From the top of a platform 5 m high, the angle of elevation of a tower was 30°. If the platform was positioned 40√3 m away from the tower, how tall was the tower?
45 m
Calculating Tower Height using Angle of Elevation
This problem involves using trigonometry to find the height of a tower, given the height of a platform, the distance from the platform to the tower, and the angle of elevation from the top of the platform to the top of the tower.
Understanding the Scenario
Imagine a vertical tower and a vertical platform. A horizontal line connects the base of the platform to the base of the tower. Another horizontal line can be imagined from the top of the platform towards the tower. The angle of elevation is measured upwards from this horizontal line to the top of the tower. This setup forms a right-angled triangle where:
- The horizontal distance between the platform and the tower is one leg of the triangle.
- The vertical distance from the top of the platform to the top of the tower is the other leg (the height above the platform).
- The line of sight from the top of the platform to the top of the tower is the hypotenuse.
The total height of the tower is the height of the platform plus the calculated vertical distance above the platform.
Given Information
- Height of the platform = $5 \text{ m}$
- Horizontal distance from platform to tower = $40\sqrt{3} \text{ m}$
- Angle of elevation from the top of the platform to the top of the tower = $30^\circ$
Finding the Height Above the Platform
Let $h$ be the vertical distance from the top of the platform to the top of the tower. In the right-angled triangle formed, the horizontal distance ($40\sqrt{3} \text{ m}$) is adjacent to the angle of elevation ($30^\circ$), and the height $h$ is opposite to the angle of elevation.
We can use the tangent function, which relates the opposite side and the adjacent side in a right-angled triangle:
$\tan(\text{angle}) = \frac{\text{Opposite side}}{\text{Adjacent side}}$
Plugging in the given values:
$\tan(30^\circ) = \frac{h}{40\sqrt{3}}$
We know that the value of $\tan(30^\circ)$ is $\frac{1}{\sqrt{3}}$.
So, the equation becomes:
$\frac{1}{\sqrt{3}} = \frac{h}{40\sqrt{3}}$
To find $h$, we can multiply both sides by $40\sqrt{3}$:
$h = \frac{1}{\sqrt{3}} \times 40\sqrt{3}$
$h = \frac{40\sqrt{3}}{\sqrt{3}}$
$h = 40 \text{ m}$
This is the height of the tower above the top of the platform.
Calculating the Total Height of the Tower
The total height of the tower is the sum of the height of the platform and the height calculated above the platform.
Total Height of Tower = Height of Platform + Height above Platform
Total Height of Tower = $5 \text{ m} + 40 \text{ m}$
Total Height of Tower = $45 \text{ m}$
Therefore, the total height of the tower is $45 \text{ m}$.
Summary of Calculations
| Parameter | Value |
|---|---|
| Platform Height | $5 \text{ m}$ |
| Distance to Tower | $40\sqrt{3} \text{ m}$ |
| Angle of Elevation | $30^\circ$ |
| $\tan(30^\circ)$ | $\frac{1}{\sqrt{3}}$ |
| Height above Platform ($h$) | $h = \tan(30^\circ) \times 40\sqrt{3} = \frac{1}{\sqrt{3}} \times 40\sqrt{3} = 40 \text{ m}$ |
| Total Tower Height | $5 \text{ m} + 40 \text{ m} = 45 \text{ m}$ |
Revision Table: Key Trigonometric Ratios
| Angle ($\theta$) | $\sin(\theta)$ | $\cos(\theta)$ | $\tan(\theta)$ |
|---|---|---|---|
| $0^\circ$ | $0$ | $1$ | $0$ |
| $30^\circ$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| $45^\circ$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | $1$ |
| $60^\circ$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| $90^\circ$ | $1$ | $0$ | Undefined |
Additional Information: Angles of Elevation and Depression
Angles of elevation and depression are key 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 a point above the observer. In this problem, the angle of elevation is from the horizontal line at the top of the platform to the top of the tower.
- Angle of Depression: This is the angle measured downwards from a horizontal line to a point below the observer.
Both angles are always measured with respect to a horizontal line. These angles are often used with trigonometric ratios (sine, cosine, tangent) in right-angled triangles to find unknown lengths or angles.
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