First year students attending Hogwarts ride in boats from the train station (point H) to the castle (point C). The most direct route is 760 meters and the boats travel at 100 m/min. Unfortunately, one boat got off course by 17° and traveled on this path for 5 min. until Hagrid shouted at them to change course. a. Through what angle (round to the tenths) should the students turn in order to head directly to the castle? b. How long (to the nearest tenth of a minute) will it be before they reach the castle?
Through what angle should the students turn in order to head directly to the castle?
How long (to the nearest tenth of a minute) will it be before they reach the castle?
To find the angle the students should turn to head directly to the castle, we can use trigonometry.
a. The boat traveled off course by 17° for 5 minutes.
To find the angle they should turn, we can use the Law of Sines:
sin(A)/a = sin(B)/b
In this case, we know that Angle A is 17° (the angle they traveled off course), and side a is the distance traveled off course, which is 760 meters.
sin(17°)/760 = sin(B)/100
To find Angle B, we can rearrange the equation:
sin(B) = sin(17°) * 100 / 760
B = arcsin(sin(17°) * 100 / 760)
Using a calculator, B is approximately 1.825°.
Therefore, the students should turn by approximately 1.8° to head directly to the castle.
b. To find the time it will take for them to reach the castle, we can use the equation:
time = distance / speed
The distance from the off-course point to the castle is the same as the direct route distance, which is 760 meters. The speed of the boat is 100 m/min.
time = 760 meters / 100 m/min
time = 7.6 minutes
Therefore, it will take approximately 7.6 minutes for them to reach the castle.