Two ships leave a harbor at the same time, traveling on courses that have an angle of 140 degrees between them. If the first ship travels at 26 miles per hour and the second ship travels at 34 miles per hour, how far apart are the two ships after 3 hours?
I drew the vector for ship # 1 @ 0 deg
and ship # 2 @ 140 deg.
d1 = 26 mi / h * 3 h = 78 mi @ 0 deg.
d2 = 34 mi / h * 3 h = 102 mi @ 140 deg.
d = 102 mi @ 140 deg - 78 mi @ 0 deg.
X = hor = 102 * cos140 - 78cos0,
= -78.1 - 78 = - 156.1 mi.
Y = ver = 102sin140 - 78sin0
= 65.6 - 0 = 65.6 mi.
d^2 = x^2 + y^2,
= 24367.2 + 4298.7,
= 28665.9,
d = sqrt(28665.9) = 169.3 miles apart.
To find the distance between the two ships after 3 hours, we need to calculate the distances traveled by each ship individually.
First, we need to find the distance traveled by the first ship. We can use the formula: distance = rate × time.
For the first ship, the rate (speed) is 26 mph and the time is 3 hours. So, the distance traveled by the first ship is 26 mph × 3 hours = 78 miles.
Now, let's find the distance traveled by the second ship. Again, using the formula distance = rate × time, we have a rate of 34 mph and a time of 3 hours. So, the distance traveled by the second ship is 34 mph × 3 hours = 102 miles.
Now, we can use these distances to find the distance between the two ships. We can consider the distances traveled by the ships as two sides of a triangle, and we need to find the length of the third side.
To do that, we can use the law of cosines. The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of their lengths multiplied by the cosine of the angle between them.
In this case, the lengths of the two sides are 78 miles and 102 miles, and the angle between them is 140 degrees.
Using the law of cosines, we have:
Distance^2 = 78^2 + 102^2 - 2 × 78 × 102 × cos(140 degrees)
Now, we can calculate the distance:
Distance^2 = 6084 + 10404 - 2 × 78 × 102 × cos(140 degrees)
Using a calculator to evaluate this expression, we find:
Distance^2 ≈ 9630.5
Taking the square root of both sides, we get:
Distance ≈ 98.1
Therefore, after 3 hours, the two ships are approximately 98.1 miles apart.