Question 1170834: The angle between the coastline and
Johnson’s boat at the first lighthouse is 43°
From the second lighthouse, the angle between the coastline and Johnson’s boat is 55° How much shorter is the direct distance from the boat to the second lighthouse than the direct distance to the first lighthouse? [4]
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! To solve this problem, we need to use trigonometry and the Law of Sines. Let's break it down step by step.
**1. Visualize the Problem**
Imagine a triangle formed by:
* The boat's position
* The first lighthouse
* The second lighthouse
Let's label the points:
* B = Boat
* L1 = First lighthouse
* L2 = Second lighthouse
We are given:
* ∠BL1L2 = 43°
* ∠BL2L1 = 55°
* The distance between the lighthouses (L1L2) is 1.5 miles.
**2. Find the Third Angle**
The sum of the angles in a triangle is 180°. Therefore:
* ∠L1BL2 = 180° - 43° - 55° = 82°
**3. Use the Law of Sines**
The Law of Sines states:
a / sin(A) = b / sin(B) = c / sin(C)
Where:
* a, b, c are the sides of the triangle
* A, B, C are the opposite angles
Let's find the distances from the boat to each lighthouse:
* **Distance BL1 (distance to the first lighthouse):**
* BL1 / sin(55°) = L1L2 / sin(82°)
* BL1 = (L1L2 * sin(55°)) / sin(82°)
* BL1 = (1.5 * sin(55°)) / sin(82°)
* BL1 ≈ (1.5 * 0.81915) / 0.99027
* BL1 ≈ 1.2407 miles
* **Distance BL2 (distance to the second lighthouse):**
* BL2 / sin(43°) = L1L2 / sin(82°)
* BL2 = (L1L2 * sin(43°)) / sin(82°)
* BL2 = (1.5 * sin(43°)) / sin(82°)
* BL2 ≈ (1.5 * 0.68199) / 0.99027
* BL2 ≈ 1.0326 miles
**4. Find the Difference**
* Difference = BL1 - BL2
* Difference ≈ 1.2407 - 1.0326
* Difference ≈ 0.2081 miles
**5. Convert to Feet**
* 1 mile = 5280 feet
* 0.2081 miles * 5280 feet/mile ≈ 1098.768 feet
**6. Round to the Nearest Foot**
* 1099 feet
**Answer:**
The direct distance from the boat to the second lighthouse is approximately 1099 feet shorter than the direct distance to the first lighthouse.
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