Answer each of the following questions independently.Two cars C1 and C2 start their journey from two different points at 4 kmph and 3 kmph respectively and meet after 5 hours at a point where their paths make an angle of 135°. After the cars met, they interchange their paths. Each car first travels to the other car's starting point (i.e. C1 travels to C2's starting point along the path C2 was travelling on and C2 travels to C1's starting point along the path C1 was travelling on ) and then travels along the straight line path (i.e. the shortest distance) joining the starting points of the two cars, finally reaching it's original starting point. Find the approximate time interval between the return of C1 and C2 to their starting positions.5-6 hours11-12 hours8-9 hours3-4 hours
Question
Answer each of the following questions independently.Two cars C1 and C2 start their journey from two different points at 4 kmph and 3 kmph respectively and meet after 5 hours at a point where their paths make an angle of 135°. After the cars met, they interchange their paths. Each car first travels to the other car's starting point (i.e. C1 travels to C2's starting point along the path C2 was travelling on and C2 travels to C1's starting point along the path C1 was travelling on ) and then travels along the straight line path (i.e. the shortest distance) joining the starting points of the two cars, finally reaching it's original starting point. Find the approximate time interval between the return of C1 and C2 to their starting positions.5-6 hours11-12 hours8-9 hours3-4 hours
Solution
The problem involves a bit of geometry and speed-distance-time calculations. Here's how you can solve it:
Step 1: Calculate the distance each car traveled before they met. Since they met after 5 hours, the distance traveled by C1 = speed * time = 4 kmph * 5 hours = 20 km. Similarly, the distance traveled by C2 = 3 kmph * 5 hours = 15 km.
Step 2: Calculate the distance between the starting points of the two cars. The cars met at a point where their paths make an angle of 135°. So, the distance between their starting points can be calculated using the law of cosines in a triangle, which states that c² = a² + b² - 2abcosC, where a, b, and c are the sides of the triangle and C is the angle opposite side c. Here, a = 20 km, b = 15 km, and C = 135°. So, c² = 20² + 15² - 22015*cos(135°) = 400 + 225 + 600 = 1225. Therefore, c = √1225 = 35 km.
Step 3: Calculate the time taken by each car to reach the other's starting point after they met. The time taken by C1 = distance / speed = 15 km / 4 kmph = 3.75 hours. The time taken by C2 = 20 km / 3 kmph = 6.67 hours.
Step 4: Calculate the time taken by each car to travel from the other's starting point to its own starting point along the straight line path. The time taken by C1 = 35 km / 4 kmph = 8.75 hours. The time taken by C2 = 35 km / 3 kmph = 11.67 hours.
Step 5: Calculate the total time taken by each car to return to its starting position. The total time taken by C1 = 5 hours (before they met) + 3.75 hours + 8.75 hours = 17.5 hours. The total time taken by C2 = 5 hours + 6.67 hours + 11.67 hours = 23.34 hours.
Step 6: Calculate the time interval between the return of C1 and C2 to their starting positions. The time interval = 23.34 hours - 17.5 hours = 5.84 hours, which is approximately between 5-6 hours.
So, the answer is 5-6 hours.
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