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The Pacific halibut fishery has been modeled by the differential equation$ \frac {dy}{dt} = ky (1 - \frac {y}{M}) $where $ y(t) $ is the biomass (the total mass of the members of the population) in kilograms at time $ t $ (measured in years), the carrying capacity is estimated to be $ M = 8 \times 10^7 $ kg, and $ k = 0.71 $ per year.(a) If $ y(0) = 2 \times 10^7 $ kg, find the biomass a year later.(b) How long will it take for the biomass to reach $ 4 \times 10^7 $ kg?
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Calculus 2 / BC
Chapter 9
Differential Equations
Section 4
Models for Population Growth
Matthew T.
November 1, 2021
where y(t) is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be M = 7 ? 107 kg, and k = 0.72 per year.
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Lectures
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A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.
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Okay, so for this problem would start by figuring out what's given and then what's required. It sounds like a difficult problem, but if you just focused own what's given in what's constant, even if he don't completely understand how population grows works, you can actually figure it out because you don't really need to know much detail unless you know you're in that area. So what's given here is ah Wai dt, which is actually there. Wade of change of the population mass and why is an kg and kilograms in the time on years? And here Kay is a constant and m is also a constant. Okay, so em here is given in Kay. Here's Devon and we're also told that of the population mas at times zero is two times ten to the seven kg. So we're going to use this whole thing and to figure out first, what is the way of tea, which is the mass of the population at a given time. So the given time here as for one year. So first, let's rearrange the actual equation s. So what I did here is take ah, this here and ah basically re arrange it. So when we rearrange that you get one over why? Times one minus y over mg y equals Katie. So I'm just putting, you know, the y's on one side, Dana the T And once I did just same variables on the same end. So again, we a wrench a little bit far there. I'm just putting, you know, just fucked. Like them to the top basic factory ization. And that will give you here at the bottom one of her. Why plus one over m minus Y d y equals Keitt. So now it's in a form that we recognize and we can actually integrate it. So remember that the integral of one aware Why is Elin if y or the absolute value olynyk wide. So here at the top of the page, I have the result, which is in La Garza Mick form. Not that I put the negative here so that we can we arrange things in a way that would be easier to Seoul. So the negative it would allow me to put it, and a fraction for him, which is what we want here. See, that s O. C. Is a constant that I just got it. Okay, so now let's take the from the log to lease. Take it, Tio Exponential form so and minus y over, Why equals plaza minus e too? The constant and then e to the negative, Katie. So because here, each of minus C times puzzle minus, that's just a constant. I can just abbreviate it by and later. Eight. Okay, so next that is just rewriting it. I'm just simplifying it and to an easier for him to look at. So, by the end, when you we organize the whole equation, you're going to get way of T equals over one plus eighty through the minus. Katie. So now we have of it equation for the bio mas at a given time so we can go ahead and find the actual while you for a year one. So we're given before we do that way, we have to find a because we don't know a but to do that, we're given the volume that the mass at times zero. So we use that to find a So if you rearrange this equation here, you're going to end up was a equals that I'm over y zero, which is a time zero The massive times you're a minus one so you will find a equal suite. All right, so now finally, we can find the muss At year one. We're just plugging in numbers Know that M and K there constants in time equals one. So when you put all the numbers you will find out the mass equals three point two times cinches the seven kg. So this's part a The next fort is finding the time when why of tea or the mass equals four times ten to the seventh. So we have already done the hard part and the previous portrait. We already know the equation for the bio mash to point the fire must have given time. But here we're told of exact on my own, the mass at a given time. We don't know the time. So when you plug in the numbers here, you're going to end up was the only unknown variable here is teak. Okay, so when you when you put the numbers and you're gonna end up was e to the minus, Katie equals one over three, which is a simple number that you can just do a quick Well, Artemis ticks and find t. So I just rearranged a little bit here on the next step. Then now you can see time equals a line of three. Divided by KK is ah, consciousness, which is like point seven one. I believe so. That would give you the final answer for the time, which is one point five five years to reach about four times since was a seven kg. Okay, so hope this is clear. I know I wrote everything out beforehand because I don't have ah, you know, writing up. Hopefully this is clear. Okay. Thank you.
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