A person standing at a distance looks at a building having a height of 1000 metres. The angle between the top of the building and the ground is 30°. At what approximate distance (in metres) is the person standing away from the building.
1732
Calculating Distance Using Trigonometry
This problem involves finding the distance between a person and a building given the building's height and the angle of elevation to the top of the building. This scenario forms a right-angled triangle, where:
- The building's height is the side opposite the angle of elevation.
- The distance from the person to the building is the side adjacent to the angle of elevation.
- The angle of elevation is given as 30 degrees.
Understanding the Trigonometric Relationship
In a right-angled triangle, the relationship between the opposite side, the adjacent side, and the angle is described by trigonometric functions. The tangent function ($\tan$) relates the opposite side to the adjacent side:
$$ \tan(\text{angle}) = \frac{\text{Opposite side}}{\text{Adjacent side}} $$
Applying Trigonometry to the Building Problem
In this case:
- Angle = 30°
- Opposite side (Height of the building) = 1000 metres
- Adjacent side (Distance from the person to the building) = Let's call this 'd'
So, the equation becomes:
$$ \tan(30^\circ) = \frac{1000 \text{ m}}{d} $$
Solving for the Distance (d)
To find the distance 'd', we can rearrange the equation:
$$ d = \frac{1000 \text{ m}}{\tan(30^\circ)} $$
We know that the value of $\tan(30^\circ)$ is $\frac{1}{\sqrt{3}}$.
$$ d = \frac{1000}{\frac{1}{\sqrt{3}}} = 1000 \times \sqrt{3} $$
The approximate value of $\sqrt{3}$ is 1.732.
$$ d \approx 1000 \times 1.732 $$
$$ d \approx 1732 \text{ metres} $$
Step-by-Step Calculation
- Identify the known values: Building height (Opposite) = 1000 m, Angle of elevation = 30°.
- Identify the unknown value: Distance (Adjacent) = d.
- Choose the appropriate trigonometric function: $\tan$ because it relates Opposite and Adjacent sides.
- Set up the equation: $\tan(30^\circ) = \frac{1000}{d}$.
- Solve for d: $d = \frac{1000}{\tan(30^\circ)}$.
- Substitute the value of $\tan(30^\circ) = \frac{1}{\sqrt{3}}$: $d = 1000 \times \sqrt{3}$.
- Use the approximate value $\sqrt{3} \approx 1.732$: $d \approx 1000 \times 1.732$.
- Calculate the distance: $d \approx 1732$ metres.
Comparing with Options
The calculated approximate distance is 1732 metres, which matches one of the provided options.
| Option | Value (metres) | Match? |
|---|---|---|
| 1 | 1000 | No |
| 2 | 936 | No |
| 3 | 1732 | Yes |
| 4 | 1542 | No |
Therefore, the approximate distance the person is standing away from the building is 1732 metres.
Revision Table: Key Concepts
| Concept | Description | Relevance to Problem |
|---|---|---|
| Angle of Elevation | The angle between the horizontal line of sight and the line of sight upward to an object. | Given as 30° in the problem. |
| Right-Angled Triangle | A triangle with one angle equal to 90°. | Formed by the building, the ground, and the line of sight to the top. |
| Tangent Function ($\tan$) | In a right triangle, the ratio of the length of the opposite side to the length of the adjacent side for a given angle. | Used to relate the building's height (opposite) to the distance (adjacent). |
| Special Angles | Common angles (like 0°, 30°, 45°, 60°, 90°) with known trigonometric values. | $\tan(30^\circ) = \frac{1}{\sqrt{3}}$ is a standard value. |
Additional Information: Trigonometric Ratios
For a right-angled triangle with an angle $\theta$:
- Sine ($\sin \theta$): Ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine ($\cos \theta$): Ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent ($\tan \theta$): Ratio of the length of the opposite side to the length of the adjacent side ($\tan \theta = \frac{\sin \theta}{\cos \theta}$).
In problems involving height and distance where the hypotenuse is not directly involved (or needed), the tangent function is often very useful.
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