A boat leaves a dock at 2:00 P.M. and travels due south at a speed of 15 km/h. Another boat has been heading due east at 20 km/h and reaches the same dock at 3:00 P.M. How many minutes past 2:00 P.M. were the boats closest together?
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To find the time when the two boats were closest together, we need to determine the distance traveled by each boat at different times and compare them.
Let's start by calculating the distance traveled by the first boat (traveling south) at different times. Since the boat left at 2:00 P.M., we can use the equation: distance = speed × time.
At 2:00 P.M., the first boat has been traveling for 0 hours, so its distance traveled is 15 km/h × 0 h = 0 km.
At 3:00 P.M., the first boat has been traveling for 1 hour, so its distance traveled is 15 km/h × 1 h = 15 km.
Next, let's calculate the distance traveled by the second boat (traveling east) at different times. Since the second boat reached the dock at 3:00 P.M., we can use the same equation: distance = speed × time.
At 2:00 P.M., the second boat hasn't started yet, so its distance traveled is 0 km.
At 3:00 P.M., the second boat has been traveling for 1 hour, so its distance traveled is 20 km/h × 1 h = 20 km.
Now, we need to find the time when the distances traveled by both boats are closest, which means the two distances are equal.
Let t represent the time elapsed since 2:00 P.M. for the first boat. At this time, the first boat will have traveled 15t km.
For the second boat, the time elapsed since 2:00 P.M. is t - 1 because it reached the dock 1 hour before 3:00 P.M. At this time, the second boat will have traveled 20(t - 1) km.
Setting the two distances equal to each other, we can solve for t:
15t = 20(t - 1)
15t = 20t - 20
5t = 20
t = 4
Therefore, the boats were closest together 4 hours after 2:00 P.M.
Since we want to find the time in minutes, we can convert 4 hours to minutes:
4 hours × 60 minutes/hour = 240 minutes.
So, the boats were closest together 240 minutes past 2:00 P.M.