A radar gun was used to record the speed of a car (in feet per minute) during selected times in the first 2 minutes of a race. Use a trapezoidal sum with 4 intervals to estimate the distance the car covered during those 2 minutes.
t 0 0.3 1.0 1.6 2
v(t) 0 24.5 27.8 28.3 29.0
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To estimate the distance the car covered during the first 2 minutes of the race using a trapezoidal sum with 4 intervals, we first need to calculate the average velocity for each interval.
Interval 1: t = 0 to t = 0.3
Average velocity for this interval = (v(0) + v(0.3))/2 = (0 + 24.5)/2 = 12.25
Interval 2: t = 0.3 to t = 1.0
Average velocity for this interval = (v(0.3) + v(1.0))/2 = (24.5 + 27.8)/2 = 26.15
Interval 3: t = 1.0 to t = 1.6
Average velocity for this interval = (v(1.0) + v(1.6))/2 = (27.8 + 28.3)/2 = 28.05
Interval 4: t = 1.6 to t = 2.0
Average velocity for this interval = (v(1.6) + v(2.0))/2 = (28.3 + 29.0)/2 = 28.65
Now, we can calculate the distance covered during each interval using the average velocity and the width of the interval. Since the units for velocity are feet per minute and the time intervals are given in minutes, the distance will be in feet.
Interval 1: Distance = Average velocity * Time = 12.25 * 0.3 = 3.675 feet
Interval 2: Distance = Average velocity * Time = 26.15 * (1.0 - 0.3) = 16.89 feet
Interval 3: Distance = Average velocity * Time = 28.05 * (1.6 - 1.0) = 44.88 feet
Interval 4: Distance = Average velocity * Time = 28.65 * (2.0 - 1.6) = 45.84 feet
Finally, to estimate the total distance covered during the 2 minutes of the race, we sum up the distances covered in each interval:
Total Distance = Distance in Interval 1 + Distance in Interval 2 + Distance in Interval 3 + Distance in Interval 4
= 3.675 + 16.89 + 44.88 + 45.84
= 111.285 feet
Therefore, the estimated distance covered by the car during the first 2 minutes of the race is approximately 111.285 feet.