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Numerade Educator

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Problem 13 Medium Difficulty

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.

Answer

Lower Estimate: 34.7 $\mathrm{ft}$
Upper Estimate: 44.8 $\mathrm{ft}$

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Video Transcript

So in this particular problem you have to find the lower and upper estimates for the distance that this runner traveled. Now, I don't have the graph so I don't know what these values are going to be, but I hopefully can give you a general idea. So speed time, what we're actually going to do is we're going to be using Riemann sums, we're going to do areas of rectangles and those areas of our rectangles then are going to give us estimates. So remember rectangle, the area of a rectangle is length times with and then in this case it would be speed. So we could even say height times with so speed times time. And remember distance is rate times time. So the area of the rectangle will give us that total distances, we add them up. So find out what your whatever your why values are for each of these. So I'm going to just say a B. See, Mhm. Actually, I don't know if they want you, maybe you need to use all of them. So let me just change this. So A. B. C. D. E. And F. So basically what we're gonna do is we're gonna start by adding these up finding the area of each of our rectangles. But when we do the lower and the upper values were actually going to leave one of the ends off. So I think I'm just gonna put in some random values. Let's just say this is 345678. So are lower estimate we're going to start with Technically zero I guess. Um So at zero she's not going anywhere. Um the distance between each of these. So we'll do a zero is 0.5. So each of these is going to have a width of 0.5. So the height of my first one is zero 0.5 times three Plus 05 times four plus 0.5 times five. Now it's actually easier because all these are the same width to simply take the width and then add up all the Y. Values that correspond. So if I was to do that I would get a lower estimate in my example of approximately 12.5 feet meters, whatever the unit of measurement is. So the upper one then notice I didn't use this one for the upper or for the lower. Now I'm not going to use the zeros for the upper. So I will I will have 0.5 and then I will do three plus four plus five was six, last seven plus eight. And that gives me In my example 16.5 Yeah.