Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from the ships are 45 ° and 60° respectively. If the lighthouse is 81 m high, then the distance between two ships is:
Calculating Distance Between Ships Using Lighthouse Angles
This problem involves using trigonometry, specifically the concept of angles of elevation, to find the distance between two ships on opposite sides of a lighthouse. We can model this situation using right-angled triangles.
Understanding the Setup
Imagine a vertical lighthouse standing on a horizontal base. Let the top of the lighthouse be point A and the base be point B. The height of the lighthouse is given as 81 m. Let the two ships be located at points C and D, such that B is between C and D on a straight line representing the sea level. The angles of elevation from the ships (C and D) to the top of the lighthouse (A) are 45° and 60° respectively.
We have two right-angled triangles: \(\triangle ABC\) and \(\triangle ABD\). The lighthouse AB is the common perpendicular side to the base line CD.
- Height of the lighthouse, AB = 81 m.
- Angle of elevation from ship C, \(\angle ACB = 45^\circ\).
- Angle of elevation from ship D, \(\angle ADB = 60^\circ\).
- The distance between the ships is CD = BC + BD.
Applying Trigonometry (Tangent Function)
In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
We can use the tangent function to find the distances BC and BD.
Step 1: Find the distance of the first ship (C) from the base of the lighthouse (B).
In right-angled triangle \(\triangle ABC\):
\(\tan(\angle ACB) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{AB}{BC}\)
\(\tan(45^\circ) = \frac{81}{BC}\)
We know that \(\tan(45^\circ) = 1\).
\(1 = \frac{81}{BC}\)
This gives us:
\(BC = 81\) m.
Step 2: Find the distance of the second ship (D) from the base of the lighthouse (B).
In right-angled triangle \(\triangle ABD\):
\(\tan(\angle ADB) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{AB}{BD}\)
\(\tan(60^\circ) = \frac{81}{BD}\)
We know that \(\tan(60^\circ) = \sqrt{3}\).
\(\sqrt{3} = \frac{81}{BD}\)
Solving for BD:
\(BD = \frac{81}{\sqrt{3}}\) m.
Step 3: Calculate the total distance between the two ships.
Since the ships are on opposite sides of the lighthouse, the distance between them is the sum of their distances from the base of the lighthouse:
Distance CD = BC + BD
\(CD = 81 + \frac{81}{\sqrt{3}}\)
Step 4: Simplify the expression to match the options.
To combine the terms and match the provided format, we can find a common denominator or factor out 81.
Let's factor out 81:
\(CD = 81 \left( 1 + \frac{1}{\sqrt{3}} \right)\)
Combine the terms inside the parenthesis:
\(CD = 81 \left( \frac{\sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}} \right)\)
\(CD = 81 \left( \frac{\sqrt{3} + 1}{\sqrt{3}} \right)\)
\(CD = \frac{81(1 + \sqrt{3})}{\sqrt{3}}\) m.
This expression matches one of the given options.
Summary of Distances
| Ship | Angle of Elevation | Distance from Base of Lighthouse |
|---|---|---|
| Ship C | 45° | 81 m |
| Ship D | 60° | \(\frac{81}{\sqrt{3}}\) m |
The total distance between the ships is the sum of these two distances.
Final Answer Derivation
The distance between the two ships is \(81 + \frac{81}{\sqrt{3}}\) m.
Simplifying this expression:
\(81 + \frac{81}{\sqrt{3}} = \frac{81\sqrt{3}}{\sqrt{3}} + \frac{81}{\sqrt{3}} = \frac{81\sqrt{3} + 81}{\sqrt{3}} = \frac{81(\sqrt{3} + 1)}{\sqrt{3}} = \frac{81(1 + \sqrt{3})}{\sqrt{3}}\)
So, the distance is \(\frac{81(1 + \sqrt3)}{\sqrt3}\) m.
Revision Table: Key Trigonometry Values
| Angle (θ) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|
| 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 Angles of Elevation
The angle of elevation is the angle between the horizontal line from the observer's eye to an object and the line of sight from the observer's eye to the object, when the object is above the horizontal line. In problems like this, the horizontal line is usually the ground or sea level, and the observer is implicitly at that level, looking up at the top of the object (like a lighthouse or tower). The triangle formed is a right-angled triangle where the height of the object is opposite the angle of elevation, and the horizontal distance from the observer to the base of the object is adjacent to the angle of elevation. The tangent function is particularly useful here because it directly relates the opposite and adjacent sides to the angle.
Important Questions from Heights and Distances
The horizontal distance between two towers is 40√3 m. The angle of depression of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 130 m, find the height of the first tower.
The angle of elevation of a ladder leaning against a house is 60° and the foot of the ladder is 6.5 metres from the house. The length of the ladder is
A kite is flying at a height of 50 m. If the length of the string is 100 m then the inclination of the string to the horizontal ground in degree measures is:
A. 90
B. 45
C. 60
D. 30
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A. 11 m
B. 12 m
C. 13 m
D. 14 m
An observer 2 m tall is 150\(\sqrt3\) m away from a tower. The angle of elevation from his eye to the top of the tower is 60°. The height of the tower is:
