Question

A boat heading out to sea starts out at Point AA, at a horizontal distance of 734 feet from a lighthouse/the shore. From that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon-light from that point to be 14degrees

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. At some later time, the crew measures the angle of elevation from point BB to be 6degrees
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. Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

Answer 1

The distance from point AA to point BB is approximately 75 feet.

Step-by-step explanation:

To find the distance from point AA to point BB, we can use trigonometry and the concept of similar triangles.

Step 1: Let's consider the triangle formed by the boat, the lighthouse, and the line connecting point AA and point BB. Let's call this triangle ABC.

Step 2: We have the angle of elevation from point AA to the lighthouse as 14 degrees, and the angle of elevation from point BB to the lighthouse as 6 degrees. We can use these angles to find the angle of depression from point C (the lighthouse) to the line connecting point AA and point BB.

Step 3: The angle of depression from point C to the line connecting point AA and point BB can be found by subtracting the angle of elevation from 90 degrees. So the angle of depression is 90 - 14 = 76 degrees.

Step 4: Now we can use the concept of similar triangles.

Step 5: In triangle ABC, the angle at point C is 76 degrees, which is the same as the angle at point A in triangle ABB. Therefore, we have two similar triangles, and we can set up a proportion to find the distance from point AA to point BB:

(Distance from AA to BB) / (Distance from AA to lighthouse) = tan(6 degrees)

Step 6: Rearranging the equation, we get:

Distance from AA to BB = (Distance from AA to lighthouse) * tan(6 degrees)

Step 7: Plugging in the given values, we have:

Distance from AA to BB = 734 feet * tan(6 degrees)

Step 8: Calculating the value, we find that the distance from AA to BB is approximately 75 feet (rounded to the nearest foot).