To solve this problem, we can use trigonometry to find the distances traveled by each car in the horizontal and vertical directions. Then, we can use the Pythagorean theorem to calculate the direct distance between the two cars.
Let's assume that the starting point of the two cars is the origin (0,0) on a coordinate plane. The car traveling at 35 km/h will have covered a distance of:
Distance_1 = Speed_1 * Time = 35 km/h * 3 hours = 105 km
Similarly, the car traveling at 40 km/h will have covered a distance of:
Distance_2 = Speed_2 * Time = 40 km/h * 3 hours = 120 km
Now, let's calculate the horizontal and vertical distances covered by each car. Considering the angle between the roads is 63 degrees, the horizontal distance traveled by the first car can be calculated as:
Horizontal_1 = Distance_1 * cos(63°)
Similarly, the horizontal distance traveled by the second car can be calculated as:
Horizontal_2 = Distance_2 * cos(63°)
The vertical distances covered by each car can be calculated in the same way, but using the sine function:
Vertical_1 = Distance_1 * sin(63°) Vertical_2 = Distance_2 * sin(63°)
Now, we can find the horizontal and vertical displacements between the two cars. The horizontal displacement is the difference between the horizontal distances covered by the two cars:
Horizontal_displacement = |Horizontal_2 - Horizontal_1|
Similarly, the vertical displacement is the difference between the vertical distances covered by the two cars:
Vertical_displacement = |Vertical_2 - Vertical_1|
Finally, we can calculate the direct distance between the two cars using the Pythagorean theorem:
Direct_distance = sqrt(Horizontal_displacement^2 + Vertical_displacement^2)
Let's calculate the values:
Horizontal_1 = 105 km * cos(63°) ≈ 45.813 km Horizontal_2 = 120 km * cos(63°) ≈ 52.230 km Vertical_1 = 105 km * sin(63°) ≈ 92.684 km Vertical_2 = 120 km * sin(63°) ≈ 106.684 km
Horizontal_displacement = |52.230 km - 45.813 km| ≈ 6.417 km Vertical_displacement = |106.684 km - 92.684 km| ≈ 14 km
Direct_distance = sqrt((6.417 km)^2 + (14 km)^2) ≈ 15.4 km
Therefore, after 3 hours, the two cars will be approximately 15.4 kilometers apart by a direct route.