cos(45^\circ) = 1/\sqrt{2}\)
• \(\tan(45^\circ) = 1\)
• \(\sin(60^\circ) = \sqrt{3}/2\)
• \(\cos(60^\circ) = 1/2\)
• \(\tan(60^\circ) = \sqrt{3}\)

Always ensure your calculations are accurate and pay attention to the units of measurement.