Final answer:
To calculate the distance between two ships spotted from a lighthouse, we use trigonometry, finding individual distances to each ship using the tangent of the known angles of depression and the lighthouse's height. We then subtract the distance to the closer ship from the distance to the farther ship.
Step-by-step explanation:
To determine the distance between the two ships that are spotted from a 156-foot high lighthouse, we can use trigonometry to find the horizontal distances from the lighthouse to each ship and then find the distance between the ships. The angles of depression to the first and second ship are 27° and 7° respectively.
Let's denote the distances from the lighthouse to the first and second ship as d1 and d2, respectively. Since the lighthouse's height (156 feet) and the angle of depression are known, we can use the tangent of the angle to calculate these distances. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
For the first ship:
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- tan(27°) = 156 / d1
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- d1 = 156 / tan(27°)
For the second ship:
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- tan(7°) = 156 / d2
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- d2 = 156 / tan(7°)
The distance between the two ships can then be found by subtracting the distance to the first ship from the distance to the second ship: distance between ships = d2 - d1.
Once we find d1 and d2, we will have the distance between the two ships, which will help the lighthouse operator determine if the ships are at risk to collide, thanks to the vigilant observation from the lighthouse.