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.
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.
1 answer