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On the dock: The tip of the fishing rod_ 15 feet above the fish is being reeled in by person sitting perfect right triangle. The fishing line was reeling the water;...

Question

On the dock: The tip of the fishing rod_ 15 feet above the fish is being reeled in by person sitting perfect right triangle. The fishing line was reeling the water; with the fish and the water surface form (negative rate because the fishing line is getting shorter = and fish at rate of negative feet per second shorter): is the distance from the fish to the dock getting shorter at the instant when there is a total of How fast 25 feet of fishing line out? (5 points) At that moment as above what ra

on the dock: The tip of the fishing rod_ 15 feet above the fish is being reeled in by person sitting perfect right triangle. The fishing line was reeling the water; with the fish and the water surface form (negative rate because the fishing line is getting shorter = and fish at rate of negative feet per second shorter): is the distance from the fish to the dock getting shorter at the instant when there is a total of How fast 25 feet of fishing line out? (5 points) At that moment as above what rate (in radians per second) is the angle of elevation (the angle formed between the line and the water) changing? (5 points) Also, what is the rate change ofthe area of the right triangle at that moment? points) point if not) and label for the vertical Note: To get full credit vou must draw the picture side/the height and B for the horizontal side /the basc, and € for the hypotenuse of the right point = if sides are labeled differently and -1 point for triangle, and 0 as the angle of elevation (-1 missing unit in the answer)



Answers

Standing on the shore of alake, a fisherman sights a boat far in the distance to his left. Let $x,$ measured in radians, be the angle formed by the line of sight to the ship and a line due north Assume due north is 0 and $x$ is measured negative to the left and positive to the right. (See Figure $6.53 .$ The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance $d(x),$ in kilometers, from the fisherman to the boat is given by the function $d(x)=1.5 \sec (x)$
a. What is a reasonable domain for $d(x)$ ?
b. Graph $d(x)$ on this domain.
c. Find and discuss the meaning of any vertical asymototes on the gaph of $d(x)$ .
d. Calculate and interpret $d\left(-\frac{\pi}{3}\right)$ . Round to the second decimal place.
e. Calculate and interpret $d\left(\frac{\pi}{6}\right)$ . Round to the second decimal place.
f. What is the minimum distance between the fisherman and the boat? When does this occur?

Here We have a fishing pole on the tip of the fishing police stand Faith from the water level, a fish is reeled at the rate of one foot per second. Um, this means that the length of the line if we call these X is decreasing at this with so the next 80 is negative one foot per second. Now we wait. You know, the the angle made by the line in the water level with data, and we want to find the rental rate of change of detail. So t t t when the length of the line is 25 feet. Here we have a right triangle. We know they're so opposite side to dangle, Tita and um, we're also know the rate of change of X so we can use sign the sine function because this will involve the opposite side which went in always 10 and of X. So sign off the cycle 10 over x both accent. It'll change with respect to the time teeth so we can different share takes DDT of scientific hm equal d d t of 10 over x here. When we differentiate sign of Tito, we should keep in mind. He ties a functional tea. So we have to use the general first with different shapes. Stein and that is co sign Tita. Then multiply with the teeth are dating and on this side we have 10 over extend over existent times, one over act. So we have 10 and then they leave a tip of one over X. We get minus one over X squared times the ecstasy on the right side. We know access 25 the exit is negative one. So here we obtain a negative one time. So negative we get rid of the negatives. So have 10 over X squared, which is stand over 25 squared on. That is 625 and simply prize will be right by five. We get to 0 125 Now if we know cause Santita then we will be able to determine t t talk. Get it? Coastline did come. We look at our travel again Is gonna be the a Jason side to the angle, Tita or divided by the high purchase. So let's call this side. Why? So we have why Divided by X which is 25. So we want to find Why? Why is one of the sides of this right triangle where we know X is 25? And they saw the other side This tent So we can use the pit agrarian theorem on and we get y squared plus squared equal 25 square. And then we have y squared equal to 625 minus 100. Why square these 525 and square? The 525 which it is five squared of 21. So now we can go back here and fight co sign Tika is five squared of 21 over 25. We can't simply fine. And we get squared 21 over. Now we can go back here cause Santy ties squared them too. 21 over five so we can solve for de tete a tete so didn't eat at the tea is too over 125 times square Root off 21 over five We divide by that soul Be flipped We get five over Scripture 21 And then it was simply fine And we're left with two over 25 School took 21 But this with if we plug it in a calculator, we will see. That is approximately equals 0.17 And this is measured. This is the rate of change of the angle is measured in radiance. You got a second?

This is a related rates question based off of a figure and it scenario introduced by the book. So what you have to do is end up making your physical model for this, which is gonna look at the triangle. So here we have the structure with a hook at the top, and then over here is water going down to the boat. Then we have our partners here. L as given in the book. L they appear. And then we have 10 right here. And then here's our angle data. So for part A here, waiting to show what the data over D l is equal to. Well, to do that, we need to first get a derivative data. But to do that, we need to get a function for data in terms of L. Anyway. Well, we're gonna use this with Trip. We're gonna use Trig for this. So we have 10 and l corresponding to the opposite side from data and the iPod news. Well, what, you're gonna metric function involved? He's gonna be signed. Sign of data is equal to the opposite over the hot news, which is going to be equal to 10 over l and then to get a function data in terms of l. We just take the inverse sign of both sides. The sign on the left side cancels out, and then we have sine inverse of 10 over l and I will take the derivative of say that would expect l to do this. We have to use a chain rule since we have 10 over L inside of here. But first off, we have to have our derivative of inverse ein. That's gonna be one over the square root of one minus. Whatever we had in print, disease squared. What we have is 10 over l. We have 10 over l squared, Then we multiply by the derivative of 10 over L. But we can say 10 over l is equal to 10 Oh, to the negative one power. Then taking a derivative of this, we end up getting negative 10 l to the negative to power by the power rule. So that's what we'll have inside of here. So that would simplify this a little bit. I'm going to have one over the square root of one minus 100 over l squared since we'll do the square on the top and the bottom times Negative 10 over l squared, moving L back into the denominator. So now we're gonna cross multiply I'm not cross multiply able to play across here to get negative 10 on top And then we'd have l squared times one minus 100 over l squared. Well, we can split up l squared into l times l and then l itself l is equal to the square root of l squared. So this is equal to negative 10 over L A Times Square root. We can put it inside of here now. L squared one minus 100 over elsewhere. You can put it inside based on the laws of square roots. So when we have negative 10 over, no times the square root l squared minus 100 sent. The else weighed on the numerator and denominator will cancel out. And so now we found the correct enter here the data over D eligible to negative 10 over l Times Square root of L squared minus 100. Now for part B. We're just gonna be plugging in a few terms here, So we'll have de Sade over d l. When l is equal to 50. So I'm gonna write this of 50 here that is going to be equal to negative 10/50 times 50 squared, minus 100. So 50 squared his point of equal to 2500. Since we multiply five by five and then we'll have two zeros. So this will be negative. 10/50 times square root of 2500 minus 100 which is equal to negative. 10/50 times square roots of 2400. Then we can divide the top and the bottom by 10 to get negative 1/5 times square, root of 2400. And then this is not quite the simplest form, but this is the answer. With this plugged in, now we're poking in the value 20 we'll get the fate of the l of 20 is going to be equal to negative 10 over 20 times square root of 20 squared minus 100 is going to be equal to negative 10/20 tens of square root of 400 minus 100 which equals negative one divided by 10 Again the top and bottom negative 1/2 times the square root of 300. Our last value is 11. So plugging in 11 for defeat over D L. You get negative. 10/11 times 11 squared, minus 100 11 squared is 1 21 minus 100. His equally negative 10/11 times square root of 21. Now, for part C wanted to evaluate a specific limit for the limit as l approaches 10 from the positive direction. So look at this. Were 1st 1 to just look at our function here. So we see where the problem is. Is this l right here equals 10 l squared is 100 then we have 100 minus 100. So therefore a zero in the denominator making this fraction undefined. So that's why we're willing to look at this limit. Well, if it goes from this increasing right here, then we would dio two numbers like 151 125 105 101 getting closer and closer. So we're getting a positive values here. They're getting closer and closer to 100. So that means when we take l squared, l squared minus 100 the limit as l approaches 10 in a positive direction would end up being zero just because of what we can see there, just plugging it in in this more simple case. And so as the denominator here is going to approach zero, the numerator goes to a larger and larger number or, in this case, a smaller, smaller number. Since we have a negative on top and we're having a positive number in the denominator at all time. And so then this limit as the data approaches 10 the positive of de Seda over D l would be equal to negative infinity. So for part B, were wondering why the data over the L is negative when the boat is moving toward the dock, and so that doesn't make sense. That is going to be because they'd actually increases when the boats being goes. The doc. So why do we have that are derivative is going to be negative always since we always have positive values of the L. Well, if we think about it when the boat is moving closer to the dock, as in this we're going this way. All right. Pot news is decreasing in size R l and This is the derivative that the pin done L So when l is increasing, then we would end up having that data over D. L is de treating.

All right, we've got a river here. So I'm going to draw the bank of the river. The river banks. The current is 5.33 m per second. So V sub c v sub C is 5.33 m per second and the with is 127 m. So I'm going to write X final is 127 meters. Now, velocity of the motorboat is 17 0.5 m per second. I want to cross the river at a 90 degree angle. So if I want to cross at a 90 degree angle, then I'm going to have to two point the motorboat upstream. That will give me a horizontal component of the velocity. And this is V sub M and a vertical component of the velocity. But the vertical component of the velocity must equal V subsea. So I'm going to do put feta here and I'm going to write these sub. I am times the sine of theta must equal This is part a V sub C. So therefore data would be the inverse sine of V sub C over V sub M. Let me put that into a calculator and I set this up so that you can see my calculator, drag it over inverse sine of V sub C. 5.33 over V sub M. Which is 17.5. Right, okay. And I've verified that that is the correct answer according to the back of the book. Okay. So let's move on to be how long is it gonna take to cross the river? B well, X is going to equal X initial which is zero plus VX initial which would be VM co signed data. That would be VX initial times T. So T. Is going to be X final over VM co sign data. So we'll put that into the calculator. X final is 127 m. Mhm. V sub m that would be 17.5. Um And then co sign of data. I'm going to write fi here because it doesn't accept data. Then I'm just gonna put PFI in here. Okay, 7.6 two seconds. Let's move on to see in which direction, direction for minimum crossing time straight across. Because then V Sub X would equal uh 17.5. There will be a Visa. But why? But straight across will be the maximum vis of X. D. How long will it take the to cross the river? Well, X. We're going um X final equals V. Times T. So all I have to do is X final over Visa Becks. So let me do that. X final is going to be 100 and 27 m divided by Visa. X. Which is 17.5 7.26 seconds. E. The minimum speed to be able to cross the river. If you go 90 degrees to the riverbank, I'm not sure I understand the question. For e minimum speed that will enable you to cross across the river 90 degrees to the riverbank. Well, at any speed you'll eventually cross because the flow of the river is perpendicular to your speed. Oh, but this means if we're pointed 90 degrees. Okay. All right. I mean if the actual movement is 90 degrees. So what was question A again? If you so Visa M sine Theta would have to equal V subsea. V sub M New science data would have to equal V sub C. So if we let sign data equal one and it can't quite equal one. But V sub M New would have to be a little bit larger. Then V sub C. It would just have to be we just had to be greater than V Sub C and V sub C is 5.33 It couldn't be exactly 5.33 but just a little bit greater then 5.33 So thank you for watching.

Given that we are pulling in a fish with on a dock that's 10 ft high. We're interested at the moment that this length of fishing line is 25 ft and we know that it's being reeled in. We'll go ahead and call that Z that green line that's being reeled in that negative one her second because the length is decreasing, which is why it's negative. So that's really important. And then we could set up first, just a relationship between the angle Fada so we could do sign, which is opposite tennis Fixed. That's a constant Z is going to change that green fishing line length and then another way to rewrite that is just 10 times. See to the negative one. It's downstairs. So then let's take the derivative of both sides So derivative of Sign is co sign. But we're differentiating with respect to time, so it's gonna be the data over DT. Then we're gonna move the negative one in front, so that becomes a negative 10 z to the negative, too. Mhm, the negative two. And then all of that times dizzy 80. But the good news is we know what thes e V T is just like every related great problem. We have to know one rate to solve for the other one. But that's a negative one. Let's just go and replace that with a negative one the leave the units off for just a minute. We also know that Z at this instant is 25. And so let's just rewrite that as all divided by 25 squared is the same thing is Z to the negative, too Negative. 10 stays up top and then we're going to divide the co sign over to the other side. So that's going to go in the denominator cosign of guns, Bread data. Okay, so at this point, we've now gotten rid of because I'm there and now this will allow us to solve or the data duty. The only thing we need to know now is what data is. But we could set up a relationship Sign of data is equal to 10 over 25 and then move the sign over. That would be an inverse sign or take the universe sign of both sides. And that is what we're going to substitute birth data there, so it's gonna be negative times a negative. So these will actually turn both into positive now, which makes sense because the angle should be, uh, increasing. It should be getting bigger here, so that makes sense. Why it's positive. And so 10 divided by 25 squared cosine of signed in verse 10/25 and that should give us the data. BT is equal to zero 0.0 17 and then it keeps going 45 or for six if we ran. Radiance is the unit for data divided by time, which is, and seconds so radium for one second. Now, if you wanted to convert that, of course you could do that times, um, 1 80 over pie. So let's do that over time. And that basically tells us the angle is changing by that one degree. Every second there is the same idea. Okay, so that's it.


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