A boat is heading towards a lighthouse, whose beacon-light is 136 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 6degrees
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, before they draw closer. They measure the angle of elevation a second time from point BB at some later time to be 19degrees
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. Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.
Let's consider the right triangle formed by the boat, the lighthouse, and the point where the boat first measures the angle of elevation (point A).
In this triangle, the height of the lighthouse (H) is 136 feet and the angle of elevation (θ1) is 6 degrees.
Using trigonometry, we can relate the opposite side (H), adjacent side (x), and angle of elevation (θ1) as follows:
tan(θ1) = H / x
To find x, we can rearrange the equation:
x = H / tan(θ1)
Plugging in the given values, we have:
x = 136 / tan(6 degrees)
Calculating this on a calculator, we find:
x ≈ 136 / 0.105104
x ≈ 1295.01 feet
Now, let's consider the right triangle formed by the boat, the lighthouse, and the point where the boat second measures the angle of elevation (point B).
In this triangle, the height of the lighthouse (H) is still 136 feet, but the angle of elevation (θ2) is now 19 degrees.
Using trigonometry, we can relate the opposite side (H), adjacent side (y), and angle of elevation (θ2) as follows:
tan(θ2) = H / y
To find y, we can rearrange the equation:
y = H / tan(θ2)
Plugging in the given values, we have:
y = 136 / tan(19 degrees)
Calculating this on a calculator, we find:
y ≈ 136 / 0.326264
y ≈ 416.98 feet
Finally, to find the distance from point A to point B, we subtract the distance from point A to the lighthouse (x) from the distance from point B to the lighthouse (y):
distance AB = y - x
distance AB ≈ 416.98 - 1295.01
distance AB ≈ -878.03
Since distances can't be negative, we round the answer to the nearest foot:
distance AB ≈ 878 feet