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. Give a 2 decimal place answer and include units.
t 0 0.3 1.0 1.6 2
v(t) 0 24.5 27.8 28.3 29.0
A = 0.5(b1 + b2)h
h = dx so the change in x
A = 0.5(v(0) + v(0.3))(0.3 - 0) + 0.5(v(0.3) + v(1))(1 - 0.3) + 0.5(v(1) + v(1.6))(1.6 - 1)
+ 0.5(v(1.6) + v(2))(2 - 1.6)
A = 0.5(0 + 24.5)(0.3) + 0.5(24.5 + 27.8)(0.7) + 0.5(27.8 + 28.3)(0.6)
+ 0.5(28.3 + 29.0)(0.4)
A = 3.675 + 18.305 + 16.83 + 11.46
A = 50.27 feet
The car covered approximately 50.27 feet during those 2 minutes.
Well, I have to say, this car sounds like it's really in a hurry! Let's see if we can estimate the distance it covered during those 2 minutes.
To do that, we'll use a trapezoidal sum with 4 intervals. Each interval will have a width of 0.5 minutes (2 minutes divided by 4 intervals).
First, let's find the average velocity for each interval. We'll do that by taking the sum of the two velocity values at the endpoints of each interval and dividing it by 2.
For the first interval (t = 0 to t = 0.5 minutes):
Average velocity = (0 + 24.5) / 2 = 12.25 feet per minute.
For the second interval (t = 0.5 to t = 1.0 minutes):
Average velocity = (24.5 + 27.8) / 2 = 26.15 feet per minute.
For the third interval (t = 1.0 to t = 1.5 minutes):
Average velocity = (27.8 + 28.3) / 2 = 28.05 feet per minute.
For the fourth interval (t = 1.5 to t = 2.0 minutes):
Average velocity = (28.3 + 29.0) / 2 = 28.65 feet per minute.
Now, let's calculate the total distance covered by adding up the areas of the trapezoids formed by each interval.
For the first interval (t = 0 to t = 0.5 minutes):
Area = 0.5 * (12.25 + 0) = 6.125 feet.
For the second interval (t = 0.5 to t = 1.0 minutes):
Area = 0.5 * (26.15 + 24.5) = 25.825 feet.
For the third interval (t = 1.0 to t = 1.5 minutes):
Area = 0.5 * (28.05 + 27.8) = 27.925 feet.
For the fourth interval (t = 1.5 to t = 2.0 minutes):
Area = 0.5 * (28.65 + 28.3) = 28.475 feet.
Finally, let's add up the areas of all the intervals to get the total distance covered:
Total distance = 6.125 + 25.825 + 27.925 + 28.475 = 88.35 feet.
So, using a trapezoidal sum with 4 intervals, the car covered approximately 88.35 feet during those 2 minutes.
To estimate the distance the car covered during the first 2 minutes of the race, we can use a trapezoidal sum with 4 intervals. The formula for the trapezoidal sum is:
∆d ≈ (∆t/2) * (f(t0) + 2f(t1) + 2f(t2) + ... + 2f(tn-1) + f(tn))
Where ∆t is the difference in time between each interval and f(t) represents the speed of the car at time t.
Given the following data points:
t: 0 0.3 1.0 1.6 2
v(t): 0 24.5 27.8 28.3 29.0
We can calculate the intervals:
∆t1 = t2 - t0 = 0.3 - 0 = 0.3
∆t2 = t3 - t1 = 1.0 - 0.3 = 0.7
∆t3 = t4 - t2 = 1.6 - 1.0 = 0.6
∆t4 = t5 - t3 = 2 - 1.6 = 0.4
Now, let's calculate the trapezoidal sum for each interval:
∆d1 = (0.3/2) * (0 + 2 * 24.5 + 27.8) = 8.175
∆d2 = (0.7/2) * (27.8 + 2 * 28.3 + 29.0) = 22.715
∆d3 = (0.6/2) * (28.3 + 2 * 29.0 + 0) = 13.98
∆d4 = (0.4/2) * (29.0 + 0) = 5.8
Now, let's find the total distance covered by summing up all the intervals:
Total Distance = ∆d1 + ∆d2 + ∆d3 + ∆d4
= 8.175 + 22.715 + 13.98 + 5.8
= 50.67
Therefore, the estimated distance the car covered during the first 2 minutes of the race is approximately 50.67 feet.
To estimate the distance covered by the car during the first 2 minutes of the race, we can use a trapezoidal sum with 4 intervals. This involves approximating the area under the velocity-time graph by dividing it into trapezoids.
First, calculate the width of each interval by subtracting the previous time from the next time. In this case, the intervals are:
Interval 1: (0.3 - 0) = 0.3 minutes
Interval 2: (1.0 - 0.3) = 0.7 minutes
Interval 3: (1.6 - 1.0) = 0.6 minutes
Interval 4: (2 - 1.6) = 0.4 minutes
Next, calculate the average velocity within each interval by adding the velocity at the beginning and end of the interval and dividing by 2:
Interval 1: (0 + 24.5) / 2 = 12.25 feet per minute
Interval 2: (24.5 + 27.8) / 2 = 26.15 feet per minute
Interval 3: (27.8 + 28.3) / 2 = 28.05 feet per minute
Interval 4: (28.3 + 29.0) / 2 = 28.65 feet per minute
Now, multiply the average velocity of each interval by the width of that interval to find the area of each trapezoid:
Area of trapezoid 1: 0.3 * 12.25 = 3.68 square feet
Area of trapezoid 2: 0.7 * 26.15 = 18.31 square feet
Area of trapezoid 3: 0.6 * 28.05 = 16.83 square feet
Area of trapezoid 4: 0.4 * 28.65 = 11.46 square feet
Finally, add up the areas of all four trapezoids to estimate the total distance covered by the car during the 2 minutes:
Total distance = 3.68 + 18.31 + 16.83 + 11.46 = 50.28 feet
So, the car covered approximately 50.28 feet in the first 2 minutes of the race.
Did you make your sketch?
then
Area (or distance) = (1/2)(.3)(0 + 24.5) + (1/2)(.7)(24.5+27.8) + (1/2)(.6)(27.8+28.3) + (1/2)(.4)(28.3+29.0)
= ....
I will let you do the arithmetic.