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What is an Example of a Problem on a Lighthouse?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
A problem on a lighthouse usually involves using trigonometry to find unknown distances or heights. It often relates to the angle of elevation (looking up) or angle of depression (looking down) from the lighthouse to a ship or object. These problems help us understand how to measure things that are hard to reach directly.
Simple Example
Quick Example
Imagine a lighthouse keeper sees a ship in the distance. If the lighthouse is 50 meters tall and the keeper knows the angle of depression to the ship is 30 degrees, they can calculate how far the ship is from the base of the lighthouse. It's like finding the distance to your friend's house if you know your height and the angle you look down to see their gate.
Worked Example
Step-by-Step
PROBLEM: A lighthouse is 75 meters tall. From the top of the lighthouse, a ship is observed at an angle of depression of 45 degrees. How far is the ship from the base of the lighthouse? (Assume the base of the lighthouse is at sea level and the ship is on the surface of the water.)
STEP 1: Draw a right-angled triangle. The lighthouse is the vertical side (opposite the angle of depression in the alternate interior angle). The distance from the lighthouse to the ship is the horizontal side (adjacent).
---STEP 2: Identify the given values: Height of lighthouse (Opposite side) = 75 m. Angle of depression = 45 degrees. (Remember, the angle of depression from the top of the lighthouse to the ship is equal to the angle of elevation from the ship to the top of the lighthouse, by alternate interior angles. So, the angle inside our triangle at the ship's position is 45 degrees).
---STEP 3: We need to find the distance from the lighthouse to the ship (Adjacent side). The trigonometric ratio that relates Opposite and Adjacent is Tangent (tan).
---STEP 4: Set up the equation: tan(angle) = Opposite / Adjacent. So, tan(45 degrees) = 75 / Distance.
---STEP 5: We know that tan(45 degrees) = 1.
---STEP 6: Substitute the value: 1 = 75 / Distance.
---STEP 7: Solve for Distance: Distance = 75 / 1 = 75 meters.
---ANSWER: The ship is 75 meters away from the base of the lighthouse.
Why It Matters
Understanding lighthouse problems helps in navigation, surveying, and even designing drones for delivery. Engineers use these principles to calculate distances and heights in construction, while in AI/ML, similar geometric calculations are used for object detection and self-driving cars to map their surroundings.
Common Mistakes
MISTAKE: Confusing angle of elevation with angle of depression. | CORRECTION: Angle of elevation is looking UP, angle of depression is looking DOWN. Remember they are equal when considering alternate interior angles in the triangle formed.
MISTAKE: Using the wrong trigonometric ratio (e.g., sine instead of tangent). | CORRECTION: Always identify the known sides (Opposite, Adjacent, Hypotenuse) and the side you need to find relative to the given angle. Then choose the correct SOH CAH TOA ratio.
MISTAKE: Not drawing a clear diagram. | CORRECTION: Always draw a simple, clear right-angled triangle. Label the lighthouse, ship, angles, and unknown distances. This makes it much easier to visualize the problem and apply the correct formula.
Practice Questions
Try It Yourself
QUESTION: From the top of a lighthouse 100 meters high, the angle of depression to a boat is 30 degrees. How far is the boat from the base of the lighthouse? (Use tan(30) = 1/sqrt(3) or approx 0.577) | ANSWER: Distance = 100 / tan(30) = 100 * sqrt(3) approx 173.2 meters.
QUESTION: A ship is 200 meters away from the base of a lighthouse. If the angle of elevation from the ship to the top of the lighthouse is 60 degrees, what is the height of the lighthouse? (Use tan(60) = sqrt(3) or approx 1.732) | ANSWER: Height = 200 * tan(60) = 200 * sqrt(3) approx 346.4 meters.
QUESTION: A lighthouse is 60 meters tall. Two ships are on the same side of the lighthouse. The angles of depression to the ships are 60 degrees and 30 degrees respectively. Find the distance between the two ships. | ANSWER: Let D1 be distance to ship 1, D2 to ship 2. D1 = 60/tan(60) = 60/sqrt(3) = 20*sqrt(3) approx 34.64m. D2 = 60/tan(30) = 60*sqrt(3) approx 103.92m. Distance between ships = D2 - D1 = 60*sqrt(3) - 20*sqrt(3) = 40*sqrt(3) approx 69.28 meters.
MCQ
Quick Quiz
From the top of a 50m high lighthouse, a ship is seen at an angle of depression of 60 degrees. What is the distance of the ship from the base of the lighthouse?
50 * sqrt(3) meters
50 / sqrt(3) meters
100 meters
25 meters
The Correct Answer Is:
B
Using tan(angle) = Opposite/Adjacent, we have tan(60) = 50 / Distance. Since tan(60) = sqrt(3), Distance = 50 / sqrt(3) meters.
Real World Connection
In the Real World
These calculations are vital for coastal guards and navy personnel who use radar and optical instruments to track ships. Even in modern applications like drone delivery, the drone's camera uses similar trigonometry to calculate its distance and height from a delivery spot, ensuring safe and accurate drops, much like a lighthouse guides ships.
Key Vocabulary
Key Terms
ANGLE OF ELEVATION: The angle measured upwards from a horizontal line to an object above. | ANGLE OF DEPRESSION: The angle measured downwards from a horizontal line to an object below. | TRIGONOMETRY: The branch of mathematics dealing with the relations between the sides and angles of triangles. | SOH CAH TOA: A mnemonic for the three basic trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
What's Next
What to Learn Next
Next, you can explore problems involving multiple triangles or more complex scenarios like finding the height of a building with two observation points. This will strengthen your understanding of applying trigonometry in various real-world situations, preparing you for advanced physics and engineering concepts.
