Chapter 2.7, Problem 62PS

Solution

To find: In which year did the population first reach $722,000$ ?

The population first reached $722,000$ in the year $1966$ .

Given information:

From $1890\text{and }2000,$ the American Indian, Eskimo, and Aleut population P (in thousands) can be modeled by the function $P=0.0035t^3-0.235t^2+4.87t+243$ , where t is the number of years since $1890$ .

Formula used:

The real zeros of a function are the points where graph crosses the x -axis.

Calculation:

To find the year, in which population first reached $722,000$ , use $P=722$ in the given function and simplify as:

  $\begin{array}{c}P=0.0035t^3-0.235t^2+4.87t+243\\ 722=0.0035t^3-0.235t^2+4.87t+243\\ 0=0.0035t^3-0.235t^2+4.87t-479\end{array}$

Upon graphing this function, the required graph would look like as shown below:

  

From the above graph, it is visible that the graph crosses the x -axis at $x=75.532$ , so the zero of the function would be $75.532$ . Since the range of years is $1890\text{to }2000,$ so the answer would be 76^th year after $1890$ .

Therefore, the population first reached $722,000$ in the year $1966$ .