A boat heading out to sea starts out at Point AA, at a horizontal distance of 590 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 11degrees
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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 tenth of a foot if necessary.
Let's assume that the distance from point AA to point BB is x feet.
We can form a right triangle with the lighthouse beacon-light as the top vertex, point AA as the bottom left vertex, and point BB as the bottom right vertex.
In this triangle, the angle of elevation from point AA to the lighthouse beacon-light is 11 degrees, and the angle of elevation from point BB to the lighthouse beacon-light is 6 degrees.
Since we have a right triangle, we can use the tangent function to find the lengths of the triangle's legs.
In triangle AAB (where AB is the base and the angle opposite AB is 11 degrees), we can use the tangent function:
tan(11) = opp/adj = AB/590
Solving for AB:
AB = 590 * tan(11)
In triangle ABB (where AB is the base and the angle opposite AB is 6 degrees), we can use the tangent function:
tan(6) = opp/adj = AB/(AB + x)
Solving for x:
x = AB * tan(6) / (1 - tan(6))
Substituting the value of AB:
x = (590 * tan(11)) * tan(6) / (1 - tan(6))
x ≈ 152.51
So, the distance from point AA to point BB is approximately 152.51 feet.