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Running In 1987, Canadian Ben Johnson set a world record in the 100-m sprint. (The record was later taken away when he was found to have used an anabolic steroid to enhance his a. Use the information in the table and left endpoints to estimate the distance that Johnson ran in miles. You will first need to calculate for each interval. At the end, you will need to divide by 3600 (the number of seconds in an hour), since the speed is in miles per hour.b. Repeat part a, using right endpoints.c. Wait a minute; we know that the distance Johnson ran is 100 m. Divide this by 1609, the number of meters in a mile, to find how far Johnson ran in miles. Is your answer from part a or partb closer to the true answer? Briefly explain why you think this answer should be more accurate.

a) 0.049 $\mathrm{mi} \quad$ b) 0.063 $\mathrm{mi} \quad$ c) $\mathrm{b}$ is more accurate

Calculus 1 / AB

Chapter 15

Integration

Section 3

Area and the Definite Integral

Integrals

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this question were given a table of a runner speed over time in the time it takes them to around 100 meters and we are asked to figure out how far they have bread using very Monson's. So we have their speed. I know that the interval of speed gives me distance. So by integrating this using a reminds on approximation, we're gonna end up with distance. In part, they were asked to use left endpoints. We're gonna use the left endpoints as the heights of our rectangles for a remand some. And then we need to find the with a view track tangled based on the table before we can calculate any areas. So the width of this first interval is just 1.84 minus zero, which gives me 1.84 and then I'm gonna go ahead and capital it with the rest of the intervals as well. So 3.8 minus 1.84 is 1.96 and we have a with off 2.58 seconds, 0.85 seconds, 1.73 seconds and finally 0.87 seconds. So these air the wits that we're gonna use for when we find the area of the chart individual rectangles. Then we need to multiply the use by the height of the truck tangle and someone together together approximation. So in part, we were asked to use left endpoints said, this means a lot that our first interval from 0 to 1.84 seconds gonna look at the left, most speed, which would be zero. I'm gonna have 1.84 times zero for a second interval. You know, we have a with of 1.96 This time. The left most speed is 12.9. I'm just continuing down. So 2.584 or with times 23.8 quit 85 seconds times 26.3 MPH, 1.73 seconds times 26.3 MPH. And finally, for our last honorable, we have a width of 0.87 seconds times 26.2 MPH. So this gives me a sum of 1 77.162 But this isn't our final answer. We need to convert units, so I know that To calculate the area between these rectangles, I did a speed times of time. So I did MPH. Time seconds. Now if I had multiplied by hours, I will get MPH, times, hours, which gives me simply miles. So I need to convert those seconds into ours. We do this by dividing by 3600. There's 3600 seconds in an hour and I get that you left endpoints. The runner travels approximately 0.0 for nine miles. Mom, Part B were asked to repeat this this time using right endpoints. So the width of the intervals is still the same. We can still use the numbers and green, but now I want to use my great most endpoints. So for the first interval, we have a with of 1.84 seconds. But now the right speed is 12.9 and we'll go ahead and look at the rest of the interval. So 1.96 times 23.8 in this keys plus a with off 2.58 times 26.3 was the wife of 1.73 seconds. Sorry, I forgot being with a 0.8 size living with a 0.85 seconds times 26.3. Now that interval with A with Bob, 1.73 seconds times 26 MPH. And finally, that last interval with the with of 0.87 And that right most speed is 25.7 MPH based on the table. So this some works out to be to 27.932 Again, we need to convert to get this two miles. So divide by 3600 and this tells us the that the runner has wrong approximately 0.63 miles. So now in part C, we're asked to look at these answers and see if they're reasonable. So we know that the runner ran 100 meters and we're told that there are 1609 meters per mile. So by dividing by that, I will get the distance that their room and miles. So we know that there in exactly 0.62 miles, our answer in part B is closer to this. And this is because when we looked at that left most honorable The first interval, they were using the left most speed of zero MPH. So the distance up they ran in the 1st 1.84 seconds wasn't accounted for. We did account for using the right interval between names be accounted for more of the area under the quote unquote curve or move the distance that they ran. So Part B is gonna be more accurate and closer to the distance that the runner actually agreeing in this case.

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