From the top of a lighthouse, 100 m high, the angle of depression of a boat is \({\tan ^{ - 1}}\left( {\frac{5}{{12}}} \right).\) What is the distance between the boat and the lighthouse?
240 m
Understanding the Problem: Lighthouse, Boat, and Angle of Depression
The problem asks us to find the distance between a boat and the base of a lighthouse, given the height of the lighthouse and the angle of depression from the top of the lighthouse to the boat.
The angle of depression is the angle between the horizontal line of sight and the line of sight downwards to an object. When looking from the top of the lighthouse down at the boat, this angle is formed.
Relating Angle of Depression and Elevation
In this scenario, the angle of depression from the top of the lighthouse to the boat is equal to the angle of elevation from the boat to the top of the lighthouse. This is because the horizontal line from the top of the lighthouse is parallel to the water surface (where the boat is), and the line of sight acts as a transversal. These angles are alternate interior angles.
Let the angle of depression (and thus the angle of elevation) be \(\theta\). We are given that \(\theta = {\tan ^{ - 1}}\left( {\frac{5}{{12}}} \right)\).
This means that the tangent of the angle \(\theta\) is \(\tan(\theta) = \frac{5}{12}\).
Setting up the Right Triangle
We can visualize this problem as a right-angled triangle. The vertices of the triangle are:
- The top of the lighthouse.
- The base of the lighthouse.
- The position of the boat.
In this right triangle:
- The height of the lighthouse (100 m) is the side opposite to the angle of elevation (\(\theta\)) from the boat.
- The distance between the boat and the base of the lighthouse is the side adjacent to the angle of elevation (\(\theta\)) from the boat. This is what we need to find.
Using Trigonometry: The Tangent Ratio
The tangent function in a right-angled triangle relates the opposite side to the adjacent side:
\(\tan(\text{angle}) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)
In our case, with the angle of elevation \(\theta\) at the boat:
\(\tan(\theta) = \frac{\text{Height of Lighthouse}}{\text{Distance between boat and lighthouse}}\)
Step-by-Step Calculation
We know that \(\tan(\theta) = \frac{5}{12}\) and the Height of the Lighthouse is 100 m. Let 'd' be the distance between the boat and the lighthouse.
So, we have the equation:
\(\frac{5}{12} = \frac{100}{d}\)
To find 'd', we can cross-multiply:
\(5 \times d = 12 \times 100\)
\(5d = 1200\)
Now, divide by 5 to isolate 'd':
\(d = \frac{1200}{5}\)
\(d = 240\)
So, the distance between the boat and the lighthouse is 240 meters.
Final Answer
The distance between the boat and the lighthouse is 240 m.
| Quantity | Value |
|---|---|
| Height of Lighthouse | 100 m |
| Angle of Depression (\(\theta\)) | \({\tan ^{ - 1}}\left( {\frac{5}{{12}}} \right)\) |
| \(\tan(\theta)\) | \(\frac{5}{12}\) |
| Distance (d) | ? |
Revision Table: Lighthouse Distance Calculation
| Concept | Explanation | Application |
|---|---|---|
| Angle of Depression | Angle between horizontal line of sight and downward line of sight. | Used to determine angle of elevation at the boat. |
| Trigonometric Ratios | Relationships between angles and side lengths in right triangles (SOH CAH TOA). | Tangent (TOA): \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\). |
| Right Triangle | Formed by lighthouse height, distance on water, and line of sight. | Allows use of trigonometric ratios to solve for unknown side. |
Additional Information: Angles of Elevation and Depression
Angles of elevation and depression are crucial concepts in trigonometry, often used in problems involving heights and distances.
- Angle of Elevation: When you look *up* at an object, the angle between your horizontal line of sight and your line of sight up to the object is the angle of elevation.
- Angle of Depression: When you look *down* at an object, the angle between your horizontal line of sight and your line of sight down to the object is the angle of depression.
It's important to remember that the horizontal line of sight is always parallel to the ground (assuming the ground is flat). This parallelism is why the angle of depression from point A to point B is equal to the angle of elevation from point B to point A, forming alternate interior angles when the line of sight acts as a transversal.
These concepts are widely applied in surveying, navigation, physics, and engineering to calculate distances and heights indirectly.
Similar Questions
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?
The shadow of a tower is found to be x meter longer, when the angle of elevation of the sun changes from 60° to 45°. If the height of the tower is 5(3 + √3) m, then what is x equal to?
If a flag-staff of 6 m height placed on the top of a tower throws shadow of 2√3 m along the ground, then what is the angle that the sun makes with the ground?
At what height is the top of the tower above the ground level ?
A ladder 9 m long reaches a point 9 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the flagstaff is 60°. What is the height of the flagstaff?
The angle of elevation of a tower of height h from a point A due South of it is x and from a point B due East of B is y. If AB = z, then which one of the following is correct?
A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge form the balloon is 48°. If the height of the balloon above the bridge is 122 m, then what is the length of the bridge?
The top of a hill observed from the top and bottom of a building of height h is at angles of elevation π/6 and π/3 respectively. What is the height of the hill?
A spherical balloon of radius r subtends an angle α at the eye of an observer, while the angle of elevation of its centre is β. What is the height of the centre of the balloon (neglecting the height of the observer)?
What is the height of the lamp post ?
Important Questions from Heights and Distances
A peacock sitting at the top of a 3 meter high pole saw a snake approaching towards pole at a distance three times of the height of the pole. Then it jumping from pole will catch the snake at what distance from the pole if both are running with same speed ?
The foot of a ladder 25 m long is 7 m from the base of the building. If the top of the ladder slips by 4 m, then by how much distance will the foot of the ladder slide?
Two hotels stand 25 m apart. One of them is 70 m high and the angle of depression of the top of other as observed from the top of this hotel is 45°. Height of the other hotel is:
If the angles of elevation of a balloon from two consecutive kilometer-stones along a straight road are 30° and 60° respectively, then the height of the balloon above the ground will be:
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?
