Dayaan M. answered • 9d
Earned A’s Twice in Precalculus | 5 Years of Tutoring Experience
In order to estimate distance from a table of speeds, we can use the fact that distance is the area under the velocity curve. We only have five data points instead of a formula, so we approximate that area with trapezoids, one between each pair of consecutive times.
Before setting anything up, look carefully at your t values, because this problem has a trap built into it. Your times are 0, 0.4, 1.0, 1.5 and 2, so the gaps between them are 0.4, 0.6, 0.5 and 0.5. They are not all the same. That matters because the trapezoid formula most books give you, the one that looks like (h/2)(v0 + 2v1 + 2v2 + 2v3 + v4), only works when every interval has the same width h. If you use it here you get 12.15, which is wrong.
Instead we go back to what a trapezoid actually is. The area of one trapezoid is its width times the average of its two heights, so we do each one separately and add them up. For each interval, the width is the change in time and the two heights are the two speeds.
From t = 0 to 0.4, the width is 0.4 and the speeds are 0 and 4, so the average height is 2:
0.4(2) = 0.8
From t = 0.4 to 1.0, the width is 0.6 and the speeds are 4 and 7.2, so the average height is 5.6:
0.6(5.6) = 3.36
From t = 1.0 to 1.5, the width is 0.5 and the speeds are 7.2 and 8.6, so the average height is 7.9:
0.5(7.9) = 3.95
From t = 1.5 to 2.0, the width is 0.5 and the speeds are 8.6 and 9.0, so the average height is 8.8:
0.5(8.8) = 4.4
Adding all four pieces together:
0.8 + 3.36 + 3.95 + 4.4 = 12.51
Remember to include units, since the question asks for them. Speed was in meters per second and time was in seconds, so multiplying those gives meters, and the answer is 12.51 meters.
If you notice, you can sanity check this without doing any of the work over. The swimmer never goes faster than 9 meters per second, so in 2 seconds they cannot possibly have gone more than 18 meters, and they were speeding up from a dead stop so it should be well under that. 12.51 sits comfortably in that range. If you had gotten something like 25 you would know immediately that something went wrong.
One last thing about this estimate. The speeds are increasing the whole way, and the curve is bending over as the swimmer approaches top speed, which means it is concave down. Trapezoids cut across underneath a concave down curve, so this answer is slightly less than the true distance. Knowing whether your estimate is an over or an under is a common follow up question on these.
So, our final answer is 12.51 meters.
