By the time the car reached a speed of 120 mph, it had traveled an estimated distance of 2965 miles.
To estimate the distance traveled by the car when it reached a speed of 120 mph, we can use trapezoids to approximate the area under the velocity curve.
First, let's calculate the time intervals between each speed change. We can do this by subtracting the time at one speed from the time at the previous speed:
Interval 1: 3.5 sec - 2.2 sec = 1.3 sec
Interval 2: 5.1 sec - 3.5 sec = 1.6 sec
Interval 3: 6.8 sec - 5.1 sec = 1.7 sec
Interval 4: 9.0 sec - 6.8 sec = 2.2 sec
Interval 5: 11.7 sec - 9.0 sec = 2.7 sec
Interval 6: 14.5 sec - 11.7 sec = 2.8 sec
Interval 7: 18.1 sec - 14.5 sec = 3.6 sec
Interval 8: 23.0 sec - 18.1 sec = 4.9 sec
Interval 9: 28.9 sec - 23.0 sec = 5.9 sec
Interval 10: 40.1 sec - 28.9 sec = 11.2 sec
Now, we can calculate the average speed for each interval by taking the sum of the speeds at the start and end of the interval and dividing by 2:
Average speed for interval 1: (0 mph + 20 mph) / 2 = 10 mph
Average speed for interval 2: (20 mph + 30 mph) / 2 = 25 mph
Average speed for interval 3: (30 mph + 40 mph) / 2 = 35 mph
Average speed for interval 4: (40 mph + 50 mph) / 2 = 45 mph
Average speed for interval 5: (50 mph + 60 mph) / 2 = 55 mph
Average speed for interval 6: (60 mph + 70 mph) / 2 = 65 mph
Average speed for interval 7: (70 mph + 80 mph) / 2 = 75 mph
Average speed for interval 8: (80 mph + 90 mph) / 2 = 85 mph
Average speed for interval 9: (90 mph + 100 mph) / 2 = 95 mph
Average speed for interval 10: (100 mph + 110 mph) / 2 = 105 mph
Next, we can calculate the distance traveled during each interval by multiplying the average speed by the time interval:
Distance for interval 1: 10 mph * 1.3 sec = 13 miles
Distance for interval 2: 25 mph * 1.6 sec = 40 miles
Distance for interval 3: 35 mph * 1.7 sec = 59.5 miles
Distance for interval 4: 45 mph * 2.2 sec = 99 miles
Distance for interval 5: 55 mph * 2.7 sec = 148.5 miles
Distance for interval 6: 65 mph * 2.8 sec = 182 miles
Distance for interval 7: 75 mph * 3.6 sec = 270 miles
Distance for interval 8: 85 mph * 4.9 sec = 416.5 miles
Distance for interval 9: 95 mph * 5.9 sec = 560.5 miles
Distance for interval 10: 105 mph * 11.2 sec = 1176 miles
Finally, we can sum up the distances for each interval to get the total distance traveled by the car:
Total distance = 13 miles + 40 miles + 59.5 miles + 99 miles + 148.5 miles + 182 miles + 270 miles + 416.5 miles + 560.5 miles + 1176 miles
Total distance = 2965 miles
Therefore, by the time the car reached a speed of 120 mph, it had traveled an estimated distance of 2965 miles.