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Woman launches her boat from pointOnbank of a straight river; 3km wide, and wants to reach point B, Skm downstream on the opposite bank quickly as possible She coul...

Question

Woman launches her boat from pointOnbank of a straight river; 3km wide, and wants to reach point B, Skm downstream on the opposite bank quickly as possible She could row her boat directly across the river to point C and then run to B. or" she could row directly to B; Or she could row to some point D between C and B and then run to B. If she can rOW 6kmyh and run Skmyh, where should she land to reach B as SOon as possible? (We assume that the speed of the water is negligible compared with th

woman launches her boat from point On bank of a straight river; 3km wide, and wants to reach point B, Skm downstream on the opposite bank quickly as possible She could row her boat directly across the river to point C and then run to B. or" she could row directly to B; Or she could row to some point D between C and B and then run to B. If she can rOW 6kmyh and run Skmyh, where should she land to reach B as SOon as possible? (We assume that the speed of the water is negligible compared with the speed at which the woman rOws: ) dy (b If y = € + tanz, show that cos? _ 2y + 2x = 0. dr



Answers

A woman who can row a boat at 6.4 $\mathrm{km} / \mathrm{h}$ in still water faces a long, straight river with a width of 6.4 $\mathrm{km}$ and a current of 3.2 $\mathrm{km} / \mathrm{h}$ .
Let $\hat{\mathrm{i}}$ point directly across the river and $\hat{\mathrm{j}}$ point directly down stream. If she rows in a straight line to a point directly opposite her starting position, (a) at what angle to i must she point the boat and
(b) how long will she take? (c) How long will she take if, instead. she rows 3.2 $\mathrm{km}$ down the river and then back to her starting
point? (d) How long if she rows 3.2 $\mathrm{km}$ up the river and then back
to her starting point? (e) At what angle to \hat{i} ~ s h o u l d ~ s h e ~ p o i n t ~ t h e ~ boat if she wants to cross the river in the shortest possible time? (f)
How long is that shortest time?

Okay, so this question gives us a north flowing river where the speed of the water north is given by this function f of X and it asks us some questions about a boat. So the first question part, eh? Says that the velocity of the boat is five meters per second, west, east. So the boat is going directly across the river this way, and it asks how long we're gonna end up in the wider action. Because, of course, even though we're going in the extraction five meters per second, we're under this water flow force f of X. So how do we do this? Well, let's construct our vector v A t. And that's equal to the sub X and visa by or our velocity And the ex direction is always positive. Five. But in the wider action, it's this floor eight f of axe. But we have a function of tea. So we want to get rid of this X somehow, and we can do that. Using the fact at visa backs equals five meters per second. So that means that ex of tea is just equal to five t. And you could get that by integrating because since we started X equals zero, our integration constant zero. So this gives us the relation X equals five t and now we have a substitution that we can make in the F of X. So f of X is the same thing as f of five t because X equals five t so we can just plug in five T everywhere we see an X to get our f of tea. So effort five T is just equal to three over 400 times five t times 40 minus five t. So now our updated vector V of tea becomes five because the ex velocities unchanged and then our new function that is solved for in terms of time. So now we have a velocity and we want to go to position. So to do that, we need to know. When does the boat reach the shore? Well, since X and wire independent directions, we know that ex of tea equals five t, and the river is 40 meters long, so 40 equals five t or it takes eight seconds to reach the end of the river. So that means that Delta Y is just the integral from zero to a of the y velocity with respect to time. So to calculate this displacement, we just integrate the why component of our velocity, which we said those three over 400 times five t times 40 minus five t tt and that you can just punch this into your calculator to get Delta y equals 16 meters and we'll just write that a little neater because remember, integrating velocity gives us displacement. So now we can move on to part B that says No, we're in the same river that's still flowing north, but we're moving at an angle Fada, but still at five meters per second and it wants us to know it's asking us what angle should we go at so that we end directly across from where we started. So we want Delta X to be 40 meters still, but we want Delta y two now be zero. You want to go directly across the river so we can start with our vector v of tea again, which is piece of X and visa. Why? And now let's figure out what these components are. So no, we have to realize that we have components of velocity because we're not going in a straight line and the river velocity to consider. So if we draw RV boat right here at an angle Fada, no. We'll get velocity in each direction from the boat and using trigonometry. We see that the boat velocity and the ex direction is just five co sign data and the Y velocity for the boat is just five signed data. But that's not the whole story because we still have to consider the water flow. So v of tea is still equal to five co signed data in the ex direction because the boat velocity is not here. Sorry. The river velocity doesn't affect that, But we get a five scientific data from the boat velocity plus our function f of X from the river flow. So now we're gonna do the same thing as we did in the last problem and turn this f of X into an f of tea and data in this case. So since we said that visa backs was five coastline data, ex of tea is just five co sign of theta times t because again we're starting at X equals zero. So this gives us the relation X equals five coastline data times t so we can plug in tow f of X so fo five t for sorry. Effort five co signed data T is equal to three over 400 times five co sign data T Times 40 minus five co sign Fada Tiu And now we can plug back into our velocity one last time to get our expression for V A T, which is just five co sign data and then five signs data plus f of tea which is three over 400 times five co sign Tha T times 40 minus five co sign Veda Times t So now we're almost ready to integrate. We just need to figure out we just need to figure out how long we're going once again So we know that Delta X equals 40 or ex of T equals 40 at the end time, so 40 equals except T or 40 equals five co sign of theta times t four. The end time is equal to eight divided by co signed data so we can eliminate our end time. So sweet. So now we have another substitution. We can make it the end. So why of tea is the integral of Visa? Why? See? Or we can answer Great our function which is the Y velocity from the boat plus effort five Co sign data T DT And we know what both of these are. It's just a lot to write So this gives us the function. Why of tea? When you integrate this out is equal to five. Sign of fate a T plus 3/4 co sign Fada t squared minus 1/16 co sign squared of fada t cute and our integration constant zero because we're again starting at y equals zero And now we want to find why have tea at the end, which we know has to be equal zero four We found that T end is equal to eight divided by coastline data which is plugging everything in zero Cols Why of eight over coastline data or five science data over coastline data times eight plus 3/4 times eight times co signed data because one of the coastline Tate is divide out. We're sorry coz and data over coast t squared this eight should be squared OK, I see over co signed data Squared CO signed data, and then our last term is minus 1/16. A cube co sign squared over, co sign cute. And then we can actually just get rid of these denominators because we have a zero. So just multiply through by by our one over co sign, and this will just reduce it so much easier. Equation to solve, which is just 640 times signed data plus 2 56 equal zero or sine theta equals negative 2 56 over 6 40 for signed data equals negative 2/5 or fada equals the inverse sine of negative 2/5 which is equal to negative 23 0.58 degrees.

So here the spirit has given us, It is given as V as equals two 36 km/h. It is statistics kilometers per hour. Let us convert it into meet a person can so we will write 36 multiplied where 1000 divided by 3600. So it will give us a value of 10 m purse again. No, We have been given that the spirit of the flow of the river is equals two. Fear of the flow of the river as is given or you is given as or two Square root of nine plus 0.1 X. So at maximum at limiting case you will be equals to be. So we can write, we can write who is square root of nine plus 0.1 X. This will be equals to 10. Or let us square both sides. So we will get four multiplied with nine plus 0.1 X. This will be equal to 100. Or we can write it as nine plus 0.1 X. This will be 100 divided food. That is 25, but X will be equal to 25 -9. You were 0.1 or it will be equals two. It will be close to 100 60 m. Mhm. The latest sick option, so after taking off some VC option B is correct. Mhm.

Being given our river in this problem, who's Richt is 1.20 mines. This is the rip. The reward 1.20 My eyes. You. Sheena wants to cross the river from point A and she starts rowing her boat with the speed three MPH in a direction upstream. 60 degree up. Scream the river like this. This is the direction off the velocity of the board and the speed off water current has been given as 1160 miles are are This is 1.60 miles but are which is represented as we are the speed off water current. And here this is we be the speed off boat in Stillwater. Three miles. But, uh, as a result, Sheena will reach at the line. Some there like this. Let it be. See, she never reach at a point c With the religion velocity this we off being it is that you are to find it in order to find the resultant velocity, I must have the value off angle between these cool directors. Maybe, and we are in this angle can be found just a fall. We will find the anger in right angle. Triangle A B D in this right angle triangle, ABC Using angle simulation This angle comes out to be 30 degree, so using linear pair, this angle comes out to be 1 50 degree. So for the first part of the problem, we have to find related velocity off Sheena with respect to her original position. Eight. So it is given with the help off triangle Law off Better addition which say's that we'd be our is given as the square root off we be square unless we are square less wise off. Maybe we are going to co sign off angle between them. And here this is three square 1.62 sweat the wise off Levi Ministry into 1.6 into close. I know 1 50 degree. It will come out Toby nine plus to find 56 plus 9.6 and call sign off 1 50 degree, which will come out to be negative. So actually it will be subtracted from the addition off nine and who bind 56 So the final value which we get is Spy Rudolf, 3.246 miles per our, which comes out to be 1.8 miles. Are are This is the answer for the first part of the problem? No. For the second part, we need toe calculate time taken by Sheena in order to reach from Lucy. So I find this time most of all, we must have the distance a c. In order to find this distance a c, we will use right angle triangle ABC. But then we must have the value of this angle, Peter. But we don't have it. Then we should have the value off angle. L far which the result end we are, is making the baby. So for it, we will use relation for this angle off result, and better then it for which will become we are sign 1 50 degree divided by read. Be less. We are course 1 50 degree. This is the standard formula. You find the angle which the resultant makes with any off the so it could become 1.6. It will sign 1 50 degree divided by three blurs 1.6 course 1 50 degree, and it will come out to the 0.8 on one point 614 because being off use, angle or sign off. 1 50 degree will come out to be negative. So it will be subtracted from this tree and this re shore will become 0.496 So finally, this angle l far is even as then. Iwas 0.496 Or I can say this angle is 26 point or degree. So finally I can find the single beater. So angle beater is 60 degree minus 26 point for degree, so it comes out to be equal to 33.6 degree. So now I can apply any Dignam metric ratio in this right angle triangle. ABC, as I have to find a sandwich is high protein IUs and I have been given this a b which is based. So I use co sign off angle. So using right and gold trying girl it B C course beater, maybe equal to based baby divided by hypertension was a C. So a sea will become be a phone cause beater, you know? Yeah, be is 1.20 miles and cause sign off Beatem in school. Sign off 33.6 degree, which comes out to be 1.20 miles divided by 0.833 means final value of this s e means total distance traveled by the boat in order to reach from eight To see it becomes 1.4 for mine's. So this is their distance. So finally I can find the time taken So trying you can by the board to cross the river is Chinese was bogus Stands upon a speed distance a C speed we off beat respecto are here This distance is 1.44 miles and we are we already have found in the first part 1.8 miles Where are so here? I will get this answer as 0.8 us. So the answer for second part is zero point it bus? No, For the hard part we need to find the distance between the point at which machina wants to cross the river. She wants to go to a point Exactly It oppose it, Tow her starting point means be but she reaches at sea. So we have to find this distance BC so easily we can find this distance which will be again in this triangle. ABC in, right and girls trying girl Yeah B c This time will you sign Beater here. Sign Vita become for perpendicular BC a bone hyper tennis isi BC upon ISI Mickum So B c is given by the a c into sign beat up which comes out to be e c is 1.44 into sign off 33 Why? I'm six DVD So 1.44 into sign 33.6 degree If I may, I will get a final answer as 0.8 miles So in place off reaching at exactly opposite wine She know Reges at a distance 0.8 miles upstream Finally in the final part If she now wants to go to the exactly opposite point B for on E we wanted to find at what angle? Exactly At what angle? She should start ruing our boat. This can be found again with the help of this right angled triangle in which this is the perpendicular we are And this is the high party news we be So the sandal will be given by Sign Peter Equal school we are on BB means 1.6 on tree which comes out to be 0.533 So Tita is sign in worse 0.533 which comes out to be 32.2 degree up the stream. So finally, if she wants to cross the re very straight, she shoot, start rowing her boat at an anger 32.2 degree off the stream.

Here we have a scenario, Sheena can row a boat at three mph in still water. And she needs to cross a river that is 1.2 miles wide. But it has a current flowing at 1.6 mph. Mhm. Since she doesn't have a calculator ready, she guesses the to go straight across. She needs to head upstream at an angle of 60° in order to form a direction that goes straight across the river. So we're gonna ask ourselves a couple questions to this scenario. Let's set up the problem first, before we ask these problems or ask these questions. So her speed. The velocity. Uh huh. Uh Sheena. Which I mean I can recall yes with respect to the water. S. and W. is equal to three MPH. The river is flowing velocity of the water with respect to the ground which are called G. V. W. G. Is equal to one 0.6 miles per hour. Okay the distance across the Is 1.2 Girl Miles. Yeah in the angle that she travels at first 60 degrees. So let's draw this out. We have a river mantra is two parallel lines. Of course this distance here Is that 1.2 miles? Mhm. If it's flowing downstream again that the speed is 1.6 mph. Sheena guesses that she needs to travel in an angle of 60° from the direction straight across the river like this or the angle with respect to the direction strip across is 60 degrees. Again that speed here just three mph. Mhm. Yeah. So now in order to find her speed with respect to the grounds we just have to sum up these factors and so let's give the maximum components. Yeah for the speed of Sheena with respect to the water we're going to have three miles per hour times. You get the X. Component that's going to be this adjacent side. Right here we're going to do the co sign of 60 degrees. That's again going to be any X. Direction. Of course this is going to be negative because she's moving to the left in our case. Okay. And for the why we're going to have this sign of 60°.. Okay. That's in the Y direction. Yeah, I just want to play this out three times the co sign of 60. Get a value of negative 1.5 mph in the X. And for the way, Three mph times the sine of 60° we get a value of 2.5 98 MPH with respect to the wife. Now let's take a look at the velocity of the water with respect to the ground victor. That's going to be directed In the Y direction, perfectly done. And so it's going to have a value of just negative 1.6 MPH in the Y. Direction because again strict it straightly down. So now let's add these two vectors together to find the velocity Sheena or the respect to the ground. Again, that's going to be by adding velocity of Sheena with respect to the water plus the velocity of the water with respect to the ground. When we do this we get negative 1.5 mph, the X. Component from Xining with respect to the water. And then we add the two that we get from both for the wine Subjecting 1.6 from 2.598. We find that speed to be positive zero point 998 MPH. Why? And so there we have her speed with respect to the starting point. The bank is now the magnitude of this new vector we've created. So I'm gonna do the square root Have 1.5 squared plus 0.998 squared to find the magnitude of that speed. So the velocity of Sheena with respect to the ground is 1.8 zero MPH. Now we ask, how long does it take her to cross the river? Remember we're only concerned about the horizontal component because this river, it goes on, you know, hypothetically forever north and south, but moving directly to the left is the only part that matters in order to cross the river. So let's take our horizontal component negative 1.5 mph. We're gonna divide that Into the distance to cross again. That distance across is 1.2 miles. We're gonna divide by the magnitude because I'm not really concerned about direction right now, although technically this would be negative if we were to take displacement into consideration, I'm gonna do 1.2 miles, divided by 1.5 mph in the horizontal direction. Find the time 1.2 divided by 1.5. And I see that it takes is your .8 hours. If we want to do minutes, that's 48 minutes to cross. Given the path that she took. Now we ask how far upstream or downstream from her starting point, will she reach the opposite bank? And for that, we are concerned about the way We see that the y component is your .998 miles per hour. And so in order to figure out how far up we went, we're going to multiply that by the amount of time 0.8 hours this time I do care about the sign because I want to know if that displacement is positive or negative. In this case, obviously it's going to be positive because she's going up the river 0.998 mph time, 0.8 hours. We find that she goes a distance of 0.798 miles. That is her distance that she travels upstream. As a final challenge. We want to know if she wanted to go straight across, what angle should she have headed? In other words, I'm going to take the original vector from up here. Remember that down here? All right, take our original water inspected the ground vector. I'm gonna move it down here this time. We're going to give her a new angle. Let's call that angle data. And we want it such that why from both of these perfectly cancel out. So all of her movement is directly across the river. So once again had these factors. It's the new vector, the vector of Sheena with respect to the ground which is the sum of the two of these is going to be -3 mph times the co sign with data with respect to the X. Yes. three mph. Times the sine of data -1.6 mph with respect to the wife. Yeah We want this value to be zero. In other words three times the sine of data Is equal to 1.6. So the data is going to be equal to the inverse sine arc sine of one point 6/3. The first sign of 1.6 divided by three mean value of 32 right two degrees. This is the angle that she should have traveled if she wanted to go directly across. And that angle again, this one from the horizontal. Thank you


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