Chapter 4.6, Problem 44E

To determine

To find: the year did the number of infested inland lakes first reach 120.

Answer to Problem 44E

After 9 years lake first reach 120.

Explanation of Solution

Given information:

Over a period of 14 years, the number $N$ of inland lakes infested with zebra mussels in a certain state can be modeled by,

  $N=-0.0284t^4+0.5937t^3-2.464t^2+8.33t-2.5$

Where $t$ is time.

Calculation:

Substituting $N=120$ , the equation becomes,

  $\begin{array}{l}-0.028t^4+0.5937t^3-2.464t^2+8.33t-2.5=120\\ -0.028t^4+0.5937t^3-2.464t^2+8.33t-122.5=0\end{array}$

Because negative real roots and imaginary roots doesn’t make sense, so need to check for only positive real root.

By the Descartes’s rule, the equation has four or two positive real root.

Input $Y_1=-0.028X^4+0.5937X^3-2.464X^2+8.33X-122.5$ using key

  $-0.0284X\wedge 4+0.5937X\wedge 3-2.464X{x}^2+8.33X-122.5$ .

The display is shown below:

  

Press The GRAPH key.

  

To find the zero, press $2\text{ndTRACE}2\text{ENTER}$ .

  

From the above display, the function has two zero at approximately $t=9$ and $t=15.5$ .

Therefore, after 9 years lake first reach 120.