The weather bureau reports that a cold front has just passed Austin and is heading toward San Antonio at 42kph. Since you know that Antonio is 126 km from Austin, you first think you have 126/42=3 hours before the front arrives there. However, you realize that triangles and vectors are involved. You find out the following info: Antonio's bearing from Austin is 217 degrees, measured clockwise from north. The cold front lies along a line whose bearing is 63 degrees measured clockwise from north, and its movement is perpendicular to that line.
How long will it REALLY be until the front reaches Antonio?
I have absolutely no clue how to do this. Thanks for all the help!
To solve this problem, we need to use trigonometry and vector analysis. Let's break down the problem step by step:
1. Find the distance the cold front has to travel to reach San Antonio:
Since San Antonio is 126 km away from Austin and the cold front is moving at a speed of 42 km/h, we can calculate the time it takes for the front to reach San Antonio by dividing the distance by the speed:
Time = Distance / Speed = 126 km / 42 km/h = 3 hours
2. Determine the direction of the cold front relative to north:
The bearing of the cold front is 63 degrees measured clockwise from north. To find the direction vector, we can use trigonometry and convert the angle to radians:
Direction_radians = (90 - 63) * (π/180) = 27π/180 radians
3. Calculate the displacement vector of the cold front:
The displacement vector is the vector representing the distance and direction traveled by the cold front. It can be obtained by multiplying the speed of the cold front by time and the direction vector:
Displacement_vector = Speed * Time * Direction_vector
4. Find the bearing of San Antonio from Austin:
The bearing of San Antonio from Austin is 217 degrees measured clockwise from north. Again, we convert the angle to radians:
San_Antonio_bearing_radians = (90 - 217) * (π/180) = -127π/180 radians
5. Calculate the displacement of San Antonio in the x and y components:
By using trigonometry, we can determine the x and y components of the displacement vector of San Antonio from Austin:
Displacement_x = Displacement * cos(San_Antonio_bearing_radians)
Displacement_y = Displacement * sin(San_Antonio_bearing_radians)
6. Determine the time it will take for the cold front to reach San Antonio:
Since the displacement of San Antonio consists of both the x and y components, we can calculate the magnitude of the displacement as follows:
Displacement = √(Displacement_x^2 + Displacement_y^2)
Finally, we divide the displacement by the speed to find the time it will take for the cold front to reach San Antonio:
Time = Displacement / Speed
By following these steps, we can calculate the exact time it will take for the cold front to reach San Antonio.