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A boatman wants to cross a canal that is 3 km wide and wants to land at a point 2 km upstream from his starting point. The current in the canal flows at 3.5 km/h and the speed of his boat is 13 km/h.(a) In what direction should he steer?(b) How long will the trip take?

a) The boatman should steer along the direction which makes an angle of $43.4^{\circ}$with the vertical, toward upstream (that is $+$ ve $y$ direction.)b) 20.2 minutes

05:39

Wen Z.

03:05

Carson M.

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 2

Vectors

Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

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Question were given that it's an interesting one. Boardman wants across the canal, There is three km wide and want to learn at the .2 km upstream from the starting position. So, let's try to draw how this looks like. So, let's say this is the canal and this is three km wide. And the Boardman starts from a point, let's see over here and he wants to land are two km upstream from this point. So he wants to learn somewhere over here. All right, now, the canal is flowing. Uh, canal is flowing over here that 3.5 with the speed of 3.5 kilometer per r. And the speed of the board is 13 kilometers per are, But we don't know the direction. So in which direction should the board steer? All right. So, uh, let's say uh, the board steers somewhere in this direction and this is the anger tita. And we know this speed already is given. That's 13. So, if you make the components and there will be two components. A horizontal one will be 13 costs cheetah And the vertical one will be 13 sign Tita. All right. Uh, now, uh, we can divide this whole thing in two parts. One is the war tickle motion. The vertical motion which will just depend on the vertical velocity. So vertical velocity is only 13 scientist to And the distance is three. So, where does the time taken time taking this distance over speed? Distance is three And the speed is 13 scientific to. So this is the required time. Let's call this equation one. And now we talk about the horizontal motion, horizontal motion is given by uh again the horizontal distance over time. So horizontal horizontal speed will be the horizontal distance over speed. Speed will be the sum of these two. So that time is gonna be distance which is two kilometers over speed which is 3.5 plus 13 Cause cheetah. And let's call this equation too definitely both. The time has to be seen because that's the same object which is traveling. So we are going to equate one and two. So it's gonna look like 3/13 scientific to has to be equal to two or 3.5 plus 13. Cause cheetah, this is a proportional list Cross multiply. This will be 3.5 times three which is 10.5 10.5 plus 39 Cause Cheetah is equal to 39 sign Cheetah. So uh let's bring this over to the right. So we have 39 scientific to minus cause cheetah is equal to 10.5 Which implies that the value of scientists uh minus cost cheetah is 10.5 or 39. Uh that's where I think we can use a graphically calculator to find the value of tree to So let me do that and get the value. So this is coming at 0.977 Radiance. And if you want to film writers and degrees. So we have .977. Let's convert that and degrees In 20 over by. So that's coming at 55.98°. So the angle is 55.98°. So this is the direction with respect to the horizontal which the which the board should have. So that that answers part a part A and for part bill Stewart over here for part B. We just need to find the time. So that will be relatively easy because we already have the equation of time and we already got the value of tree to so it's just a matter of substitution of Tita which is 55.98. So sign of 55 points 98°. So let me find that A sign of this times 13 and three should be divided with. So that's zero to a zero point where seconds up to uh seconds. Uh not really, it won't be seconds. We have to be careful about the units because everything was in kilometer per are and we have not made any conversion. So the unit has to be an are. So it's zero point 28 or this is up to two decimal places. So this is the time which it will take. Thank you

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