The angle of elevation of the top of a tower from a point 20 m away from its base is 45 °. What is the height of the tower?
20 m
Calculating Tower Height using Angle of Elevation
This problem requires us to find the height of a tower given the angle of elevation from a point on the ground and the distance of that point from the base of the tower. This is a classic application of trigonometry, specifically dealing with right-angled triangles.
Understanding the Problem Setup
Imagine the tower standing vertically on the ground. The point on the ground from where the angle of elevation is measured, the base of the tower, and the top of the tower form a right-angled triangle. Let's define the parts of this triangle:
- The height of the tower is the side opposite the angle of elevation.
- The distance from the point on the ground to the base of the tower is the side adjacent to the angle of elevation.
- The line of sight from the point on the ground to the top of the tower is the hypotenuse.
In this problem, we are given:
- Distance from the base (adjacent side) = 20 m
- Angle of elevation (\(\theta\)) = 45°
- Height of the tower (opposite side) = ?
Applying Trigonometry to Find Tower Height
We need a trigonometric ratio that relates the opposite side and the adjacent side of a right-angled triangle. The tangent function (\(\tan\)) is defined as the ratio of the length of the opposite side to the length of the adjacent side for a given angle in a right-angled triangle.
The formula is:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
In our case:
- Opposite side = Height of the tower (let's call it \(h\))
- Adjacent side = Distance from the base (20 m)
- Angle (\(\theta\)) = 45°
So, we can write the equation as:
\[ \tan(45^\circ) = \frac{h}{20 \text{ m}} \]
Solving for the Height
We know the value of \(\tan(45^\circ)\) from trigonometric tables or common trigonometric values. \(\tan(45^\circ) = 1\).
Substitute this value into the equation:
\[ 1 = \frac{h}{20} \]
To find \(h\), multiply both sides of the equation by 20:
\[ h = 1 \times 20 \]
\[ h = 20 \text{ m} \]
Therefore, the height of the tower is 20 meters.
Final Answer
Based on the calculation using the angle of elevation and the distance from the base, the height of the tower is 20 m.
| Given Information | Value |
|---|---|
| Distance from Base (Adjacent) | 20 m |
| Angle of Elevation (\(\theta\)) | 45° |
| Trigonometric Ratio Used | \(\tan(\theta) = \text{Opposite} / \text{Adjacent}\) |
Revision Table: Trigonometric Ratios
| Ratio | Definition (in a Right Triangle) |
|---|---|
| Sine (\(\sin(\theta)\)) | \(\text{Opposite} / \text{Hypotenuse}\) |
| Cosine (\(\cos(\theta)\)) | \(\text{Adjacent} / \text{Hypotenuse}\) |
| Tangent (\(\tan(\theta)\)) | \(\text{Opposite} / \text{Adjacent}\) |
Additional Information: Angles of Elevation and Depression
Angle of Elevation: The angle formed by the line of sight and the horizontal line when the observer is looking upwards at an object.
Angle of Depression: The angle formed by the line of sight and the horizontal line when the observer is looking downwards at an object.
Both angles are measured from a horizontal line. In geometry 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 they are in the same vertical plane.
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