Ports A and B are 400 miles apart. One boat starts from port A for port B at 6:05 and travels with a constant speed of 12 mph. If it leaves port A at 8:20, what speed musta a second boat have to arrive at port B at the same time? Explain your reasoning.
The first boat takes 400/12 hrs, or 33 1/3 hrs.
the second boat has to get there in 33 1/3 - 2 1/4 hrs.
speed second boat: 400/time above.
To find the speed of the second boat, we first need to calculate the time it took for the first boat to travel from port A to port B.
The first boat started at 6:05 and traveled with a speed of 12 mph. We need to convert the time of departure, 6:05, to hours.
6:05 is equivalent to 6 hours and 5 minutes. Since there are 60 minutes in an hour, we can express 5 minutes as 5/60 = 1/12 hours.
So, the first boat's departure time in hours is 6 + 1/12 = 73/12 hours.
Now, let's calculate the distance traveled by the first boat from port A to port B. We are given that the distance between the two ports is 400 miles, and the first boat traveled with a speed of 12 mph.
Using the formula: distance = speed × time, we can rearrange the formula to find the time of travel: time = distance / speed.
So, using the distance of 400 miles and the speed of 12 mph, the time taken by the first boat is 400 / 12 = 33.33 hours.
Now, let's find the arrival time of the first boat at port B. The departure time was 73/12 hours, and it took 33.33 hours of travel time.
Therefore, the arrival time of the first boat at port B is (73/12) + 33.33 = 135.58 hours.
Now, we need to find the time of travel for the second boat. The departure time for the second boat was given as 8:20. We need to convert this time to hours as we did before.
8:20 can be expressed as 8 hours and 20 minutes. Converting 20 minutes to hours, we get 20/60 = 1/3 hours.
Therefore, the departure time of the second boat in hours is 8 + 1/3 = 25/3 hours.
Next, we subtract the arrival time of the first boat from the departure time of the second boat to find the time difference between the two boats:
(25/3) - 135.58 = -110.58 hours.
Since the time difference is negative, it means that the second boat departed before the first boat arrived. Therefore, the second boat must travel faster than the first boat to arrive at the same time.
To calculate the speed of the second boat, we divide the distance between the two ports (400 miles) by the negative time difference (-110.58 hours):
Speed of the second boat = 400 miles / (-110.58 hours) = -3.62 mph.
Note that the negative sign indicates that the second boat is traveling in the opposite direction of the first boat. In this case, the magnitude of the speed is 3.62 mph.
So, the second boat must travel at a speed of 3.62 mph in the opposite direction to arrive at port B at the same time as the first boat.