The velocity in the table is increasing, . Find an upper estimate for the total distance traveled using
(a) n=4
(b) n=2
Which of the two answers is more accurate?
0 3 6 9 12
32 36 39 42 46
1.632
To find an upper estimate for the total distance traveled using the given table, we can use the trapezoidal rule. The trapezoidal rule involves estimating the area under the velocity-time graph by dividing it into trapezoids.
First, we need to find the width of each trapezoid. In this case, the width is the difference between consecutive time values, which is 3 (from 0 to 3, 3 to 6, 6 to 9, and 9 to 12).
Next, we need to find the height of each trapezoid. To do this, we average the velocity values at the two time points that define each trapezoid. So, for (a) n=4, we have the following trapezoids:
Trapezoid 1: Average height = (32 + 36) / 2 = 34
Trapezoid 2: Average height = (36 + 39) / 2 = 37.5
Trapezoid 3: Average height = (39 + 42) / 2 = 40.5
Trapezoid 4: Average height = (42 + 46) / 2 = 44
To find the area of each trapezoid, we multiply the average height by the width:
Area of Trapezoid 1 = 34 * 3 = 102
Area of Trapezoid 2 = 37.5 * 3 = 112.5
Area of Trapezoid 3 = 40.5 * 3 = 121.5
Area of Trapezoid 4 = 44 * 3 = 132
Finally, to find the upper estimate for the total distance traveled, we sum up the areas of all the trapezoids:
Total distance traveled = Area of Trapezoid 1 + Area of Trapezoid 2 + Area of Trapezoid 3 + Area of Trapezoid 4
Total distance traveled = 102 + 112.5 + 121.5 + 132 = 468
For (b) n=2, we only have two trapezoids:
Average height for Trapezoid 1 = (32 + 36) / 2 = 34
Average height for Trapezoid 2 = (42 + 46) / 2 = 44
Area of Trapezoid 1 = 34 * 3 = 102
Area of Trapezoid 2 = 44 * 3 = 132
Total distance traveled = 102 + 132 = 234
In this case, the answer for (a) n=4 is more accurate because we have a larger number of smaller trapezoids that better approximate the curve. As we increase the number of trapezoids, the approximation becomes more accurate. So, using more trapezoids gives us a more accurate estimate for the total distance traveled.