I am in a car and travel for 12 minutes. Below are the speeds in mph, recorded every two minutes. Use trapezoids, right-bound rectangles, and midpoint rectangles to estimate the distance I traveled.
Min 0 2 4 6 8 10 12
Speed 20 22 35 46 50 50 20
Trapezoids:
Right-bound rectangles:
Midpoint rectangles:
Thank you very much for your time and help.
To estimate the distance traveled using trapezoids, right-bound rectangles, and midpoint rectangles, we need to use the recorded speeds and approximate the area under the speed-time graph.
Here's how you can calculate the estimates for each method:
1. Trapezoid rule:
In the trapezoid rule, we approximate the area by dividing the graph into trapezoids and calculating the area of each trapezoid.
To estimate the distance using the trapezoid rule:
- First, divide the time interval into smaller subintervals. In this case, we have intervals of 2 minutes.
- Next, calculate the average speed within each subinterval by taking the sum of the speeds at the beginning and end of each subinterval and dividing by 2.
- Multiply each average speed by the time interval to calculate the area of each trapezoid.
- Sum up the areas of all the trapezoids to get the estimated total distance traveled.
Using the given data, the estimated distance using the trapezoid rule would be:
(2/2) * (20 + 22) + (2/2) * (22 + 35) + (2/2) * (35 + 46) + (2/2) * (46 + 50) + (2/2) * (50 + 50) + (2/2) * (50 + 20)
= 42 + 57 + 81 + 96 + 100 + 35
= 411 miles
2. Right-bound rectangles:
In the right-bound rectangle rule, we approximate the area under the curve by using rectangles with heights equal to the value of the function at the right endpoint of each subinterval.
To estimate the distance using the right-bound rectangle rule:
- Divide the time interval into smaller subintervals.
- Take the speed recorded at the end of each subinterval as the height of each rectangle.
- Multiply the height of each rectangle by the width (time interval) to calculate the area.
- Sum up the areas of all the rectangles to get the estimated total distance traveled.
Using the given data, the estimated distance using the right-bound rectangle rule would be:
2 * 22 + 2 * 35 + 2 * 46 + 2 * 50 + 2 * 50 + 2 * 20
= 44 + 70 + 92 + 100 + 100 + 40
= 446 miles
3. Midpoint rectangles:
In the midpoint rectangle rule, we approximate the area under the curve by using rectangles with heights equal to the value of the function at the midpoint of each subinterval.
To estimate the distance using the midpoint rectangle rule:
- Divide the time interval into smaller subintervals.
- Take the speed recorded at the midpoint of each subinterval as the height of each rectangle.
- Multiply the height of each rectangle by the width (time interval) to calculate the area.
- Sum up the areas of all the rectangles to get the estimated total distance traveled.
Using the given data, the estimated distance using the midpoint rectangle rule would be:
2 * 21 + 2 * 28.5 + 2 * 40.5 + 2 * 48 + 2 * 50 + 2 * 35
= 42 + 57 + 81 + 96 + 100 + 70
= 446 miles
So, using both the right-bound rectangle rule and the midpoint rectangle rule, we estimate that the total distance traveled is 446 miles. Using the trapezoid rule, we estimate the distance to be 411 miles.