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Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.

$117 \frac{3}{3} \mathrm{ft}$

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Caleb E.

Baylor University

Michael J.

Idaho State University

Boston College

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racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car in miles per hour during the 1st 10 seconds of the race. Use the midpoint rule T to estimate how much further Kelly travels, then Chris does during the 1st 10 seconds. Okay, Tee's the midpoint rule. We're going to want to break up our time intervals s so that we can actually use data at the midpoint s. So what do I mean by that? If we break up our time intervals into to second intervals. So, for example, 0 to 2. 2 to 4. 4 to 66 to eight and 6 to 10. This allows us to use made points because in the first interval, from 0 to 2, we have a midpoint At T equals one. In the next interval, we have a midpoint of T equals three, and so on and so forth. We have 57 and nine. So we're going to be using data at the mid points to play the midpoint rule. So the idea here is that we want to figure out how much farther Kelly travels, then Chris does in each of these each of these intervals, and then we're just going to add them up at the end. So to start, let's look at the first central zero, less than or equal to t lesson equal to two. So here we want to figure out How much farther does Kelly traveled in Chris in this interval? So how do we figure out distance, the total, The difference in their position that's going to be velocity times time. So let's call this D one. Like the difference in their distances. I am the first interval, so that's going to be velocity times time. But since we're looking at the difference in their positions, we're going to look at the difference in their velocities. So that's going to be V K. Minus V. C. But we're looking at the midpoint, which is Time T equals one. So okay, let's write her a V K minus V C. Then we're going to multiply by Delta T. That's going to give us the total difference in their distance for this time interval. So we use the midpoint time key Close one. Let's write the note. T equals one good. And what we're going to do is we're going to God used the values from t equals one. So at T equals one V K is 22 minus R V. C is 20. Noticed these earn miles per hour are delta T. Here is two seconds. So we're actually going to want to convert our seconds to hours. So how do we convert seconds to hours? Well, there's 60 seconds in one minute. Oh, we want to write it in the opposite way so that our units cancel. Okay, there's 60 seconds in one minute and 60 minutes in one hour. So 60 seconds in one minute and 16 minutes in one hour. Since this is miles per hour, we see your seconds in seconds. Cancel minutes and minutes cancel. And then this is miles per hour. Hours cancels with ours. So we're going to be left with the distance in terms of miles. So multiplying this out, we have a two times two over 60 times 60 which is for over 3600 miles. Ah, which is equal to two over 1800 or one over 900 miles. So this is the difference in their distance during the first time interval from 0 to 2. So what we want to do is we want to do the same thing for each, um time interval. So two less than or equal to four less cynical time lets a race that let's write it neater to less than equal to t listening. Go to four we're going to do for less than equal to t lessner equal to six six to AIDS and then finally 8 to 10. Okay, so let's let's do it. D two is going to be equal to now. We're looking at the second time in trouble, so we're going to use Time t equals three. So it's going to be V K minus V C times Delta T So 52 minus 46 miles per hour times two seconds. Uh, times this whole piece here. Two seconds. But we converted to, um, hours. So it's going to be to over 60 times sixties 360. So I had 102 money here. No, I didn't. This is correct. Times to over 3600. So what we have here is six times one over 1800 which is one over 300 miles. So we look at this next time interval D three, we're going from 4 to 6. So the midpoint is that Time t equals five. So if he came on, his VC is 71 minus 62 multiplied by the time which is one over 1800 hours. So here, 71 minus 62 is nine. So nine over 1800 is one over 200 d four is equal to well, dif for We're looking at the time interval 6 to 8. So our mid point is that time is seven. So he came on. This VC is 86 minus 75 multiplied by a delta T, which is two seconds or a one over 1800 hours. So this is equal to 11 over 1800 miles. Again, it's always important to keep track of units. It's It's usually a good indicator that you're doing something either correctly or incorrectly. Okay? D five is from Tom Goes from 8 to 10. Sorry. Midpoint is nine. So he came on. His VC is 98 minus 86 98 minus 86. Uh, times won over 1800 which is equal to 12 over 1800 or we can simplify it a little bit more. We could divide by half by two six over 900 which is to over 300 miles. Okay, so now, in the total distance that Kelly is ahead of Chris at the end is just going to be the sum. So we take D one plus D two plus 33 plus 84% E five, so total distance is equal to won over 900 plus won over 300 plus one of 200 plus 11 over 1800 plus to over 300. All of this is in miles, so let's make note of that. This isn't miles, but we can converted two feet by using a conversion factor. How many miles are how many feet are in one mile? There is 5280 feet in one mile. So after we compute all of this, we get our final answer in terms of feet. So we find that at the end of the 10 seconds, Kelly is ahead of bike eyes ahead of Chris by a total distance of 117 and 1/3 feet

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Caleb E.

Baylor University

Michael J.

Idaho State University

Boston College

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