Biologieal Researeh Stephanie, a biologist who does research...

Biologieal Researeh Stephanie, a biologist who does research for the poultry industry, models the population $P$ of wild turkeys, $t$ days after being left to reproduce, with the function $$ P(t)=-0.00001 t^{3}+0.002 t^{2}+1.5 t+100 $$ (a) Graph the function $y=P(t)$ for appropriate values of $t$ (b) Find what the maximum turkey population is and when it occurs. (e) Assuming that this model continues to be accurate, when will this turkey population become extinct? (d) Writing to Learn Create a scenario that could explain the growth exhibited by this turkey population.

Biologieal Researeh Stephanie, a biologist who does research for the poultry industry, models the population $P$ of wild turkeys, $t$ days after being left to reproduce, with the function
$$
P(t)=-0.00001 t^{3}+0.002 t^{2}+1.5 t+100
$$
(a) Graph the function $y=P(t)$ for appropriate values of $t$
(b) Find what the maximum turkey population is and when it occurs.
(e) Assuming that this model continues to be accurate, when will this turkey population become extinct?
(d) Writing to Learn Create a scenario that could explain the growth exhibited by this turkey population.

Key Concepts

Optimization

Optimization in mathematics refers to the process of finding the best solution or extremum (maximum or minimum) of a function within a given domain. In the context of population dynamics, determining the maximum population size and the time at which it occurs is a typical optimization problem, often solved by finding the derivative of the function and setting it to zero to identify critical points.

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents. In modeling real-world phenomena such as population dynamics, polynomial functions can capture both growth and decline, with the degree of the polynomial determining the potential number of turning points and overall behavior of the system.

Graphing Functions

Graphing a function involves plotting its output values against its input values on a coordinate system to visualize trends and critical features like intercepts, peaks, and troughs. In applications like biological population models, an accurate graph enables the analysis of when and how the population increases, reaches a maximum, or declines over time, providing a visual tool for understanding complex behavior.

Root Finding

Root finding is the process of determining the values of the variable that make the function equal to zero. This concept is crucial in population modeling when determining events such as extinction, which is represented by the function equaling zero. Techniques ranging from factoring and applying the quadratic formula to numerical methods are used depending on the complexity of the function.

Mathematical Modeling

Mathematical modeling involves using mathematical constructs to represent real-world systems and predict their behavior under various conditions. By building a model with a function, researchers can simulate scenarios, analyze trends over time, and explain outcomes such as population growth, peak, decline, or extinction, making it an essential tool in fields like biology and economics.

Transcript

00:01 Hello, hope you're doing well.

00:03 So before we dive into this problem, let's just review what the steps that we have to take for each part of this problem.

00:10 So for part a, we just need to plug in a few values of t and then find out what p of t corresponds to them.

00:18 So get a few points for this graph and then graph to get a good idea of what this graph looks like.

00:23 Then for part b, we're going to, our graph should give us some sort of maximum point like that.

00:29 And we can find the maximum term turkey population using that point.

00:34 And then for part c, we need to figure out at some point the graph is going to go down and hit the x -axis.

00:41 And that's the point in which the turkey population becomes extinct.

00:44 So we need to find that t value.

00:49 And then lastly, we need to come up for some sort of explanation as to why that would be the case.

00:54 So first for part a, let's plug in a few t values and see what p of t values we get.

00:59 So let's kind of start with t -e -0 and go.

01:01 By 100.

01:03 So if we have t is equal to 0, you plug in 0 for t, and everything becomes 0 except for this 100.

01:09 So we're left with 0 to 0, the 0 .00.

01:13 Then let's go to t is equal to 100.

01:16 So you plug in 100 for t, and if you put all that into your calculator, you should end up with a value of 100, 260.

01:26 So 260 is what you should get for the p of t.

01:28 Then let's move on to t is equal to 200.

01:31 So plugging t, is equal to 200 into here, you should end up with a value, p value of 400.

01:38 Moving on to t is equal to 300, plugging that into this equation, you should end up with a p of t value of 460.

01:47 Then for t is equal to 400, if you plug that into your p of t equation, you should end up with a p of t value of 380.

01:57 And then for t is equal to 500, plugging that into your equation, you shouldn't end up with a value of 100.

02:09 Okay, so now we can go and graph all of these points...

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