Chapter 4.6, Problem 43E

To determine

To find: the year which did the population first reach 722000.

Answer to Problem 43E

After 68 years from 1890 or in year 1958, the population first reaches 722000

Explanation of Solution

Given information:

1890 to 2000, the A.I, E, and A population $P$ can be modeled by the function

  $P=0.004t^3-0.24t^2+4.9t+243$ , where $t$ is the number of years since 1890.

Calculation:

Substituting $P=722$ , the equation becomes,

  $\begin{array}{l}0.004t^3-0.24t^2+4.9t+243=722\\ 0.004t^3-0.24t^2+4.9t-479=0\end{array}$

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 one positive real root.

Input $Y_1=0.004X^3-0.24X^2+4.9X-479$ using key

  $0.004X\wedge 3-0.24X{x}^2+4.9X-479$ .

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 one zero at approximately $t=68$ .

Therefore, after 68 years from 1890 or in year 1958, the population first reach 722000.