Two ships leave a harbor at the same time, traveling on courses that have an angle of 140 degrees between them. If the first ship travels at 26 miles per hour and the second ship travels at 34 miles per hour, how far apart are the two ships after 3 hours?
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To find the distance between the two ships after 3 hours, we need to find the distances traveled by each ship and then use trigonometry to find the distance between them.
Let's start by finding the distance traveled by the first ship after 3 hours. Since the ship travels at a constant speed of 26 miles per hour, we can use the formula: Distance = Speed × Time.
Distance traveled by the first ship after 3 hours = 26 miles/hour × 3 hours = 78 miles.
Now, let's find the distance traveled by the second ship after 3 hours using the same formula.
Distance traveled by the second ship after 3 hours = 34 miles/hour × 3 hours = 102 miles.
Next, we need to use trigonometry to find the distance between the two ships. We have an angle of 140 degrees between their courses and the distances we just calculated for each ship.
The distance between the two ships can be found using the Law of Cosines, which states that in a triangle, the square of the length of one side equals the sum of the squares of the lengths of the other two sides, minus twice the product of those side lengths multiplied by the cosine of the angle between them.
In this case, we can use the Law of Cosines to find the distance between the two ships as follows:
Distance^2 = First Ship's Distance^2 + Second Ship's Distance^2 - 2 * First Ship's Distance * Second Ship's Distance * cosine(angle between their courses).
Plug in the values we have:
Distance^2 = 78^2 + 102^2 - 2 * 78 * 102 * cos(140 degrees).
Calculating this, we get:
Distance^2 ≈ 6084 + 10404 - 16044 * -0.76604444311.
Distance^2 ≈ 6084 + 10404 + 12275.3173555.
Distance^2 ≈ 28763.3173555.
Taking the square root of both sides, we get:
Distance ≈ √28763.3173555.
Distance ≈ 169.79 miles.
Therefore, the two ships are approximately 169.79 miles apart after 3 hours.