Two trains leave a station at 11:00am. One train travels north at a rate of 75 mph and another travels
east at a rate of 60 mph. Assuming the trains do not stop, about how many minutes will it take for the
trains to be 250 miles apart?
V1 = 75i mi/h.
V2 = 60 mi/h
Tan A = V1/V2=75/60 = 1.2500, A = 51.3o.
d1 = 250*sin51.3 = 195 mi.
V1*T = 195, 75 * T = 195, T = 2.60 h. =
156 min.
To find the time it takes for the trains to be 250 miles apart, we can use the Pythagorean theorem since the trains are moving in perpendicular directions (north and east).
Let's call the time it takes for the trains to be 250 miles apart as "t."
The distance traveled by the northbound train can be calculated using the formula: distance = speed * time.
For the northbound train, the distance traveled is 75t miles.
For the eastbound train, the distance traveled is 60t miles.
According to the Pythagorean theorem, the sum of the squares of the distances traveled by each train will be equal to the square of the total distance between them. In this case, it would be:
(75t)^2 + (60t)^2 = 250^2
Simplifying the equation:
5625t^2 + 3600t^2 = 62500
9225t^2 = 62500
To find the value of t, divide both sides by 9225:
t^2 = 62500/9225
t^2 ≈ 6.77
Taking the square root of both sides:
t ≈ √6.77
t ≈ 2.6
Therefore, it would take approximately 2.6 hours for the trains to be 250 miles apart.
To convert hours to minutes, multiply by 60:
2.6 hours * 60 minutes/hour = 156 minutes
So, it would take approximately 156 minutes for the trains to be 250 miles apart.
To find out how many minutes it will take for the trains to be 250 miles apart, we need to calculate the time it takes for the trains to cover this distance.
Let's break down the problem into two dimensions: north (vertical) and east (horizontal).
- The train traveling north is moving at a rate of 75 mph, so in one hour it covers a distance of 75 miles. Therefore, the northward component of the distance covered by the trains after t hours will be 75t.
- The train traveling east is moving at a rate of 60 mph, so in one hour it covers a distance of 60 miles. Hence, the eastward component of the distance covered by the trains after t hours will be 60t.
We can use the Pythagorean theorem to find the total distance between the trains after t hours. The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
Applying the theorem to our problem, we have:
distance^2 = (northward distance)^2 + (eastward distance)^2
distance^2 = (75t)^2 + (60t)^2
distance^2 = 5625t^2 + 3600t^2
distance^2 = 9225t^2 + 3600t^2
distance^2 = 12825t^2
Now, we set the equation equal to the square of the desired distance, which is 250 miles:
250^2 = 12825t^2
62500 = 12825t^2
To solve for t, we divide both sides of the equation by 12825:
t^2 = 62500 / 12825
t^2 = 4.8662
Now, take the square root of both sides to solve for t:
t = sqrt(4.8662)
t ≈ 2.204
Therefore, it will take approximately 2.204 hours for the trains to be 250 miles apart.
To convert this time to minutes, we multiply by 60 (since there are 60 minutes in an hour):
t_minutes = 2.204 * 60 ≈ 132.24
So, it will take approximately 132.24 minutes for the trains to be 250 miles apart.