From a point P on a level ground, the angle of elevation of the top of a tower is 30°. If the tower is \(110\sqrt 3\) m high, what is the distance (in m) of point P from the foot of the tower?
330
Understanding the Angle of Elevation Problem
The problem asks us to find the horizontal distance from a point on the ground to the foot of a tower, given the height of the tower and the angle of elevation from the point to the top of the tower. This scenario forms a right-angled triangle, where the tower's height is the opposite side, the distance from the point to the foot is the adjacent side, and the angle of elevation is the angle at the point on the ground.
Setting up the Geometry for Distance Calculation
Let's represent the situation with a diagram.
- Let A be the top of the tower.
- Let B be the foot of the tower on the level ground.
- Let P be the point on the level ground.
The tower AB is perpendicular to the ground BP. Thus, triangle ABP is a right-angled triangle with the right angle at B.
- The height of the tower AB is given as \(110\sqrt 3\) m.
- The angle of elevation from P to A (angle APB) is given as 30°.
- We need to find the distance BP.
Applying Trigonometry to Find the Distance
In the right-angled triangle ABP, we have:
- Opposite side to angle APB (30°) = AB (Height of the tower)
- Adjacent side to angle APB (30°) = BP (Distance from P to the foot of the tower)
The trigonometric ratio that relates the opposite side and the adjacent side is the tangent function:
\(\tan(\text{angle}) = \frac{\text{Opposite}}{\text{Adjacent}}\)
In our case:
\(\tan(30^\circ) = \frac{AB}{BP}\)
Calculation Steps for Point P Distance
We know the value of \(\tan(30^\circ)\).
\(\tan(30^\circ) = \frac{1}{\sqrt 3}\)
Substitute the given height of the tower (\(AB = 110\sqrt 3\) m) and the value of \(\tan(30^\circ)\) into the equation:
\(\frac{1}{\sqrt 3} = \frac{110\sqrt 3}{BP}\)
Now, we solve for BP:
Multiply both sides by BP:
\(BP \times \frac{1}{\sqrt 3} = 110\sqrt 3\)
Multiply both sides by \(\sqrt 3\):
\(BP = 110\sqrt 3 \times \sqrt 3\)
Since \(\sqrt 3 \times \sqrt 3 = 3\):
\(BP = 110 \times 3\)
\(BP = 330\)
The distance of point P from the foot of the tower is 330 m.
Conclusion
Using the angle of elevation and the height of the tower within a right-angled triangle framework, we calculated the distance of point P from the foot of the tower to be 330 m.
| Parameter | Value |
|---|---|
| Height of the Tower (AB) | \(110\sqrt 3\) m |
| Angle of Elevation (APB) | 30° |
| Trigonometric Ratio Used | Tangent (\(\tan\)) |
| Distance from P to Foot (BP) | Calculated as 330 m |
Revision Table: Key Trigonometric Values
It's helpful to remember common trigonometric values for angles like 30°, 45°, and 60°.
| 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 on Height and Distance
Problems involving height and distance often use trigonometry to solve real-world scenarios.
- Angle of Elevation: The angle formed by the line of sight with the horizontal when the object is above the horizontal level.
- Angle of Depression: The angle formed by the line of sight with the horizontal when the object is below the horizontal level.
- These problems typically involve solving right-angled triangles using sine, cosine, or tangent, depending on the sides and angles known and needed.
Important Questions from Heights and Distances
If x is the distance of P from the bottom of the pillar, then consider the following statements :
1. x can take two values which are in the ratio 1 : 3
2. x can be equal to the height of the flagstaff
Which of the statements given above is/are correct?
What is a possible value of tan θ ?
A vertical tower standing on a levelled field is mounted with a vertical flag staff of length 3 m. From a point on the field, the angles of elevation of the bottom and tip of the flag staff are 30° and 45° respectively. Which one of the following gives the best approximation to the height of the tower?
Two poles are 10 m and 20 m high. The line joining their tops makes an angle of 15° with the horizontal. The distance between the poles is approximately equal to
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?
