97–100. Logistic growth Scientists often use the logistic grow...

Question

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model.

{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.

a. Graph P using a graphing utility. Experiment with different windows until you produce an S-shaped curve characteristic of the logistic model. What window works well for this function?

Accepted Answer

Hello. In this video we are going to be approaching the following graphing problem. We are told that a tree is planted. The maximum possible height of the tree is 12.5 ft. The height of the tree is given by the following equation Y is equal to 500 divided by 40 plus 2,500e raised to the power of negative 0.75 X. This is where X is in months and Y is in feet. We want to draw the graph of the function. OK, so in order to approach this problem, let's just go ahead and take some testing values for X. We're going to create a number table with X and the output Y. and what we are going to do is we are going to determine the shape of the graph by taking a look at the outputs for some values of X. Now, let's go ahead and plug in 8 different values for X. We can try out 02468, 1012, and 15. What we are going to do with these values of X is that we are going to take each of them and plug them into the function. And with the help of a calculator, we will go ahead and determine the behaviors of the graph based on the outputs of Y. So, if we were to take the value of zero and plug it into the given function, the output given to us would be 0.197. Repeating this process for two will give us the output 0.836. For the value of 4, we get the output 3.04. For the input of 6, we get an output of 7.378. For the input of 8, we get an output of 10.823. For 10, we get an output of 12.082. For 12, we get an output of 12.404. And finally, for 15, we get an output of 12.49. Now, if we take a look at the outputs of the function, what do the outputs tell us about the characteristics of the graph? Well, first of all, at the input X on the 00 value, we get an output of Y of 0.197. This is going to be the Y intercept for the graph, which is going to be the starting point of our function. Next, notice that we have a small increase between the interval of 0 and 2. We increase from the zap value of 0.197 to 0.836. So, on the interval from 0 to 2, we have a very slow growth. Now, on the interval from 2 to 10, notice that the values of the outputs increase exponentially. We go from 0.836 to 7.378 to 12.082. So what this means is that on the interval from 0 to 10, we have a rapid growth. And finally, on the interval from 10 to 15, notice that this rapid growth gradually starts to decrease. We go from 12.082 and stay below the value of 12.5. The maximum value that we hit is 12.49. So what this means is that the growth slows down, or in other words, we have another slow growth on the interval from 12 to 15. Furthermore, what this tells us is that since the maximum value is going to be 12.5 ft, as X approaches the value of infinity, The output Y approaches the value of 12.5. What this means is that the maximum value that this tree is going to approach is going to be 12.5. So, in other words, the bigger the X gets, the closer to 12.5 we get. Using this information, let's go ahead and draw the graph of the tree now. We will go ahead and label one, and we will label the very maximum height as 12.5. So, using the first input, we have 0.0197. So we are going to be just above the X axis for our starting point. Now at the value of 2, we have this rap slow growth that is slightly below 1. Then we have a very rapid growth going from 0 from 2 to 10. Which hits the ceiling of 12.082. And then, as we approach infinity, this growth starts to slow down and stay below the line of 12.5. And this is going to be the graph of the tree function. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

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