A sail boat on lake Huron leaves Southampton and sails 20 degrees west of north for 20km. At the same time, a fishing boat leaves Southampton and sails 30degrees west of south for 15 km. At this point, hot far apart are the boats, to the nearest kilometer?
After making your sketch, you will see that the angle between their paths
is 130°, so the cosine law would be a good idea
if d km is the distance between them, then
d^2 = 20^2 + 15^2 - 2(20)(15)cos 130°
take it from here.
On a nautical note, if you sail 15 km at 30° west of south from Southhampton, you boat would be grounded near Port Elgin.
To find out how far apart the boats are, we can use the concept of vector addition.
First, let's visualize the situation:
- The sailboat starts from Southampton and sails 20 degrees west of north for 20 km.
- The fishing boat also starts from Southampton and sails 30 degrees west of south for 15 km.
To determine the position of each boat, we will use a coordinate system with a north-south axis (y-axis) and an east-west axis (x-axis). Both boats start at the origin (0, 0).
For the sailboat:
- The distance north (y-direction) is given by: 20 km * sin(20°)
- The distance west (x-direction) is given by: 20 km * cos(20°)
For the fishing boat:
- The distance south (y-direction) is given by: 15 km * sin(30°)
- The distance west (x-direction) is given by: 15 km * cos(30°)
To find the position of each boat, we can add these components to the origin:
Sailboat position: (20 km * cos(20°), 20 km * sin(20°))
Fishing boat position: (-15 km * cos(30°), -15 km * sin(30°))
Now, let's calculate these positions:
Using a calculator, we find:
Sailboat position: (18.196 km, 6.848 km)
Fishing boat position: (-12.990 km, -7.500 km)
To find the distance between the two boats, we can use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate it:
Distance = sqrt((18.196 km - (-12.990 km))^2 + (6.848 km - (-7.500 km))^2)
Distance = sqrt((31.186 km)^2 + (14.348 km)^2)
Distance = sqrt(974.094 km^2 + 205.976 km^2)
Distance = sqrt(1180.07 km^2)
Distance ≈ 34.35 km
So, the boats are approximately 34 kilometers apart, to the nearest kilometer.