9. [0/1 Points] DETAILS MY NOTES SCALC9 13.4.034. PREVIOUS...

9. [0/1 Points] DETAILS MY NOTES SCALC9 13.4.034. PREVIOUS ANSWERS ASK YOUR TEACHER Water traveling along a straight portion of a river normally flows fastest in the middle, and the speed slows to almost zero at the banks. Consider a long straight stretch of river flowing north, with parallel banks 40 m apart. If the maximum water speed is 3 m/s, we can use the sine function, $f(x) = 3 \sin\left(\frac{\pi x}{40}\right)$, as a basic model for the rate of water flow x units from the west bank. Suppose a boater would like to pilot the boat to land at the point B on the east bank directly opposite point A. If the boat maintains a constant heading and a constant speed of 5 m/s, determine the angle (in degrees south of east) at which the boat should head. (Round your answer to one decimal place.) 67.6 $\times$ south of east Need Help? Read It Submit Answer

Transcript

00:01 In this prime, we have a river that is three meters per second to the east, and that is three meters per second relative to the shore.

00:14 That's the w stands for water, s stands for shore.

00:23 So it's the speed of the water relative to the shore, three meters per second.

00:29 V, they tell us that the boat relative to the water speed is eight meters per second.

00:34 So boat, b for boat, relative to the water is 8 meters per second.

00:39 That's the notation.

00:41 Although the width of the river is 0 .2 kilometers.

00:45 The end result is if you start at a and a b, which is directly north of your position.

00:53 Now, how do you do? this is a relative motion problem.

00:56 How do you do a problem like this? well, you're adding up.

01:02 It's like having a boat and a person walking on the boat.

01:07 And you want to know what that person is doing relative to the shore.

01:10 You add up as vectors the motion of the boat and the motion of the person, the person relative to the boat.

01:18 And that gives you what the person is doing relative to the shore.

01:21 This is a vector operation.

01:23 So this is how we do it.

01:26 So the boat, relative to the shore, is equal to what the boat is doing relative to the water plus what the water is doing relative to the shore.

01:37 Those two effects put together as vectors, gives you what the boat is doing relative to the shore.

01:43 That's how it is, but it's a vector operation.

01:46 It's a vector operation.

01:49 Okay.

01:50 Now, let's look at this, let's look at this in terms of what we're told.

01:55 We are told that relative to the shore, the velocity of the boat is going to be from a to b.

02:05 It's what ends up.

02:06 Somebody watching the shore sees it going from a to b.

02:11 So let's draw this out.

02:14 So we have this vector, vbs, that we know, we're given that.

02:20 We're also given the water vector.

02:24 So we're given this vector.

02:26 We just got to find what this first vector is to add to this to get me this vector.

02:32 So here is water route to the shore.

02:39 Now remember the triangle rule.

02:42 Tangle of the second vector is drawn to tip of the first.

02:49 You draw the resultant from the tail of the first to the tip of the second.

02:54 Now it's kind of, this is kind of unusual because you normally do it the other way around, but let me draw this in and then show you that it actually does satisfy that construction.

03:03 This would be the fee of the boat relative to the water, not the speed, but the velocity, the vector, the velocity of the boat relative to the water.

03:14 That's different.

03:17 That's different.

03:17 This is, it has as magnitude 8 meters per second, but it's, but this has got the directional aspect.

03:24 Okay, let's see how this works.

03:27 Tail of the second vector, drawn to the tip of the first.

03:32 Then you draw for the resultant from the tail of the first, tip of the second.

03:35 That looks right to me.

03:37 That's how, that's how you do it...

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