A car travels up a hill at a constant speed of 45 km/h and returns down the hill at a constant speed of 70 km/h. Calculate the average speed for the round trip.
To calculate the average speed for the round trip, we first need to find the total distance traveled. Since the car travels up and down the hill, the distance for the uphill and downhill journey would be the same.
Let's assume the distance traveled in one direction (uphill or downhill) is 'd' km.
Therefore, the total distance for the round trip is 2d km (d km uphill + d km downhill = 2d km).
Next, we can calculate the time taken for each leg of the journey.
For the uphill journey, the speed is 45 km/h, and the distance is 'd' km. Hence, the time taken for the uphill journey is d/45 hours.
For the downhill journey, the speed is 70 km/h, and the distance is also 'd' km. Therefore, the time taken for the downhill journey is d/70 hours.
Now, to find the average speed, we divide the total distance by the total time taken for the round trip:
Average Speed = Total Distance / Total Time
Since the total distance for the round trip is 2d km and the total time is (d/45 + d/70) hours, we can substitute these values in:
Average Speed = 2d / (d/45 + d/70)
To simplify the equation, we need to find a common denominator for the denominators on the right side:
Average Speed = 2d / [(70d + 45d) / (70 * 45)]
Simplifying the expression within the brackets further:
Average Speed = 2d / [(115d) / (70 * 45)]
Now, we can simplify the expression by multiplying the numerator by the reciprocal of the denominator:
Average Speed = 2d * (70 * 45) / (115d)
Next, we can cancel out the common factor of 'd':
Average Speed = 2 * (70 * 45) / 115
Finally, calculating the average speed:
Average Speed ≈ 63.47 km/h
So, the average speed for the round trip is approximately 63.47 km/h.