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Problem 6: The population of the ever popular tank pet, sed monkeys_ can be modeled by x(t) 80e 25t where t is the number of hours in the tank Answer all parts in c...

Question

Problem 6: The population of the ever popular tank pet, sed monkeys_ can be modeled by x(t) 80e 25t where t is the number of hours in the tank Answer all parts in complete 36+e 25t' sentences including units a) At what rate are the sea monkeys growing when they are in the tank for 10 hoursmonkeys during the 3"dto 7ih hour? b) What is the average number of sea

Problem 6: The population of the ever popular tank pet, sed monkeys_ can be modeled by x(t) 80e 25t where t is the number of hours in the tank Answer all parts in complete 36+e 25t' sentences including units a) At what rate are the sea monkeys growing when they are in the tank for 10 hours monkeys during the 3"dto 7ih hour? b) What is the average number of sea



Answers

Guppy Population $\mathrm{A} 2000$ -gallon tank can support no more than 150 guppies. Six guppies are introduced into the tank. Assume that the rate of growth of the population is
$\frac{d P}{d t}=0.0015 P(150-P)$
where time $t$ is in weeks.
(a) Find a formula for the guppy population in terms of $t .$
(b) How long will it take for the guppy population to be 100? 125?

If we have a function en of tea equals 12 times E to the 0.12 tea, where teas and years and entities and million's the relative rate of growth for the population would be found by writing this or thinking of this function in terms of just a regular exponential function, not of Base e. And so to answer Part E. It's to recognize that end of tea or, um, would be the same thing as 12 times what to the T power. And that's found by evaluating E to the 0.12 and e yeah, to the 0.12 Okay, yeah, is about 1.12 And we want to do this as a percentage. So we would say this is about yeah, this would be 1.12 So it's an exponential growth, and this would be growing at 1.2%. Expressing this as a percentage that is a relative rate of growth for Part B. The fish population, after five years, can be found by evaluating and of five, because tea is in years not be 12 times e to the 0.12 times five. So we pop over to a calculator 12 times E to the 0.12 times five, and that is about 12.74 And that would be 12.74 million for Part C. After how many years with a number of fish reached 30 million? Well, then we want to solve the following equation. 30 equals 12 times E to the 0.12 tea. And so we saw this equation by first dividing 12 to both sides on 30. Divided by 12 is 2.5. So 2.5 equals E to the 0.12 teeth. We can take the natural log of both sides and on the left we have the natural log of 2.5 equals the natural log of an exponential. Basie just counter acts. We get 0.12 tea, and now we divide both sides by 0.12 and we have tea right here. And so the natural log of 2.5, divided by 0.12 is 76 point. Um, we'll say about 76.4. So after how many years will it have reached? 30 million? Uh, well, it be between 76 77 years. Mhm and part d sketch a graph of this population. So we have two axes or we have tea and end of t representing the population. We know that initial amount was 12. So if you sit ups, um, axes such as going from 10 to 70 and from 0 to 30 on an A T Y axis and using some clues from the situation, we know we would have the points 0 12 because that's where it started. We know from, for instance, part B that the 0.5 12 points, you know, almost 13. And from the last one, we know that the point about 76 77. 30 yeah, would be on the function. And so we could estimate a little over 70 and 30. Just This is just a sketch on. There you have it

The rate differential of a very respected T. In standard pollution index per hour of a pollutant put into the air by a smokestack is given by differential wave respected t equal 150 over one plus 0.25 Times T -4 sq and all that plus 25 40 years time. In our after six a.m. We're going to estimate the total amount of pollutants put into the air between six am and noon is interrupts of the rule with six some infamous. So if this is the rate of pollute and put into the air then the Total amount of polluting put into the air between six and 9 will be the integral from 0 to 6 of this expression here and zero from 0 to 6. Because at zero we have zero hours after six a.m. Because we start just at six a.m. And at noon we have six hours after six a.m. So let's say that first. So you want two estimate Angel from 0 to 6 of 150 over one plus zero point 25 times T -4 sq last 25 by the episode. The rule with six sub intervals. Mhm. So we define the function f of t equal the integration here is 150 over one plus 0.25. T minus four square last 25. That's functional. 15 be defined On the interval 06. So oh then we calculate age which is the step size. That is a common distance between any two consecutive notes and ages defined as the length of the interval of integration 6 0 over The numbers of Interval six. That is one. So step sizes one with this step size we can say the notes are it's not equal zero x 1 equal one X two equal to X three girls 3 x four equal 4 X five equal five and X six equals six. And now we can say that the integral from zero 2 6 of the function. Fft they find here above Is approximately equal to T6. There is a trap cider rule with six of intervals which is fine as age half times the image of the first note. It's not and here maybe I'm going to change something and is that we are using the variable T. So maybe it's better he is better T instead of X. Doesn't change anything and very clear but he turned tough Consistency of meditation. So here is the zero T. 1 T two T. Three 84 T five and T. six. Okay So it's f. at the zero plus two F. At T wan plus two. F at T. Two Was to f. 33. List to f. Plus two f. T five Plus F 86. Just put the six better. He also saw statistics here. Okay is it let's say we arrange this you better left. Okay and this is equal to ages one so one half times F zero. Mhm Let's do F at one is to effort to was to effort three blessed to if at four was to effort five plus 76. This us one half times. Now we relate the function of this virus. Ft zero Yeah 150 over one plus 0.25. four square first course 16. Now we have 16 Time 0.25 gain and say F to your room here's five. Okay This is 55 for those two times. If at one yes then it's not equal but approximately equal because I'm here approximations to the number Of the image 71 point 1538. Okay. 461 538 4616. There is only this one here plus Two times f at two is 100 200 here. Yeah we can put that lows 200 last two times if a three 100 45 times 290. So with that also here us 290 plus. So we have a fat 1234. Yes effort for is 175 times two. 115 plus At five we get Okay. Uh huh. 45 that is again 290 plus at six Andrea 100. So except F one just an approximation with all these things. It's all the numbers. All other numbers are interesting numbers so could have been an interview but Because of F one we have an approximation so that's approximately equal to. We do all these calculations. Now let's see the number into your number. Maybe first Yes one half Yeah. Times. And now some of the interior numbers here is 1285 plus. and now two times this number here let's say is equal to 1 42 were you point 3076. Okay. 33% 69,230 769 mm three. Now we made some here. That is um 142. 142 points. Sorry we're no it's not. That is yes 40 1427 1427 point three Series 7 six. Mhm. Mhm. And two three 076 uh nine to and now this divided by two is equal to it's about 700 13 point 65 38 4615. Okay. 384. Okay 62. Mhm. And this this should be in a standard pollution index. So uh we can say that The intervals from six For its from 0 to 6 of F of t differential T. So approximately equal to 307 100 13 0.6 five Let's say 4 to stay with six figures are or digits and three decim. And then this implied that implies that the total amount of pollution put into the air total amount of for written the victors. How for you to food into the air Between six am and noon is about 713.6 54 pollution index. Standard pollution index. Yeah. Mm. And that is the answer. We have calculated numerically by using traps out of the wall with six of intervals and the internal was integral of the function. Uh huh. Which is equal to the rate the A. D. T. Of pollutant put into the air given by this expression here. And we found that the result was about 713.654 standard pollution index.

Okay, so we want to find the weight of a small fish in grams after two weeks. Actually, we're giving this. This is WG on. Do we want to find the rates that the fish is growing at Time? T is a beautiful who were taking care of it appear so that's W prime evaluated up for So that's equal to W. T. Said It's their points are actually in particular limit as T approaches four of WT and thats 0.1 Do you squared minus, um, W evaluated up for. So that's 0.1 times four to depart, too. So that's equal to 1.6 all over team on its four. So it's not that one point six. That's a, um, it's seat. It's eso in our new reading. So let's start buying factoring outs a point one. So we have 0.1, and then you have t squared minus 16 over a team when it's four. So it so it's here. We can practice into a T minus war Times T plus for and now let's council out or like terms. Now, using text sub we get is your 0.1 times four plus four. So this is equity of 0.1 times AIDS, which is 0.8 grams for a week

So if we're given that the bacteria population and million's is been by the equation here on DH, I forgot to put the one half hour So if we have that the bacteria population is given by end of T Uh, and this is equal to t Time's team by t plus nine to one half power plus twelve where our time is an hours and we want to find the rate of change of the bacterial population with respect to time. After zero hours, seven, half hours, eight hours. Remember, any time we talk about rate of change of some formula, this should tell us that we're looking for a derivative. So we need to find but in crime of Tears and then we'LL plug in zero hour, seven a half hours, eight hours into this equation here. So on the side, let's go ahead and fine. What is it so and prime of tea? So remember, I can distribute the derivative across plus signs as well as I can pull any constant since the derivative is a linear operator. So I can rewrite this like this here so two times the derivative with respect to tea of bi T plus nine to the one half power plus the derivative with respect to time of twelve. So first recall that the derivative of a constant is zero And then to take this derivative here and I dropped my tea that should be right there being well supplied by it. Now, if I want to take the derivative of this I'll have to use the product Rule says I have to function Speed multiplied together So I have tea here and I have by t plus nine to one half power so I can go ahead and write this as two times. So first all right, down t then I take the derivative Uh bye Tio plus nine to one half power nine one half Then all switched the places of these two functions. So I'll have by t plus nine phase two the one half times the derivative of team So too so the tea gets dropped down and then we called to take the derivative of five T plus nine What half It is the generalized power rule So I'll take this one. Move it out front and it's a one off of it. So like it tee times more half time's so bye plus mine But since I subtracted one off my power this will now be a negative on half power And then remember, Since this is essentially change, you'LL have to take the derivative of the inside function and multiply by it. So the derivative of five people sign is five and then I'm done with this first expression and then to take the driven ou of here, er, I should first just go ahead and write down plus nine to the one half and in the derivative of tea is one. And so at this point, all we need to do is clean this up a little bit. They can look a little bit nicer sense. We're wanting to plug in three terms at this, So let's simplify this a much as we can, so we won't have to do as much work when we plug in the numbers. So too. So actually, before I do that, notice that we have this five T plus nine common term in each of these. So what we could do is pull out the smallest power of this. So the smallest power is negative half. So I'll go ahead and pull out five plus nine razed to the negative one half. And so this first expression that will leave me with so again combined on my constant signed up with five half and then the only variable he left with his team. And then if I pull out a negative one half power from bye keep plus nine to the one half that will be like I'm adding a half to its power. Says I have to divide it by bye t plus nine to the negative half. So I'd be left with by TV plus nine. It's then I can go ahead, clean this up a little bit more all go ahead and combined my life terms by adding them. And so I have to times So five half plus five Give me fifteen over too eighteen plus nine and then changing this negative hath power into a positive. It will be five t plus nine. But remember, a half power is actually a square room. So I'll write it like this here and for the last up I can go ahead and distribute this to in and in doing so I end up with fifteen team plus eighteen all over. And I can't write any further down. Let me actually move this up a little bit. Okay? Square root of five eighteen plus no. Must have a sign in between. So plus no. So this here is the derivative of our function. And so this tells us the rate of change over the bacteria population in millions. So now we just need to take each of these times here and plugged them in. So for this first one I want to find in prime zero So this here and I'LL go ahead and change the color of my cereal So this will be fifteen times zero plus eighteen over bye time's your robe plus nine. And so in the new Maria or I'LL be left with eighteen and the denominator all get square root of nine It's where the ninety three. So this here with the six. So this here is saying that at a time of zero the bacteria population is changing at six million bacteria her our and so this is at tea is equal to zero. We'll do the same thing for seven of an hour. So in prime Oh, so then fifths. So fifteen times seven or five plus a teepee plus eighteen over that of the square roots. Uh, buy time seven over. Fine plus nine. So fifteen times seven over five gives twenty one and then twenty one. Post eighteen thirty nine. And then in the denominator off the square root of so five times seven, fifty seven plus minus sixteen. So this is thirty nine over four, and that is nine point seven. Fuck. So no time of seven fifths of an hour. So he went to seven. Five. The population is increasing at nine point five million. Her our When you go ahead and box these answers that we've got so far a bit. And for the last one, little repeat, the process will take time. But this time we're plugging in eight. So fifteen times eight plus beating over the square root of fine times. I like that a little bit clearer. Worth nine. All this is so fifteen times eight is a hundred twenty plus eighteen will give one hundred and thirty eight and numerator and then ended in on their off the square root of so five times eight is forty plus nine. Forty so this will be one hundred and thirty eight over seven, and this here is approximately equal to nineteen point seven one. So there's a a little bit. So at tea is equal to eight, we're increasing by nine point seven one million million or our and what?


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