For number forty eight for four point four were asked. Tio R. A is to find the growth function for this crane population that we're working on and when No we need. Is that tea? It's not just the time. It's the time since nineteen thirty eight. So that's something to keep in mind. Part A is just asking us to find the growth function. So from if you read back into the chapter, this is the function that models Grosso for part A. We just want to replace this with some of the variables that we have from the, um from the questions. So if we read the question, you find your K value, which is you, Colton Negative, which is equal to six point six point zero one times ten to the negative. Fifth. We were given an M, which is our maximum population, and it isthe seven eighty seven. And we're also given this bit of information that g for when the time miss nineteen fifty eight in the time since nineteen thirty eight makes this twenty and we get that from nineteen fifty eight minus nineteen. Thirty eight is equal to thirty two, and we're going to use this bit of information and part B. So I'm going to put a little bee next to this, so we know. Okay, So, back to part A. It's just asking us to plug in essentially the K and the M into the general GFT formula. So the growth formula is going to equal M, which is seven. Eighty seven times Jeannot, which we don't know what we will find over Jeannot plus seven eighty seven minus jeannot E to the negative. Six point zero one times ten to the negative. Fifth times, times seven eighty seven times. Sorry, guys. Times seven eighty seven times t. So this is part, eh? This is what our growth function needs to look like. Okay, Part B is now asking us to find Jeannot. So I'm going to take the function we did in part A. And I'm going to recall the information that G of twenty was equal to thirty two and I'm going to rewrite my equation now. So the equation for from part is now going to become G of T is going to be thirty two. So we have thirty two here. Tea is equal to twenty. Keep that in mind. Thirty two is equal to seven. Eighty seven, and then we need to solve for Jeannot, divided by Jeannot again, we need to solve for these Jeannot. So we are going to have to eventually combine them all. Gina plus seven eighty seven, minus Jeannot. All of that times e to the negative. Six point zero one times end to the negative. Fifth times seven, eighty seven times T, which we know is twenty now. So these values are coming from what's running grain and what we got from the green part in part A. Now our goal is to solve this for Jeannot. And this does take quiet. A bit of algebra for simplification. Purpose is to make this easier. I'm going to write this part right here. So this exponents, I'm going to keep it as e to the negative. Kay and tea. Okay, but that is this value right here. We know. Okay, we know, and we know T. And that's just for simplification purposes. So on the next page, I'm going to start by I'm going Tio multiply. Bring this over and bring this over. So that's gonna be my next steps. So on the next screen. That's from would it be doing? I'm going to be multiplying this over here and dividing by the thirty two to bring it over. So this is going to be by division. Okay, so if we multiply over the bottom, we're left with Jeannot plus seven eighty seven minus Jeannot e to the negative KMT Again. This is just for simplification. Purposes writing it, Brian. That exponents is equal to seven eighty seven and we divided over the thirty two Jeannot. So this is already looking a little better. I'm going to distribute this Jeannot plus seven eighty seven each. The negative KMT were came to your all known minus Jeannot E to the negative. K M t. This is still equal to seven. Eighty seven over thirty two times. Jeannot. Now we have this, and we're going to bring Yeah, RG not over. So we have all of our ji knots on one side and bring this over by adding and subtracting. So now we're going to have thiss schools Well on, it's working on writing again. Now we're back. Okay, so we're going to bring these over, so we're gonna have Jeannot minus Jeannot e to the negative. Km two minus seven. Eighty seven over thirty two. Jean not equal to negative seven eighty seven e to the negative K and tea. So now we're here. The next thing we're going to do is we're going to factor out each of our ji knots. That's got to be on the next page. So we're going to factor out the Jeannot. So we have Jeannot times one minus E to the negative. Kay and T minus seven eighty seven over thirty two. This is equal to negative seven eighty seven e to the negative KMT again. We know. Okay, we know him and we know T. So let's recall that really quick. Okay? ISS six point zero one times ten to the negative fifth and is equal to seven. Eighty seven is the maximum population tears equal to twenty. So these are all going right into here and also to there And in the final step, Jeannot is equal to negative seven eighty seven e to the negative K mt. Divided by one minus E to the negative. Okay. M T minus seven eighty seven over thirty two. We know all of these numbers. So Go ahead, take sometime. Type meant your calculator. Find the top, find the bottom and do the division and see if you can end up with an initial population equal to twelve point seventy four. This is our Jeannot. So the next part of this question is to start playing and very putting an actual years into the equations. So we have. So this is going to be kind of for part. See, we have our original equation, which all right and green now reply. Jefty is equal to seven. Eighty seven times. Jeannot. We found that to be twelve point seven four over twelve point seven four our Jeannot, which we found in part B plus seven eighty seven minus twelve point seven four times E to the negative six point zero one times ten to the negative Fifth. That's kay times, M times t. So now we have a function of just tea being the variable, which is what we want with Cem simplification. We can come up with our final GM team, which is going to look like this, and this is just by doing the multiplication that we're able to dio and this is going to make it easier to find the derivative, which is what we need in the next part to the negative point zero forty seventy. So this is what we're left with. This is what we're going to plug in the values for, um, when the question tells us to find, see should say, find the population and growth rate. So this is going to be the equation used for finding the growth rate. So we'LL quit that and this one. Sorry, this the regular fund function is toe find the population at a time t and then in green we're gonna work on the formula we need for finding the growth rate growth rate at the tea. And to do that, we're going to need Jean Prime of team so we can start working on that right here will do it. And I guess I'll keep it in green g prime of T. We're going to use the quotient rule is equal to low twelve point seven four plus seven seven four point two six e to the negative point zero forty seventy Low time's derivative of the top low D high which is zero minus. Hi Deal. Oh, so this part right here. That derivative is zero. This is the derivative we need. It's an exponential derivative. So we have the original exponential. This is a nagato fit on one page t. So the original exponential and then we're gonna come down here times the derivative, the exponents, which is negative point zero forty seven. And all of this is going to be over. What's on the bottom squared. So twelve point seven four plus seven seven four point two six e to the negative point zero forty seven t squared. So again, a recap. We were finding the derivative of this quotient. So we used the quotient rule, which is low. What's on the bottom times? The derivative of the top minus the top times the derivative of the bottom. In the derivative of the bottom is the original exponential function, which is right here times the derivative of the top. And that's where this part comes in. On the next page, we're going to continue with this, but we're going to simplify. So after Cem simplification, rg prime of tea should be equal to thirty six really thirty six four nine three eight e to the negative point zero forty seventy. And if you look back at the previous page, you'll notice that the first term cancels out because this multiplied by zero over the bottom, which is twelve point seven four plus seven. Seventy four point two six e to the negative point zero forty seven t square. So again, this is the formula that we're going to use for the rate for the next question's for the rate. Okay, now we have our formulas. Now we can actually answer part see which is finding the population and rate of growth when tea is equal to nineteen forty five so t equal to nineteen. Forty five. But that's not really the correct way to write it. If we remember, T is the year since nineteen thirty. Wait, So what we really want is that tea is just equal to seven. So and blue, we're going to find the population. We're going to find the population so the population isn't our original G equation, so g of seven should be equal to zero zero two six point three eight over again. We're just plugging in. Seven for tea. Twelve point seven four class seven, seventy four point two six, eh? To the negative point zero forty seven times seven. And that's our T value. That's for the population at this time. For the rate we're going to be plugging into the derivative, which is the green. So the rate, which is G prime of seven, should equal thirty six for nine. Three eight e six four nine three eight e to the negative point zero forty seven times seven over that denominator, twelve point seven four plus seven seventy four point two six e to the negative point zero forty seven times seven, Which is t all that squared. So these are the two formulas. Give yourself some time to make sure you can plug them into the calculator correctly. The population AT T equal seven should be seventeen point five nine in the growth rate at this time. Should be point eight zero nine Now D and E are going to follow very similarly. So I'm not going to go through the steps of plugging this in what I will do, though, So our next question is deep. Here we have the year is nineteen eighty five, so tea is equal to nineteen eighty five, minus nineteen thirty eight Because it's the time since nineteen thirty eight. So we're looking at T equal to forty seven here again in blue. You're gonna plug in the population. So you're going to find G of forty seven for the blue equation, and you're going to find G prime of forty seven, which is the green equation for the rate. That's the rate blue represents the population after given time. In this case, we're working with tea is equal to forty seven. Give yourself some time to plug those into the calculators you're on ly plugging in this for tea and this for tea, but in two separate equation. So in the year nineteen eighty five, we have the population being one hundred two point fifty six, and in this case, our growth rate is equal to four point one nine. This is definitely an increase over the previous growth rate in part, See, So now we're going to do the same thing again. This time we're doing part E here. We're given that the year is two thousand five and they get our time is the time that has passed since nineteen thirty eight. So our time is sixty seven years now two thousand and five minus nineteen. Thirty eight. That's how you get the sixty seven. This is our tea. To find the population at this time G of sixty seven into the original equation we found in part B, and for the rate of growth, you're going to plug that into G prime of sixty seven, which is the derivative of the function. So at this time again, give yourself some time posit video. See if you can plug this in for yourself at G of fifty seven, the population should be to eighteen point one nine, and you should get a growth rate of about seven point for one. That's part E. Part F is asking us about the growth rates. So I'm going to list them. We had t equals seven. We had a rate of point eight zero nine. T equaled forty seven. We had a rate of four point one nine. Auntie equaled sixty seven. We had a rate of seven point for one. So F is asking us what happens to the rate of growth over time. So his time going time is going on. It seems like the rate of growth is it. See if you guys can guess. This for yourself is increasing and it seems like it's increasing fairly rapidly. It seems like it's definitely increasing quicker to start and then it's slowly starting to level off, but it definitely is increasing.