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.