A sail boat on lake Huron leaves Southampton and sails 20degrees west of north of 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 km? ( the answer is 4139 km, but I need to know the method to understand how to get that answer)
draw direction lines NS-EW at Southampton.
draw the paths of the two boats, the angle between them should be 145°
So this time it does look like the Cosine Law
x^2 = 20^2 + 15^2 - 2(20)(15)cos 145
= 400 + 225 + 491.491
= 1116.49
x = √1116.49
= 33.41 km
Your answer of 4139 km is "absurd" , I have been to Southampton on Lake Huron and it is only about 120 km across the lake to Michigan.
Srry but how did u get 145?
To determine the distance between the two boats, we can use the law of cosines. First, let's break down the problem into two separate movements:
1. The sailboat moves 20 degrees west of north for 20 km.
2. The fishing boat moves 30 degrees west of south for 15 km.
Now, let's find the components of movement for each boat:
1. Sailboat movement:
- Northward component = 20 km * cos(20 degrees)
- Westward component = 20 km * sin(20 degrees)
2. Fishing boat movement:
- Southward component = 15 km * cos(30 degrees)
- Westward component = 15 km * sin(30 degrees)
Next, we need to find the final positions of the sailboat and fishing boat by adding their components of movement to their initial positions.
1. Sailboat final position:
- North coordinate = 0 km (as it sails north from Southampton)
- East coordinate = 0 km (no eastward movement)
- West coordinate = Westward component of sailboat movement
= 20 km * sin(20 degrees) = 6.84 km
2. Fishing boat final position:
- South coordinate = 0 km (as it sails south from Southampton)
- East coordinate = 0 km (no eastward movement)
- West coordinate = Westward component of fishing boat movement
= 15 km * sin(30 degrees) = 7.5 km
After finding the final positions of the sailboat and fishing boat, we can use the distance formula between two points to find the displacement between them:
Distance = √[(West coordinate of sailboat - West coordinate of fishing boat)^2 + (North coordinate of sailboat - South coordinate of fishing boat)^2]
Distance = √[(6.84 km - 0 km)^2 + (0 km - 7.5 km)^2]
Distance = √[(6.84 km)^2 + (-7.5 km)^2]
Distance = √[46.9056 km^2 + 56.25 km^2]
Distance = √[103.1556 km^2]
Distance = 10.1556 km
Finally, rounding the distance to the nearest kilometer, the boats are approximately 10 km apart.