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, you need to calculate the area of each trapezoid and then sum them up.
First, let's start by calculating the width of each interval:
Interval 1: Width = 0.3 - 0 = 0.3 minutes
Interval 2: Width = 1.0 - 0.3 = 0.7 minutes
Interval 3: Width = 1.6 - 1.0 = 0.6 minutes
Interval 4: Width = 2 - 1.6 = 0.4 minutes
Next, let's calculate the average speed (velocity) for each interval. To do this, we'll use the formula for the average of two values:
Average Speed = (v(t1) + v(t2)) / 2
Interval 1: Average Speed = (0 + 24.5) / 2 = 12.25 feet per minute
Interval 2: Average Speed = (24.5 + 27.8) / 2 = 26.15 feet per minute
Interval 3: Average Speed = (27.8 + 28.3) / 2 = 28.05 feet per minute
Interval 4: Average Speed = (28.3 + 29.0) / 2 = 28.65 feet per minute
Now, let's calculate the area of each trapezoid using the formula:
Area = (Base1 + Base2) * Height / 2
Interval 1: Area = (0.3 + 0) * 12.25 / 2 = 1.8375 feet
Interval 2: Area = (0.7 + 0.3) * 26.15 / 2 = 11.4925 feet
Interval 3: Area = (0.6 + 0.6) * 28.05 / 2 = 16.827 feet
Interval 4: Area = (0.4 + 0.4) * 28.65 / 2 = 11.52 feet
Finally, sum up the areas of all four intervals to get the estimated distance the car covered during the first 2 minutes of the race:
Total Distance = 1.8375 + 11.4925 + 16.827 + 11.52 = 41.676 feet
Therefore, the estimated distance the car covered during the first 2 minutes of the race is approximately 41.676 feet.