A person on a ship sailing due south at the rate of 15 miles an hour observes a lighthouse due west at 3p.m. At 5p.m. the lighthouse is 52degrees west of north. How far from the lighthouse was the ship at a)3p.m.? b)5p.m.? c)4p.m.?
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At 5 pm, the ship is 30 miles south of its 3pm position. The distance from the lighthouse at 3pm is thus
d/30 = tan52°
so, d=38.4
Now, to get the distance at any other elapsed time of t hours,
d^2 = 38.4^2 + (15t)^2
Let 'er rip!
To solve this problem, we need to use some basic trigonometry and apply it to the situation given in the problem.
Let's assume that the ship's position at 3 p.m. is point A, and the lighthouse is point L. The ship is sailing due south, so we can draw a straight line down from point A. The distance between the ship and the lighthouse at 3 p.m. is what we need to find.
Now, let's consider the situation at 5 p.m. At this time, the lighthouse is 52 degrees west of north from the ship's position. We can draw a line from point L that is 52 degrees clockwise from the north direction. Let's call this line LA.
Since we now have two lines, LA and the southward line from point A, we can use trigonometry to find the distance between the ship and the lighthouse.
a) To find the distance between the ship and the lighthouse at 3 p.m., we need to find the length of the line from point A to the line LA. Let's call this line AL. From the given information, we know that the ship's speed is 15 miles per hour. Since they observe the lighthouse due west at 3 p.m., we can conclude that point A lies on a circle centered at the lighthouse. The length of AL can be calculated by multiplying the ship's speed by the time between 3 p.m. and 5 p.m. (2 hours). Therefore, AL = 15 mph * 2 hours = 30 miles.
b) At 5 p.m., we want to find the distance between the ship's position (point A) and the lighthouse (point L). From the given information, we know that the lighthouse is 52 degrees west of north. Using trigonometry, we can apply the sine function to find this distance. Let's call this line AL2. Since we have one angle (52 degrees) and one side (30 miles), we can use the sine function: sin(angle) = opposite/hypotenuse. Therefore, sin(52 degrees) = AL2 / 30 miles. Rearranging the equation gives us AL2 = 30 miles * sin(52 degrees) ≈ 23.66 miles.
c) At 4 p.m., we need to estimate the distance between the ship and the lighthouse. Since we know that the ship's speed is 15 miles per hour, between 3 p.m. and 4 p.m., the ship would have traveled 15 miles. Therefore, at 4 p.m., the ship would be 15 miles away from its position at 3 p.m.
In summary:
a) The ship was 30 miles away from the lighthouse at 3 p.m.
b) The ship was approximately 23.66 miles away from the lighthouse at 5 p.m.
c) The ship was 15 miles away from the lighthouse at 4 p.m.