Solving the Platform Height Problem

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

Understanding the Setup: Platform, Tower, and Angle of Elevation

Imagine a vertical platform and a vertical tower standing on the ground. The horizontal distance between their bases is given. An observer is at the very top of the platform. The angle of elevation is the angle measured upwards from the observer's horizontal line of sight to the top of the tower.

• Height of the tower = 47 m
• Horizontal distance between platform and tower = 40 m
• Angle of elevation from the top of the platform to the top of the tower = 45°
• We need to find the height of the platform.

Visualizing with a Right-Angled Triangle

We can model this situation using a right-angled triangle. The vertices of this triangle would be:

1. The top of the platform (where the observer is).
2. The top of the tower.
3. A point directly below the top of the tower, but at the same horizontal level as the top of the platform.

In this triangle:

• The base is the horizontal distance between the platform and the tower, which is 40 m. This is the side adjacent to the angle of elevation.
• The vertical side is the difference in height between the top of the tower and the top of the platform. This is the side opposite to the angle of elevation.
• The angle of elevation (45°) is one of the acute angles.

Applying Trigonometry (SOH CAH TOA)

We have the angle of elevation (45°), the adjacent side (horizontal distance), and we want to find the opposite side (height difference). The trigonometric ratio that relates the opposite side and the adjacent side is the tangent function.

The formula is:

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

In our case:

\( \tan(45^\circ) = \frac{\text{Height difference}}{\text{Horizontal distance}} \)

Calculating the Height Difference

We know that \( \tan(45^\circ) = 1 \).

Let the height of the platform be \(h\) meters.

The height of the tower above the level of the top of the platform is \(47 - h\) meters.

The horizontal distance is 40 m.

Plugging these values into the tangent formula:

\( \tan(45^\circ) = \frac{47 - h}{40} \)

\( 1 = \frac{47 - h}{40} \)

Solving for the Height of the Platform

Now, we solve this equation for \(h\):

1. Multiply both sides by 40:
   \( 1 \times 40 = 47 - h \)
   \( 40 = 47 - h \)
2. Add \(h\) to both sides:
   \( 40 + h = 47 \)
3. Subtract 40 from both sides:
   \( h = 47 - 40 \)
4. Calculate the result:
   \( h = 7 \)

So, the height of the platform is 7 meters.

Final Answer for Platform Height

Based on the calculations using the angle of elevation and trigonometric principles, the height of the platform is 7 meters.

Revision Table: Key Concepts for Height and Distance Problems

Additional Information: Understanding Angles of Elevation and Depression

Angles of elevation and depression are crucial concepts in trigonometry applications, especially in problems involving heights and distances. Both angles are always measured relative to a horizontal line.

• Angle of Elevation: When you look up at an object above your horizontal line of sight, the angle between your horizontal line of sight and your line of sight to the object is called the angle of elevation.
• Angle of Depression: When you look down at an object below your horizontal line of sight, the angle between your horizontal line of sight and your line of sight to the object is called the angle of depression.

In problems like this one, drawing a clear diagram helps identify the right triangle and correctly apply the trigonometric ratios (sine, cosine, or tangent) based on which sides and angles are known or need to be found.