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:
I'm confused with the units and midpoint rectangles mostly.
just tedious work ....
I suggest you make a sketch,
and plot the points (0,20) , (2,22), ....
you will get trapezoids by joining all the y-values and drawing in all the verticals
look at the first trapezoid, its base = 2
the left vertical is 20 and the right vertical is 22
area = 2*(20+22)/2 = 42
second trap
area = 2(22+35)/2 = 57
etc
add them up
for right-bound rectangles, take the height of the right-hand side, and the base will be 2 for all of them
1st rectangle = 2(22) = 44
2nd rectange = 2(35) = 70
etc
I don't know how you were taught the method for midpoint rectangles. Look in your text or your classroom notes
For midpoint rectangles my notes say something about expanding the interval but I'm not sure how that works.. And if it's minutes and hours would I have to do something to change it?
To estimate the distance traveled using different methods, we'll divide the 12-minute interval into smaller subintervals and approximate the distance within each subinterval. The given speeds are in mph (miles per hour), but to calculate distance, we need to convert them to miles per minute.
First, let's convert the speeds from mph to miles per minute:
Speeds (mph): 20 22 35 46 50 50 20
Speeds (miles per min): 20/60 22/60 35/60 46/60 50/60 50/60 20/60
Speeds (miles per min): 1/3 11/30 7/12 23/30 5/6 5/6 1/3
Now let's calculate the distance traveled using different estimation methods.
1. Trapezoids:
In the trapezoid method, we approximate the area under the curve of speed by dividing the 12-minute interval into trapezoids. The distance traveled in each subinterval is the average of the speeds at the endpoints multiplied by the subinterval width.
Subintervals:
0-2 min, 2-4 min, 4-6 min, 6-8 min, 8-10 min, 10-12 min
Calculating distances for each subinterval:
1/2 * (1/3 + 11/30) * 2/60 = 3/180
1/2 * (11/30 + 7/12) * 2/60 = 31/360
1/2 * (7/12 + 23/30) * 2/60 = 23/360
1/2 * (23/30 + 5/6) * 2/60 = 29/360
1/2 * (5/6 + 5/6) * 2/60 = 5/360
1/2 * (5/6 + 1/3) * 2/60 = 13/360
Total distance = (3/180) + (31/360) + (23/360) + (29/360) + (5/360) + (13/360) = 104/360 = 13/45 miles.
Therefore, using the trapezoid method, you traveled approximately 13/45 miles.
2. Right-bound rectangles:
In the right-bound rectangle method, we approximate the area under the curve using rectangles with the heights of the speeds at the right endpoints of each subinterval. The distance traveled in each subinterval is the speed at the right endpoint multiplied by the subinterval width.
Subintervals (right endpoints):
2 min, 4 min, 6 min, 8 min, 10 min, 12 min
Calculating distances for each subinterval:
(11/30) * 2/60 = 11/900
(7/12) * 2/60 = 7/360
(23/30) * 2/60 = 23/900
(5/6) * 2/60 = 1/180
(5/6) * 2/60 = 1/180
(1/3) * 2/60 = 1/900
Total distance = (11/900) + (7/360) + (23/900) + (1/180) + (1/180) + (1/900) = 23/900 miles.
Using the right-bound rectangle method, you traveled approximately 23/900 miles.
3. Midpoint rectangles:
In the midpoint rectangle method, we approximate the area under the curve using rectangles with the heights of the speeds at the midpoint of each subinterval. The distance traveled in each subinterval is the speed at the midpoint multiplied by the subinterval width.
Subintervals (midpoints):
1 min, 3 min, 5 min, 7 min, 9 min, 11 min
Calculating distances for each subinterval:
(1/3) * 2/60 = 1/900
(11/30) * 2/60 = 11/900
(7/12) * 2/60 = 7/360
(23/30) * 2/60 = 23/900
(5/6) * 2/60 = 1/180
(5/6) * 2/60 = 1/180
Total distance = (1/900) + (11/900) + (7/360) + (23/900) + (1/180) + (1/180) = 46/900 = 23/450 miles.
Using the midpoint rectangle method, you traveled approximately 23/450 miles.
Therefore, using the given speeds, the estimated distances traveled are:
- Trapezoids: approximately 13/45 miles
- Right-bound rectangles: approximately 23/900 miles
- Midpoint rectangles: approximately 23/450 miles.
Note: The accuracy of the estimation methods improves as the number of subintervals increases.