Two boats leave the same place at the same time. The first sails in a straight line
N70°W at 45 miles per hour and the second goes in a straight line S23°E at 57 miles per hour. After 1 hour, how far apart are the boats, to the nearest tenth of a mile?
boat 1
north 45 cos 70
east -45 sin 70
boat 2
north -57 cos 23 opposite so add
east 57 sin 23 opposite so add
total north = 15.4 + 52.5 = 67.9
total east = +42.3 + 22.3 = 64.6
total =sqrt(67.9^2+64.6^2)
To find the distance between the two boats after 1 hour, we can use the formula: distance = speed * time.
First, let's calculate the distance traveled by the first boat in 1 hour. The speed of the first boat is 45 miles per hour, so after 1 hour, it would have traveled 45 miles.
Next, let's calculate the distance traveled by the second boat in 1 hour. The speed of the second boat is 57 miles per hour, so after 1 hour, it would have traveled 57 miles.
Now, we need to find the angle between the two boats to determine the direction in which they are moving. The first boat is sailing N70°W, which means it is moving in a northwestward direction. The second boat is sailing S23°E, which means it is moving in a southeastward direction.
Since the two boats are moving in opposite directions, the angle between them is 180 degrees.
To find the distance between the two boats, we can use the Law of Cosines:
distance^2 = (45 miles)^2 + (57 miles)^2 - 2 * 45 miles * 57 miles * cos(180 degrees).
Now, we can plug in these values and calculate the distance:
distance^2 = 2025 miles^2 + 3249 miles^2 - 2 * 45 miles * 57 miles * cos(180 degrees).
distance^2 = 2025 miles^2 + 3249 miles^2 + 1035 miles^2.
distance^2 = 6309 miles^2 + 1035 miles^2.
distance^2 = 7344 miles^2.
Taking the square root of both sides, we get:
distance = sqrt(7344) miles.
Rounding to the nearest tenth of a mile, the distance between the two boats after 1 hour is approximately 85.7 miles.