two boats left the harbour at the same time. one travelled at 15km/h on a course of 170 degrees for 30 minutes. the other travelled at 10km/h on a course of 220 degrees for 1 hour. how far apart are the boats? round the distance to the nearest kilometre.
To solve this problem, we can use the Law of Cosines. The formula is given by:
c^2 = a^2 + b^2 - 2ab*cos(C),
where c is the side opposite to the angle C, and a and b are the other two sides.
Let's assign the following variables:
- Boat 1: speed = 15 km/h, course = 170 degrees, time = 0.5 hour
- Boat 2: speed = 10 km/h, course = 220 degrees, time = 1 hour
First, let's calculate the distances traveled by each boat. Using the formula:
distance = speed * time,
we have:
- Boat 1 distance = 15 km/h * 0.5 h = 7.5 km
- Boat 2 distance = 10 km/h * 1 h = 10 km
Now, let's find the angle between the paths of the two boats. The angle A can be calculated as follows:
A = 360 degrees - (course of boat 1 + course of boat 2) = 360 degrees - (170 degrees + 220 degrees) = -30 degrees.
Note that since we are interested in the distance between the boats, the direction is not important. Thus, we can consider a positive angle of 330 degrees instead of -30 degrees.
Finally, we can calculate the distance c between the two boats using the Law of Cosines:
c^2 = a^2 + b^2 - 2ab*cos(C),
c^2 = (7.5 km)^2 + (10 km)^2 - 2 * (7.5 km) * (10 km) * cos(330 degrees).
Let's calculate this:
c^2 = 56.25 km^2 + 100 km^2 - 2 * (7.5 km) * (10 km) * cos(330 degrees)
c^2 = 156.25 km^2 + 100 km^2 - 2 * (7.5 km) * (10 km) * (-0.866)
c^2 = 286.25 km^2 + 129.9 km^2 = 416.15 km^2
The distance c is approximately the square root of 416.15 km^2, which is approximately 20.403 km.
Rounding to the nearest kilometer, the distance between the boats is 20 kilometers.