Q. 87 - Calculus

Question

In Exercises $87-89$, suppose that Leila is a population biologist with the Idaho Fish and Game Service. Wolves were introduced formally into Idaho in $1994$, but there were some wolves in the state before that. Leila has been assigned the task of estimating the rate at which the number of wolves in Idaho increased naturally, before the animals were introduced. The only information she has is a population model, which indicates that the wolf population currently satisfies the formula.

                                              $w\left(t\right)=835\left(1-e^{-.006t}\right)$

where $t$ is the number of years since$1994$.

 Leila reasons that the average rate of change of wolves from $1994$ to a given time $t$ is given by $\frac{w\left(t\right)}{t}$Why does her reasoning make sense?

Step-by-Step Solution

Answer

Since the average rate of change of wolves satisfies the mean value theorem, Leila's reasoning makes sense.

Step 1 . Given information

The Wolf population currently is,

$w\left(t\right)=835\left(1-e^{-.006t}\right)$

where $t$ is the number of years since $1994$.

The average rate of change of wolves from $1994$ to a given time $t$ is given by $\frac{w\left(t\right)}{t}$.

Step 2 . The objective is to decide whether the reason makes sense or not.

To decide it, use the mean value theorem.

From the mean value theorem,

$\frac{\Delta w}{\Delta t}=\frac{w\left(t\right)-w\left(0\right)}{t-0}$

        $=\frac{w\left(t\right)-0}{t}$  [Since, $w\left(0\right)$ is the initial population (i.e), $w\left(0\right)=0$]

         $=\frac{w\left(t\right)}{t}$

Thus, Leila reasoning makes sense.