From a point P on a level ground, the angle of elevation of the top of the tower is 30°. If the distance of point P from the foot of the tower is 510 m, then 50% of the height of the tower (in m) is:
Finding 50% of Tower Height Using Angle of Elevation
This problem involves using trigonometry 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. We are asked to find half of the calculated height.
Understanding Angle of Elevation and the Problem Setup
The angle of elevation is the angle formed by the horizontal line of sight and the line of sight upwards to an object. In this case, the object is the top of the tower.
- Let H be the height of the tower (AB).
- Let P be the point on the level ground.
- The distance from point P to the foot of the tower (B) is given as 510 m (PB).
- The angle of elevation from P to the top of the tower (A) is 30° (∠APB).
This setup forms a right-angled triangle APB, where angle B is 90°.
Applying Trigonometry to Find Tower Height
In the right-angled triangle APB, we have:
- Opposite side to angle 30° is the height of the tower, H (AB).
- Adjacent side to angle 30° is the distance from the point to the foot of the tower, 510 m (PB).
The trigonometric ratio that relates the opposite side and the adjacent side in a right-angled triangle is the tangent function.
The formula is: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
Substituting the given values:
\(\tan(30^\circ) = \frac{H}{510}\)
Calculating the Height of the Tower
We know the value of $\tan(30^\circ)$ is $\frac{1}{\sqrt{3}}$.
So, the equation becomes:
\(\frac{1}{\sqrt{3}} = \frac{H}{510}\)
To find H, we can cross-multiply:
\(H = 510 \times \frac{1}{\sqrt{3}}\)
\(H = \frac{510}{\sqrt{3}}\)
To rationalize the denominator (remove the square root from the denominator), multiply both the numerator and the denominator by $\sqrt{3}$:
\(H = \frac{510}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}\)
\(H = \frac{510\sqrt{3}}{3}\)
Now, simplify the expression:
\(H = 170\sqrt{3} \text{ m}\)
So, the height of the tower is \(170\sqrt{3}\) meters.
Calculating 50% of the Tower Height
The question asks for 50% of the height of the tower. 50% is equivalent to $\frac{50}{100} = \frac{1}{2}$.
50% of H \( = \frac{1}{2} \times H\)
\(= \frac{1}{2} \times 170\sqrt{3}\)
\(= \frac{170\sqrt{3}}{2}\)
\(= 85\sqrt{3} \text{ m}\)
Therefore, 50% of the height of the tower is \(85\sqrt{3}\) meters.
Summary of Steps
- Identify the given angle of elevation and the distance from the point to the tower's base.
- Recognize that this forms a right-angled triangle.
- Use the tangent trigonometric ratio ($\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$) to relate the height, distance, and angle.
- Substitute the known values ($\theta = 30^\circ$, Adjacent = 510 m) into the equation.
- Solve for the height of the tower (H).
- Calculate 50% of the obtained height H.
| Parameter | Value |
|---|---|
| Angle of Elevation ($\theta$) | 30° |
| Distance from Point P to Foot of Tower | 510 m |
| Height of Tower (H) - Calculated | \(170\sqrt{3}\) m |
| 50% of Tower Height | \(85\sqrt{3}\) m |
Revision Table: Key Trigonometric Values
| Angle ($\theta$) | $\sin(\theta)$ | $\cos(\theta)$ | $\tan(\theta)$ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{3}}\) |
| 45° | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{\sqrt{2}}\) | 1 |
| 60° | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) |
| 90° | 1 | 0 | Undefined |
Additional Information: Angles of Elevation and Depression
Problems involving heights and distances often use angles of elevation and depression. These concepts are fundamental in applying trigonometry to real-world scenarios.
- Angle of Elevation: As seen in this problem, it's the angle measured upwards from the horizontal line of sight to an object above the observer.
- Angle of Depression: This is the angle measured downwards from the horizontal line of sight to an object below the observer.
Both angles are always measured relative to a horizontal line. Understanding which side is opposite or adjacent to the given angle in the right-angled triangle is crucial for correctly applying the trigonometric ratios (sine, cosine, tangent).
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