A ship
A ship has been spotted by two lighthouses, A and B, as shown in the figure. What is the distance from the ship to Lighthouse A to the nearest tenth? Figure - the distance between lighthouses A and B is 40 nautical miles. From A is seen in view angle 57° and from B at 64° angle.
Correct answer:
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Dr. Math
Solution:
To find the distance from the ship to Lighthouse A, we can use the Law of Sines. The given information is:
- The distance between Lighthouse A and Lighthouse B is 40 nautical miles.
- The angle at Lighthouse A is 57° .
- The angle at Lighthouse B is 64° .
Step 1: Find the angle at the ship.
The sum of angles in a triangle is 180° . Therefore, the angle at the ship ( C ) is:
Step 2: Use the Law of Sines to find the distance from the ship to Lighthouse A.
The Law of Sines states:
Here:
- a is the distance from the ship to Lighthouse B,
- b is the distance from the ship to Lighthouse A (the value we are solving for),
- c = 40 nautical miles (the distance between Lighthouse A and Lighthouse B),
- A = 57° (the angle at Lighthouse A),
- B = 64° (the angle at Lighthouse B),
- C = 59° (the angle at the ship).
Using the Law of Sines:
Solve for b :
Final Answer:
The distance from the ship to Lighthouse A is approximately:
To find the distance from the ship to Lighthouse A, we can use the Law of Sines. The given information is:
- The distance between Lighthouse A and Lighthouse B is 40 nautical miles.
- The angle at Lighthouse A is 57° .
- The angle at Lighthouse B is 64° .
Step 1: Find the angle at the ship.
The sum of angles in a triangle is 180° . Therefore, the angle at the ship ( C ) is:
C = 180° - 57° - 64° = 59°
Step 2: Use the Law of Sines to find the distance from the ship to Lighthouse A.
The Law of Sines states:
asin A = bsin B = csin C
Here:
- a is the distance from the ship to Lighthouse B,
- b is the distance from the ship to Lighthouse A (the value we are solving for),
- c = 40 nautical miles (the distance between Lighthouse A and Lighthouse B),
- A = 57° (the angle at Lighthouse A),
- B = 64° (the angle at Lighthouse B),
- C = 59° (the angle at the ship).
Using the Law of Sines:
bsin 64° = 40sin 59°
Solve for b :
b = 40 · sin 64°sin 59°
b = ≈ 41.9
Final Answer:
The distance from the ship to Lighthouse A is approximately:
41.9 nautical miles
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Do you want to convert length units?
See also our trigonometric triangle calculator.
Try conversion angle units angle degrees, minutes, seconds, radians, grads.
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