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Automobile Velocity Two cars start from rest at a traffic light and accelerate for several minutes. The graph shows their velocities (in feet per second) as a function of time (in seconds). Car A is the one that initially has greater velocity. Source: Stephen Monk. a. How far has car A traveled after 2 seconds? (Hint: Use formulas from geometry.)b. When is car A farthest ahead of car B?c. Estimate the farthest that car a gets ahead of car B. For car A. use formulas from geometry. For car car $B,$ use $n=4$ and the value of the function at the midpoint of each interval.d. Give a rough estimate of when car B catches up with car A.

a) 9 $\mathrm{ft} \quad$ b) 2 $\mathrm{sec} \quad \mathrm{c} ) \approx 4 \mathrm{ft} \quad \mathrm{d} )$ between 3 and 3.5 $\mathrm{sec}$

Calculus 1 / AB

Chapter 15

Integration

Section 3

Area and the Definite Integral

Integrals

Missouri State University

Campbell University

Boston College

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using the graph given here of the velocity of two cars. And we want to find in part a how far car has traveled after two seconds. And so the idea there is were given. Here's two seconds were given feet her second, and the general shape when we take the integral is a rectangle. So let's just go ahead and draw a rectangle. If we're going to find the area under the curve, well, the height would be feet per second, and then we'd multiply by the with, which is seconds. But if we multiplied those two, we just end up canceling the seconds and get feet. That's why taking the integral tells us the distances because those units cancel like that and so doing so we could go ahead and split this up into a triangle. And that's triangle. Here is one half base times height on the bases, one the height of six, and so that would have an area of three. And then this rectangle is just base times, height and that would be one time six. The oh yeah, wait one time six. That's right. So that would be six. Therefore, both of those added together, those two shapes would give us this 9 ft. Okay. Part B one, his car a farthest ahead of Karbi. Well, car has a has a greater velocity this whole time all the way up until two seconds. And so that whole time it Z speed is greater. Therefore it's going to be further away. But then at this point, Carby starts toe catch up and then over well, catch up to car. So at two seconds is basically where Car A was the furthest ahead of Karbi for part C. We've now want to estimate that black area that I just shade shaded, which is how far Carrey is ahead of Karbi. Now we already know that we have the area under the curve of a from 0 to 2 and that is 9 ft. What we now want to do is calculate the area under the curve of Karbi, which is all of this area, and subtract that out. And then that subtracted out. We'll end up giving us that black area. So that's the idea. But we're gonna estimate that area using a bunch of rectangles. They each have a width of one half and so we're going to use for them. So let's just go ahead and put each of these rectangles here the first one. Sorry. I know it's hard to see them use. Green is like right here. For instance, the next one is, like right here, them here and then here. And we're using the midpoint for each of those rectangles. Estimate the area for each. So first we've got each of our widths one half all the way across. And then finally, we have the fourth one there. Now let's go back. And let's estimate the heights for these and that would end up giving us this'll. One. Looks like a height of 10.25 When we look at the midpoint there, the next one. Sorry, it looks like it has a height of about one. The next one has a height in between one and 1.5. Hmm. Maybe about 2.5. So that height there for the 3rd 12.5. And then finally, the third one, maybe about five. What could say? So that height here would be five. Now, let's just go ahead and put these in real fast, so we're going to do. I'm just gonna type this end to the calculator here. 0.5. I'm to a 0.25 and then plus the next rectangle will that be one time? 0.5. So if you just leave that plus 2.5 times 0.5 and then plus ah, five times 0.5 or just 2.5. But I'll just type it out anyway. Okay? Enter. And that gets us about 4.4 for these areas under the blue curve. And that's feet. So that's how far the blue car traveled. And therefore, if we do 9 ft and then subtract the 4.4 ft, then that will give us, um, What is that? 3.6 or 4.6? Yeah, 4.6. That's That's all right. Mental math. Not always my strongest, So nine minus that 4.6 it ahead. Okay, so I hope that help clarify what's going on with that one. And then, lastly, we have Part D. Let's go back to the graph. And so D is saying, give a rough estimate of one car be catches up with car A. I'm going to redraw the graph really quickly because it's just kind of a mess. We wanna I lights, um, important things. So we just calculated this is the distance that car A was ahead of Karbi. We want to find out when that area matches perfectly over here, and we want to then go down to what time that happens. Well, it's hard to tell based on this graph, but it looks like it's gonna happen at least between three times three and time is 3.5 somewhere in between. That gap is when the the area of part B is going to then, um, matchup back with car A completely. So I hope that explanation made sense here. And that's the idea. What velocity graphs is integrating them gives you distance.

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