Q. C - Calculus

Question

The limit of a model at infinity: Leila is interested in the effect of a stabilized wolf population on the eventual population of beavers in Idaho. The following table gives estimated beaver populations $B\left(t\right)$ for $t=0,1,2,3,4$and $5$ years after $2005$:

(a) Leila makes a plot of these values of $B\left(t\right)$and notes that the population of beavers is cyclical with diminishing amplitude. She finds that the quadratic function$M\left(t\right)=-51x^2+918x+41,389$ is a good model for the relative maximum data points at $t=0, 2,$and $4$and that $m\left(t\right)=33.25x^2-583.5x+48,122$

is a good model for the relative minimum data points at $t=1,3,$and $5.$Verify that these functions do in fact pass through the relevant data points, and graph the data for $B\left(t\right)$along with the two functions.

(b) Do the two quadratics $M\left(t\right)$and $m\left(t\right)$ever meet? If so, where? What conclusion could Leila make concerning the eventual steady population$\underset{t\to \infty }{\lim }B\left(t\right)$ of beavers in Idaho?

Step-by-Step Solution

Answer

(a) functions $M\left(t\right)=-51x^2+918x+41,389 \mathrm{and} m\left(t\right)=33.25x^2-583.5x+48,122$are a good model for the relative data points because all data points are located on a graph of functions.

(b) quadratics $M\left(t\right) \& m\left(t\right)$will not meet.

Part (a) Step 1. Given information.

Given functions are the followings.

$$\begin{gathered} M\left(t\right)=-51x^2+918x+41,389 \\ m\left(t\right)=33.25x^2-583.5x+48,122 \end{gathered}$$

The given tabular data is following.

Part (a) Step 2. Graph of functions

Plot the graph of the function $M\left(t\right)=-51t^2+918t+41,389 \& m\left(t\right)=33.25t^2-583.5t+48,122$and locate points of tabular data.

all points are located on a graph of functions so functions are a good model for the relative data points.

Part (b) Step 1. Given information.

Given functions are the followings.

$$\begin{gathered} M\left(t\right)=-51x^2+918x+41,389 \\ m\left(t\right)=33.25x^2-583.5x+48,122 \end{gathered}$$

Part (b) Step 2. The intersection of the graph of functions.

Parabola of function $M\left(t\right)=-51x^2+918x+41,389$open upward so it has minima at $(8.8,45562).$

Parabola of function $m\left(t\right)=33.25x^2-583.5x+48,122$open downward so it has maxima at $(9,45520).$

Since the minima of $M\left(t\right)$is greater than the maxima of $m\left(t\right)$so the graph of functions will not intersect.