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Chapter 3, Section 3.4, Question 34Your answer is partially correct: Try again.The tides for Toms Cove in Assateague Beach, Virginia _ on August 19, 2004 can be mod...

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Chapter 3, Section 3.4, Question 34Your answer is partially correct: Try again.The tides for Toms Cove in Assateague Beach, Virginia _ on August 19, 2004 can be modeled with the functionh(t) 1.8c05 gct-11)] 2.2where H is the height of the tide (in feet) and is the time (in hours after midnight) Find and interprett=4Round vour answer to three decimal places_feet per hourthe rate of change of the height of the tide This representsthe tolerance is +/-2%/

Chapter 3, Section 3.4, Question 34 Your answer is partially correct: Try again. The tides for Toms Cove in Assateague Beach, Virginia _ on August 19, 2004 can be modeled with the function h(t) 1.8c05 gct-11)] 2.2 where H is the height of the tide (in feet) and is the time (in hours after midnight) Find and interpret t=4 Round vour answer to three decimal places_ feet per hour the rate of change of the height of the tide This represents the tolerance is +/-2%/



Answers

The height of the water, $H,$ in feet, at a boat dock $t$ hours after $6 \mathrm{A}$. M. is given by $$ H=10+4 \sin \frac{\pi}{6} t $$ a. Find the height of the water at the dock at $6 \mathrm{A} . \mathrm{M} ., 9 \mathrm{A.M}$. noon, 6 P.M., midnight, and 3 A.M. b. When is low tide and when is high tide? c. What is the period of this function and what does this mean about the tides?

In this cushion we are given a function we're doing edge and be ethical to 10 plus four. Sign off i d Divided by six here edge it up. Height overdone in seat by He's a number of hours. Austin 6 a.m. No Part eight In party we have to find up high. Go for the head Different time interval. So I have made a demon for different diamond. No, for six in it will be he's going to zero. Why? For my name. It would be three for 12 bn Or do they will be six. Similarly, for 6 p.m. D will be well for my name. They will be 18 and for 3 a.m. will be printed. Now we have to find height at a different time. Full ed do will be 10 plus four. Same do it will be 10. Edge three will be 10 plus four. Sign by divided by toe on solving it will be 14 edge six will be then less for same by it will be done again. Edge when it could do then less for same Dubai. It is then inch 18. It could do 10 less for same today by you could do in again and edge 20. Even it will do. 10. Bless full sane salmon bite divided by two. It will be six now, after some big party via asked in fire be the time off Low tide and high day So when we see the table BC there and three in it three in height Minimal Hence no time. God's end three in right ed nine in hi date maximum tens. Hi. Date A cards EG Name E m No. In part C Via asked the Peter off the function Vino day for regional sign off E X plus B day function period It do you buy divided by a so for ethical toe 10 plus for same by D divided by six Peter will be to buy divided by by 200 by six. It will be 12 hours the be directed Well, ours. So in e completely do gonna bleed Same to you occurs hens lo tapes And hey, days happy. Why's your knee? They say the meeting off failure

Um So this question, we're looking at a phase shift and Sino Soto curve fittings. Right? Um, so the information we're given in the beginning of the problem is that the time. But we in two consecutive high tides is going to be 12 hours and 25 minutes. Right, So 12 hours and I've been to 25 minutes. Right? And according to National Oceanic and everything, uh, Saturday, April 26 2014, that's all. Uh, not that important information. High tide has occurred at 6:30 a.m. 6. 30 a. M. And that was a high tide. Right? So we know that now, at 12. 24 p. M. We've got a low tide. So we have that information as well, and then we also have information of water heights. Right. So the height of the water at high tide was 5.86 ft, and the height at low tide was negative. 0.38 ft. Yeah. And the question they're asking is, uh, when will the next high tide occur? So that is the first question. When will the next high tide occurred? Well, we know that the high tide occurs every 12 hours and 25 minutes. And we know that the first high tide occurred at 6:30 a.m. So if you take 6:30 a.m. and you add 12 hours and 25 minutes to it, Yeah, then what do you get? Well, uh, 12 hours past 6:30 a.m. will be 6. 30 PM right? So that they get rid of the 12 hours and then you simply add 25 minutes and then you'll get 6:55 p.m. So 12 hours and 27 minutes, 25 minutes after the first high tide, it will be 6. 55 PM So this is when the next high tide will occur, right? So that would be your answer for part one. And then we'll go to part two or part B. And this part is asking us to find the function in the form of Y equals a sign of Uh huh w X minus five. Our road. Uh, this is another Greek symbol, plus B, right. So we need to find a function in this form, and in order to do that, you have to know that first, eh is going to be equal to the maximum. One is the minimum over to. Right. So this is gonna be the max, which is 5.886 ft, um, minus the minimum, which is 0.38 ft. I sorry. Negative. 0.38 ft. And I divide those two by two. To get a and then to get B B is actually going to be the maximum minus a. Now you be and then to get omega or W That's an Omega. And you have to do to pi over, um, over P, which is the amount of time it takes because that you're going from one high tide to another high types from one peak to another people. So how long does it take to complete one full cycle? That's what P is. And what else do you know? Well, that's pretty much it right. Um, and I guess the other thing you need to know is that, um is your is your charge? The other thing you need to know is is your is your angle here or your value here, And that value will just simply be, um what? Your first? When was your first um, when was your first thing? Your first high tide. And that was at 6. 55 PM or sorry. 6:30 p.m. So that will be equal to 6.5. Right? So, yeah. So, um, this dis variable here is going to be how much it shifts by, um, which is basically where the when the first high tide came in. And the first high tide, of course, came in at 6. 30 AM I right? So 6.5 will be the shift there. Um, so when you plug everything in, you should get y equals A which is going to be maximum time is the minimum, and that's going to be Let's see, that was going to be a 5.86 minus negative. 0.38 Yeah. Uh, it's gonna be 6.2 4/2. It's gonna be 3.12 Time signed of Uh huh. So, yes, I sign times two pi over and R P is going to be the total time it takes, which is 12 hours and 25 minutes. Uh, 12 hours and 25 minutes is actually gonna be 12.42 um, 12.42 Right. And then all of these times x minus. Um, yeah, uh, 6.5. Right. And plus maximum, which is 586 minus eight, which is 3.12 And you get 2.74 So that's going to be basically your equation. And you can simplify that down to, like those three, uh, 3.12 times sine of two pi over 12 or 42 X minus 17184 plus 274 So that is going to be the form that your answer is going to be in mhm. All right. Okay. Okay. So, um Right, so now that you have that, uh, what you gotta do next is part seat, uh, which is actually asking us, Um uh, approximately using the function that we just made, uh, find the height of the water at 3 p.m. So 3 p.m. is going to be We can't just plug that straight into our equation here because we have to convert 3 p.m. Into military time. So three PM is actually gonna be 15, um, in military time, because you take 12 and then you add three and you get 15. 12 will be the maximum at at noon, and then you add three. Because you're going past noon to 3 p.m. And then you get 15. So then, now that we have that, we basically just want to plug in. Why? 15? And when we do that, we'll get 3.12 times sine of two pi over 12.42 Um, Times 15 minus 17184 plus 2.74 So we do that when we take the sign. Right? So to pi over 12.42 Um, that will be zero, uh, to pi over 142 there'll be mhm. Um um All right. Right. So, uh, to pi over 12.42 is 0.506 and then you multiply that by 15, and then you subtract one point 7184 get you 5.87 and you take the sign of that. Uh, you get 0.10 So you have, um, 3.12 times 0.102 and then you add 2.74 And then when you multiply by 3.12 and then add 2.74 Yeah. Um, yeah. Um Oops. Sorry. Sign of sign of 5.87 is not that sign of 5.87 is actually, uh, negative 0.399 don't forget to switch your calculator into radiant mode because we're dealing with radiance now, not degrees. Um, And then you should also get the same answer as I did here, So yeah. 0.399 And then you take that. You multiply by 3.12 you get negative 1.24 and you add that to 2.74 and then you get 1.4931 point 493 ft. So that will be your approximate answer for your height. So your height at, um, at a time, yeah, R 3 p.m. will be approximately equal to 1.493 So that will be your final answer for that one as well. So now you have the answers to all of them. Um, so, yeah, in general. So the first part was really easy. We just have to, uh, do some addition there with time. Second part. I was slightly harder. Just remember all these techniques here, um, for finding all these calculating all these things and calculating all the phase shifts, um, and fitting things into your Sinus over the old function, and then down here was just plugging in. So overall, pretty simple. Um, so that's it.

In this problem, we are given uh the time when high tide happens and the low tide low tide happens in charleston south Carolina. And we also know that the high tide occurs at every 12 hours and 25 minutes. And they and we are also given the height of the high tide and in the height of the low tide. So the question A. Is asking us, when will the next high tide occur? So we know that the first high tire occurred at 12 25 a.m. And we know that high tide occurs at every 12 hours and 25 minutes. So let's do 12 hours first. So after 12 hours is going to be at 12 25 P. M. And then you have to add 25 minutes to it because high tide occurs at every 12 hours and 25 minutes. So therefore my answer is going to be at 12:50 P. M. And question B is asking us to find this side function in the form of this that models this data. So the first thing we can go and find is the midline midline is the d. Which is the distance between the maximum and the minimum. So so we know what the maximum is. Which is the height of the high tie, which is 5.8. So that's going to be 5.8 plus the low tie high, which is 0.06 Divided by two. And that's gonna give you 2.97. And to find the amplitude, we know that's the distance between Maximum to the midline. So we know what maximum is is 5.8 minus the midline is 2.97 And uh that's gonna give you 2.91. So that's your amplitude. And the next thing we're going to find is the period. So we know the formula to find the period for sine function is two pi over B. So we know that the high tide occurs at every 12 hours and 25 minutes. So I'm gonna go go ahead and convert the 25 minutes into hours. So 25 divided by 60 will give me in hours, And then I'm going to add 12 to it. Now converted to fraction. So we know it occurs at every 149 over 12. And to solve for B, you're going to cross multiply. So you get 1 49 B equals to 20 for pie. So therefore be equals two. Okay, I'm using lower case B. So be equals two 24 pi Over 1 49. And the next day we're going to find is the phase shift. So to find the face shift, we need to find the maximum where the maximum occurs into this side function and where the maximum occurs in in in this problem. So in order to find the maximum word, uh this sign function occurred, I need to divide my period into four equal sections. So I know my period was 1 49/12. So when I divide that by four, so the length of the each interval should be 1 49 over 48. So I'm gonna go ahead, write my four intervals zero 1:49 Over 48. And then you're gonna keep adding 149 over 48. So therefore the next number is going to be 149 over 24 and then Next one is going to be 1:49 Over 16 And last one will be 149 over 12. So now I'm going to grab the function that I have so far. So go to Y equals and graph All the information we have so far. We know the amplitude which is 2.91. Then tapping sign of B. We know what B is 24. Hi, divided by 1 49 x. And we know the midline which is 2.97. So now we're gonna go to windows and change our window so X minimum should be zero and maximum is going to be 24 hours and why minimum is gonna be zero? So the high tide was Maximum was close around 2 6. And now press graph. So based on this graph let's press trace and type in 149 the right by 48. So between zero and 1 14 hour 48 the function is increasing and then it's decreasing here. So therefore the maximum for the function is located at X equals to 1 49/48. And then the maximum for the problem, the high tie occurs at 12 25 a.m. Which is same thing as x equals to zero 42 hours. Okay, So ah again what they did was 25 divided by 60. So to get more accurate answers, you can leave it as a fraction 5/12. So X equals 2 5/12. So now you're going to subtract the maximum from the data from the problem. So five over 12 minus 1:49 Over 48. So you get the answer about negative, 2.6875 equals exactly it's exactly equal to that. So now we have our face shift so so now we can write our function so it's gonna be Y equals two. The period was two 91 sign of BB was 24 pi Over 1 49 x. So another parenthesis. So because this is already negative so this is gonna be plus 2.6875. And then plus the midline which was two I think it was 2.97, Yep. Midline was 2.97 plus 2.97. So this is the function for this problem And the last one part C. It is asking us to use the function we found in part B. To predict the height of the water at five p.m. So we know that at five p.m. That's going to be um 17 hours. Okay so therefore you're gonna plug in X. here with 17. So what I'm gonna do is I'm going to type it into the function here. So add another insert. Oops insert. Okay so we have X. Plus two point six. A 75. Okay. So there's a function of tapping your calculator. So now let's press graph again and when and when we press trace Trace past 17 hours. Okay. I probably grafted somewhere. The graph looks weird and became the 2nd. 2.9. 1 24 Pi over. Oh I forgot. 1 49. There you go. Now let's try again. Grass. There you go. That that looks a lot closer than what we had earlier. Okay. So now press trace And Typing 17. We we want to know the Heidel water at 17. and according to the calculator is about one point for 797-1 fate I.

We are going to do problem number 92. Okay, discussion the function below models work at height. Okay. And with respect to the number of hours. Okay, so that given expression is hFT So this is equals two 2.91 sign. And this has given 24 pi by 2090 plus 1.360 And this is plus 2.97 No, in airport we need to find the height of water at high tide. Okay? When it is going to be high tide, when this whole function, this whole sine function is going to be one then only we we can achieve uh maximum height. Okay, so this is 2.91 multiplied to one plus 2.97 This is equals to 2.91 plus 2.97 So let us see how much this value is. We have to add up the value. That is 2.91 plus 2.97 This is equals to 5.88 5.88 ft. Okay, so this is the maximum height. Okay. Height of what threat hideout? No, we have to find the same for low tide. So in case of low tide this part this part let me just market of a different color. This part should be management. Okay. So does see how much it is coming. So height off tired. So this will be equal to 2.9 wine multiplied to management plus 2.97 So this is equals to let us just find it out. This value. There is two point 91 2.91 into minus one. This is managed 2.91 managed to 0.91 adding to 2.97 So our low tide will be 0.6 ft. Okay, so this is a managed 2.91 plus 2.97 So this is the cost to zero point 06 ft. We got a height of low tide height of high tide and see it. Is that what is the time between high and low tide? So as it is a sine function. Okay, so this is the general curve of sine function. So this is high tide and this is a low tide and the difference between these two. Okay, the gap between the dish to is half of the total time period. So let's just find all the time. But in this case that is to buy divided by 24 by divided by 29. Okay. 24 by by 149 is divided to t So you just cut it. This this is uh 149. This is 149. So piper gets cancel out and this is too little 24. So we have 149 by 12 as our time period. So there between these two will be half of that time period. This is 149. They would work too well and divided by two Okay services and 1200 to. So this is 149 by four. Okay 149 by four. So this is uh this is the time between, this is the time between high and low tide. So this is there this is in ours. This is the answer. That's all. Thank you.


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