Score n last attempt: [ = out of 4 Score gradebook: [> out of =Ca D is constantly fcct from the center of the race track and travels at a constant speed. The ang...

Running in a wind A strong west wind blows across a circular running track. Abe and Bess start at the south end of the track and at the same time, Abe starts running clockwise and Bess starts running counterclockwise. Abe runs with a speed (in units of miles/hour) given by $u(\varphi)=3-2 \cos \varphi$ and Bess runs with a speed given by $v(\theta)=3+2 \cos \theta,$ where $\varphi$ and $\theta$ are the central angles of the runners. (FIGURE CAN'T COPY) a. Graph the speed functions $u$ and $v,$ and explain why they describe the runners' speeds (in light of the wind). b. Compute each runner's average speed (over one lap) with respect to the central angle. c. Challenge: If the track has a radius of $\frac{1}{10} \mathrm{mi}$, how long does it take each runner to complete one lap and who wins the race?

So for the given problem, we're going to park our car at a trailhead. Um And FFT is gonna be the distance from the car t hours after seven a.m. On friday morning. And GFT is the distance from the car seven or two hours after 78 months sunday morning. So we want to evaluate ff zero, ff two, GF zero and GF two. So you see that from part A. Um we'll get zero because that's how far we are from the car from the very start. Um And then based on this, what we end up seeing is that the intermediate value theorem shows that there must be some point along the trail where we pass the exact same point Africa is a continuous function. So we go from zero all the way up to um the point and then we head back the next day. So we see there must be a point at the same time of the day where we passed the same point.

So for problem 31. We are asked to find whether or not the circle is correct on the graph. So we are given that Dr T equals three and we also know that this is the integral and we chose 11. But you can choose any point that you seem is right. But this point that has circled seems to be either 11 or 12. And I chose the love so we can go ahead and use the midpoint rule. And to do that, you just do Dr. T over to it caused 1.5. That would be the first midpoint so we can go ahead and right the formula, which would be delta t times the summation, uh, the May of tea of I. And in this case, I would go from one all the way to three so we can write. We can plug in t just three and do the summation, so that would give us so be of 1.5 would be 20. And then you add it with be off after 1.5 you get so it would be 1.5 plus three, which is 4.5 both three witches, Southern 30.5. So we got be of 1.5 now for V of 4.5, you would get 73 Infer V of 7.5 would get more. No. For and this equals. Hey, uh, four, five, 81. And then you must add two times 10. So two times be of 10 must be added because of the way the graphs pleased. So this equals 835. However, this is not the final answer, because this isn't mild. And we wanted in feet, so this isn't mild and you wanted in feet. So to do that, we just multiplied by 52 80 and divided by 3600 and you will get so a 35 times 50 to 80 over 3600 which is one, 22 for feet. So this would be the answer. And it does seem approximately correct with the graph. So yes,

In this question. We need to find the faith or the Taylor polynomial, center it at different points. So let's first take the purity of life in the beginning. And we need to take a dirty, up upto faith order that since it's a basic function, so it's no that if it coached to do Okay, So this is, uh, this are the narratives off if up to faith older No, If we can see that the center to be zero So zero because 20 now, first of the year it if, as euro equals to one a second curative because the 0/3 daughter directive because the minus one fourth of the theater vehicles zero in the last pieces one. And if you chose the centre to be pie, the first of them a typical zero. This, uh so the the only functioning closed zero the first of the diuretic because the minus one Sekondi generative zero third of the narrative one fourth or the narrative 0/5 or the directive you goes to find this one. So according to this, um, by the notation with the notes, the Taylor polynomial center that because zero by P five which is so we only have three terms survived. So the union term X, my nose, the cubic term 1/6 X cube and the fifth of the term process 1/1. 20 x 25. So off this coefficient comes from the fact Orioles and with the notes, the tailor Parliament was entirely pipe I Q five again, Um, only three terms survive the linear term. Que big term. The center should be high, so we have minus X minus pipe and then we'll have a cure. Big term. Um, we have a faith of that term. So this to our p 500 q 5 really me to figure out Once we have this, we can graph if the organ function safe and this to tailor porno meals on the same coordinate system. So it looks actors, let's play ball some points. So this point the purple line, that vertical line Miss X equals two pi heads. Um, so the through line, Yes, p five. Because you can see that for the through line. Um, I imagine, you know, function at X equals zero, and in the Green line is kill five. Mitch. This curve mentioned the on the on geno function. It's X equals two parts. Okay, so we can see that, um p five, the conclusions p five years better. Um, on the interval, thrown minus minus. Pi two pi half Because we can see that over this interval minus from minus pi to half the blue. The blue curve is close to the original function. The green curve has some error here. Eso obviously p five years battling cure five even over this interval. And the cure five, he spent her own. Uh, I have to two pi, so we just kind of obvious. So frump, I have to two pi the green curve. It's close to the original function and the for path three. We can show some points some different value for X And the finally of Serotta era became the original function. The tailor party nominal were used, So this is a result. So the conclusion is the p five. I respect her then Q five at x e coastal powerful because this makes sense because powerful is closer to zero. Um, so p five is sent her There's euro So p five performs better than cure five and X equals two powerful. Another observation is that when X because do I have P five and the cure five have the sin, Kara, which also makes sense because I have is the middle point of zero in the pipe. So p five sent her zero in a cure five center A pie in the meadow there will have the San Arab

So let's make sure that we set up this problem properly, so we have to get from a point p. This is where I am or where you are. So this is our point p and we have to. We're on a straight shore of a lake and there's a stranded summer who is 50 m from point. Q. On the shore, that is 50 m from you. So we have a point here. Point P. Here point Q. Here there is 50 m between here. This is on the shore, but then we have our lake and this stranded swimmer is 50 m from there. So the idea is we want to know first, the function that gives the travel time as a function of X and we know that if we were to just swim straight here, we could do that. But it's going to be slower, Um, because swimming is slower than the running. So based on this, we see that T F X is going to equal the square root of 2500 plus X squared over two. And that comes from the fact that we have 50 squared. So 50 squared is 2500. So this distance right here is going to be the Distance X that we're going to want to go. And then ultimately there's going to be this distance right here. But this distance from p two, the stranded swimmer, um, is going to be 2500 plus X squared to the square root of that. And then we divide that by two because that is how fast we can go swimming. So that's just the straight shot there. Um and then we know that it's plus 50 minus x. So X is the distance that we run. Obviously, it can't go more than 50 so we could go all the way here. Or we could go part of the way X is really what we're trying to determine that value that distance, that we're going that's over four because we can run at 4 m per second, then for part B, we want to find the critical point. Um, So what we do is we take the derivative and we set it equal to zero. So when we do that, we get cheap prime of X, we set it equal to zero, and then we solve for X and we get the X is equal to the square root of 20 five 100 over three, which is approximately 28.868 And we want to know what the minimal travel time is. So we know that this is how far we should go with X. Um, so with that being the case, we now plug this back into our original function. That gives us a more travel time. So we have t of Route 2500 over three, and that's going to give us 34.151 seconds, which means it will take about 34 seconds to get to the stranded swimmer.