two trains leave Kansas City at the same time. Train A is traveling due north at 55 mph, train B is traveling west at the rate of 65 mph. Find the distance between the two trains two hours later and the bearing of train B from train A.
To find the distance between the two trains after two hours, we can use the formula for calculating distance: distance = speed × time.
Let's start by calculating the distance each train has traveled after two hours:
Train A:
Distance_A = speed_A × time = 55 mph × 2 hours = 110 miles
Train B:
Distance_B = speed_B × time = 65 mph × 2 hours = 130 miles
Now, to find the distance between the two trains, we can use the Pythagorean theorem since the trains are traveling at right angles to each other (north and west):
Distance_between_trains = sqrt(Distance_A^2 + Distance_B^2)
= sqrt(110^2 + 130^2)
= sqrt(12100 + 16900)
= sqrt(29000)
≈ 170.39 miles (rounded to two decimal places)
Therefore, the distance between the two trains after two hours is approximately 170.39 miles.
To find the bearing of train B from train A, we use the inverse tangent function:
Bearing = arctan(Distance_B / Distance_A)
= arctan(130 miles / 110 miles)
≈ arctan(1.18)
≈ 50.19 degrees (rounded to two decimal places)
Therefore, the bearing of train B from train A is approximately 50.19 degrees.
To find the distance between the two trains two hours later, we can make use of their velocities and time. Since Train A is moving north and Train B is moving west, we can consider this as a right-angled triangle.
Let's start by calculating the distance traveled by each train in two hours. Since Train A is traveling north at a speed of 55 mph, it covers a distance of 55 * 2 = 110 miles (rate * time). Train B, moving west at 65 mph, covers a distance of 65 * 2 = 130 miles.
From the given information, we have the sides of a right-angled triangle: one side is 110 miles (the distance traveled by Train A) and the other side is 130 miles (the distance traveled by Train B). We can calculate the hypotenuse, which represents the distance between the two trains, using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. So, we can calculate the distance between the two trains as follows:
Distance^2 = (110 miles)^2 + (130 miles)^2
Distance^2 = 12100 + 16900
Distance^2 = 29000
Distance ≈ √29000
Distance ≈ 170.69 miles (rounded to two decimal places)
Thus, the distance between the two trains two hours later is approximately 170.69 miles.
Now, let's find the bearing of Train B from Train A. Bearing represents the direction in degrees relative to a reference point, usually north.
Since Train A is traveling north and Train B is moving west, the right-angled triangle formed by their paths means that Train B is located to the northwest of Train A. To find the bearing, we can use trigonometry.
We have the opposite side (130 miles) and the adjacent side (110 miles) of the right-angled triangle. By using the tangent function, we can find the angle between the line connecting the two trains and the north direction.
Tan(θ) = opposite / adjacent
Tan(θ) = 130 / 110
θ ≈ tan^(-1)(130 / 110)
θ ≈ tan^(-1)(1.18)
θ ≈ 50.88 degrees (rounded to two decimal places)
Therefore, the bearing of Train B from Train A is approximately 50.88 degrees northwest, relative to north.
65*2 to get the west and 55*2 to get the north. 2 is from the hours, to solve the distance you must use Sin130^o = 110^o/d or distance. then the answer will be the distance then to find bearing just use Tan theta (represents the bearing) = 100^o/65^o. and that's how you get the bearing the answer must be N bearing W.
sorry if I'm wrong and I also appologizing for my grammar. I'm just 8 years old kid and I need to enchance my english ability :D