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Animal populations are not capable . of unrestricted growth because of limited habitat and food supplies: Under such conditions tthe population follows logistic gro...

Question

Animal populations are not capable . of unrestricted growth because of limited habitat and food supplies: Under such conditions tthe population follows logistic growth model P(t) 1+ke where €,d,k are positive constants_ For a certain fish population in a small pond d = 1100,k = 11,€ = 0.4 and t is measured in years. The fish were introduced into the pond att = 0. How many fish were originally put in the pond? Find the population after 20 years What value does the population approach as c? Bo

Animal populations are not capable . of unrestricted growth because of limited habitat and food supplies: Under such conditions tthe population follows logistic growth model P(t) 1+ke where €,d,k are positive constants_ For a certain fish population in a small pond d = 1100,k = 11,€ = 0.4 and t is measured in years. The fish were introduced into the pond att = 0. How many fish were originally put in the pond? Find the population after 20 years What value does the population approach as c? Bonus Question: Determine whether the function 1-el/x f(x) 1+el/x is even, odd or neither



Answers

Logistic Growth Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic
growth model:
$$P(t)=\frac{d}{1+k e^{-c t}}$$
where $c, d,$ and $k$ are positive constants. For a certain fish population in a small pond $d=1200, k=11, c=0.2,$ and $t$ is measured in years. The fish were introduced into the pond
at time $t=0 .$
(a) How many fish were originally put in the pond?
(b) Find the population after $10,20$ , and 30 years.
(c) Evaluate $P(t)$ for large values of $t$ . What value does the population approach as $t \rightarrow \infty ?$ Does the graph shown $\quad$ confirm your calculations?

For this question were given a logistic growth model for a fish population in a small pond. The logistic growth model looks like this. And for this particular problem, the constants are D equals 1200 case 11 c z equals 2.2 and the variable teas in years. Pft is the population of fish in that pond forgiven at a given time T So the first question E is, um, what is the population of fish that was originally introduced into the pond? And that's the population of fish at time zero or p of zero. Let there be. You want to find the population of fish after 10 years after 20 years and after 30 years, that's be off. 10. We have 20 and p of 30. Let her see you want to find the value that p off T or the population approaches. S t approaches Infinity. Let's look at the first question. So our function is pft equal to 1200 divided by one plus 11 e raised to the power negative 0.2. I'm Steve. So p of zero would be called to 1200 if I did buy one plus 11 he raised to the power because t zero the exponents becomes zero. Eat. The zero is equal to one. So this becomes 1200 over 12 which is equal to 100 fishes. For letter B, we just need to find the values of P off 10 p 20 and p of 30. So we just did use a calculator. So for P off 10 p of tennis, equal to 1200 divided by quantity, one plus 11 times he reached to the power negative 0.2 times 10 since it is equal to 10 and that gives us the value of 482 rounded to the nearest whole number. So then, for the next one p of 20 we just need evaluate that same function, 40 equals 20 and we get 999 and it t equals 30. The value would be 1168. So that would be your answers for letter B for letters, C. We just need to find the value that pft approaches as T approaches infinity. So one way to do this is to just assign big values off T in Evaluate pft so we can let equals 100 equals 200 equals 300 determine the values of P Off T. Another way would be to use the table function off our calculator, so we type in the value off the function right. The 1200 divided by one plus 11 year raised two D power negative point toe times X X series is the variable T for time, right, and then we go into our table set up in. We want our starting value to be zero and the increments for the X variable or the time variable to be 10. And then we can go into the actual table so you'll see that when X or DE is equal to zero the population why one is 100 for X equals 10. It's 482. Mexico's 2999 Mexico's 31,168 which matches air. Answer our answers in A and B. Now, if you look at bigger values off X right at X equals 1 61 71 80 you'll see that the Y values the population values would equal to 1200. So this is the value that p off t approaches as T approaches infinity and you'll see from the graph given for this problem, that PFT actually approaches 1200 as T approaches infinity. So the graph does confirm our calculations.

Hello. Everybody wanting to solve apartment number 54. Up for uh first order linear transfer equations. Chapter of cardinals equation. Okay so Uh our question we have at # 88. The differential equation is jean. Um My duty Equal Gay n. one. and Max my nose. And okay so it's a logistic differential equation. So if the solution is and 50. Well it was lovely. Yeah. Uh And max Okay over out. Mhm. Wells and max minor stars of the flood. B. To the power Connecticut kate. And max jeez okay is that sport? Yeah. D great equal or greater than Yeah. Okay so phone number be I don't know good. Mhm. Given an ancient population are right equal 20 and a maximum population of and max with values in max equal 400. And so the logistic function becomes after substituting anti equal. Mhm. 20 multiply 400 over 20 plus 400 -20. Uh huh. Yeah I want to play eat too far negative 400 K. T. Okay so it will be equal 400 over one plus 19. Eat the about negative 400 K. Teeth. Okay so the population doubles in five years. So used and five. Mhm. Yeah. And five When people 14 as a double value to find or to get kate. Okay so we can write our functional following. Okay 14 equal 401 lost 19 B. to the Power -14 K. 400 K. Uh huh Katie. So he will be on fire. Okay. Okay so I've been doing some simplification. So it will be one loss 19 ft of our negative 14 A sort Yeah. Before negative 2000. Okay. Okay, we'll be equal then. Okay, so each part negative 40K. Will be equal 9/19 and when we get when for both sides. Yeah. Okay. So okay, we'll be equal negative Lynn mind over 19. Oh what when 22,000. Okay, so now that was just the function as and the equal 400 over one plus 19 E. To the power T blend mine 4:19 Over 19 over what? Okay, so for number 30 the whole thing For number three since M right for 43 equal some after substitution. 400 over one Lost 19. Into the power Yeah. T three limb mine lying over 19. Right? Yeah over five. And so it won't be almost deep on 388 88. Okay, so the population after 43 years as model will be approximately well 388 Fish. Mhm. Okay. So for number D. Yeah. Okay. The population is growing as this when the end oh, by Bt at maximum. Okay, so it's uh by completing the square, that's a derivative equals the end over Bt mhm. The K N month deploy mm max minus and Okay. Okay, so through the equal. Okay, negative. Okay and squirt max little bush four Los gay and and max minus gay and squirt loss. Okay. And word max over food. Okay, so will be equal. We can get Gaelle, common factors minus K. Multiplied and max over to minus. And whole sport plus gay and swirled max over or. Okay. So when and when. Okay, Greater than zero. So the maximum as we went and max Over 2 -2 equals zero. Okay. And when. Kay less than zero. The minimum is when and max Over 2 -2- and equals europe. Since K greater than zero is the maximum that lived at the moment. When the population is growing fastest is when mm max and equal M max. Sorry. Yeah. And max over equal when Uh equals m. x over two. Okay. and full number eight segment of the predicted population and people end over end of tea. So we'll be as shown draft. Okay. But that we can do that. Yeah. Okay. That's four. The population of be at my access and time in X axis. And we have the value when we start from simple type. Okay. Thanks for watching. And see you later. Endemic. A differential equation

Hello Everybody You are going to solve a problem? # 55. Yeah. Uh for first order linear differential equations. Chapter of thermal situation. Okay so we have at number eight for the question. Uh huh. The differential equation is Gm mhm over DT or by gT equal b um when mm max over and Okay. So is separable Supreme the variable and integrate. Okay so the M by the T will be called P. M. Human and max over and Okay, so we'll be well mm and limb and max minus learn and Okay okay so we can uh that and left. Right okay, so one full bush and you learned and max minus men and D. M. All the economy the teeth. Okay. So by integrating now mhm mm Uh huh. Mhm For besides and we can and took you equal when and max minus then. And so do you will be full negative one over end G mm. G M Okay, so it will be equal negative to that or one over you. Do you equal the creation of the the deep? Okay, so it will be uh negative Lynn. Okay, uh most most of you all absolute you it will be T plus C one. So what substituting? Okay, okay, so negative limb mm absolute length and max minus len and equal meaty plus C one. Okay. Okay, so some simplification you can finally get that linn and max my most limb and will be equal oh posted or negative he to the power negative pt plus a story minus. Um Okay so okay. Plus or minus the department pt want to eat the four negative one. Okay. Okay so that will be Lynn and mats minus length and equal negative C. E. To the negative B. G. Okay so we can get link and max bush and will be equal negative C. E. To the negative pt. Okay so yeah finally we can get that and equal and max experimental of C. E. To the negative the D. Okay so the solution to the front of equation will be equal that and equal and mates exponential C. E. To the negative feet. T Okay that's 40 greater than or equal zero. Okay. For number B. And the whole number the given the maximum population and max equal And makes equal 400. And the function becomes after substituting mt equal and max exponential of C. E. To the parliament. The pt. So it will be 400. Its potential C. E. To the bar negative the T. Okay. to use the initial population and zero and audio equal 20 so get see Okay so 20 equal 400 exponential Ce 240. Okay so 0.05 will be equal exponential. See okay so finally see will be equal land four point oh right okay so now the function is will be equal. Are empty Equals 1400 400. Its potential Len or 10 by E. To the government to be team. So you mentioned doubles in five years. So use and five at and why equal 40 to get me? Okay so 40 equals 400 exponential Lynn Four point oh 500. Like to the partners with five. The Okay so can that Okay. Okay. Working. Yeah. So Mhm. Um Len 0 .01 equal lamb 005 to the four negative B. Make the fight each order. Bye. The. Okay. Okay so after after that we can it sets here. Okay so finally we can get that Lynn Lynn appoint one over limb open oh five will be equal to the bar next to five feet. Okay so finally yeah he will be a B. Will be equal -1/5 limb Land 4.1 Overland Open to five. Okay. So at sea almost equal negative point 996. Okay so we will be all mostly for all points to 5-6. So the particular function is four empty Will be equal 400 exponential -2996 E. to the bar negative 4.25 26 P. Okay. Okay. A full number C. Okay. Yeah. Mhm. Well for number C. Okay. since n. 43 Mhm equals four 100 exponential. Uh We can Okay so at Pain 40 ft. Okay so t will be equal for b. Okay a specific things. That vision. Okay so Um and we'll be 2043 will be almost equal to mind three. Okay so the population after 43 years models will be approximately 2 20 293. Yes. Okay. Okay. Phone # eight. Okay. The population is growing fastest where the end is a maximum 20 n. by beauty. Its maximum and which is where delivery is equals zero. Okay so we can find a river of the N. Y. D. T. Okay so D. By D. T. Yeah D. M. Over the T. Will be equal the by the T. Or the and Lynn and max over and. Okay. Okay so it will be equal mm hmm divide et Okay. For B. M. Oh len and max minus Lynn and. Mhm. Okay. Okay. So after simplification will be equal B. D. Um over the T limb and max over and minus B. The end over by the T. Okay so family called vehicle B. The M. Y. V. T. Lynn and max over um minus one. Okay. So since neither P nor and that can equal zero. So the drift that can only occur one Lynn and max over and it will one. Okay so sold for and to find and equal N. Max who were E. Okay so at number eight or number eight story just yeah. Uh huh. Yeah. Okay. Okay. So we can grab predict population and T. For the uh greater than or equal zero and less than or equal 100. So it can shown as the following in why actors of population P. And N. X. Satisfying deal with years. Okay thanks for watching and everything before the next differential equation.

Okay, so a differential equation will contain uh dependent dependent variable and the independent variable and derivative of the independent variable with respect to the we respect to well respected independent variable. So for this problem um efficient population with harvesting, So it will be kind of capacity in the pond of 1000. And its population at any time seen modeled by a logistic equation. So there was just the equation. The end over D. C. Is equal to 0.4. Over 1000 comes and Over 1000 my understand where N. O. T. The notes the number of fifth that time teen years. So D. N. D. T minus 50 if an Is greater than 250. So for part B. So far the the possible solution curve of the equation including the solution end of zero Equal to 2 50. So if you were to graph this, um you have a slow diagram, right? So Z will be equal to the end of the D. T. So And the zero lives Lies in about 500854. Right? So the solution curve will be will be a syntactic As some tactic to end equals two And will be equal to 854


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