Chapter 4.1, Problem 23PPS

a.

To determine

The linear equation for the number of vacations(in millions) $y$ after $x$ years, if $x$ be the number of years since 2000.

Answer to Problem 23PPS

The linear equation is $y=9.575x+337.1$ .

Explanation of Solution

Given: It is given that,

In year 2000 the number of vacations were 337.1 million. And in year 2004 the number of vacations were 375.4 million.

Calculation: Consider that the year 2000 represents 0, so that the year 2004 will represent 4. Therefore the two points will be $\left(0,337.1\right)\text{ and }\left(4,375.4\right)$ .

Therefore the slope $\left(m\right)$ of the line through these points is evaluated as,

  $\begin{array}{c}m=\frac{375.4-337.1}{4-0}\\ =\frac{38.3}{4}\\ =9.575\end{array}$

Now the linear equation through the point $\left(0,337.1\right)$ having slope $\left(m=9.575\right)$ is evaluated as,

  $\begin{array}{c}y-337.1=\left(9.575\right)\left(x-0\right)\\ y-337.1=9.575x\\ y=9.575x+\mathrm{337.1....}\left(\text{Simplified}\right)\\ y=9.575x+\mathrm{337.1....}\left(1\right)\end{array}$

b.

To determine

The number of vacations that will be taken in the year 2012.

Answer to Problem 23PPS

The number of vacations taken will be 452 million in 2012.

Explanation of Solution

Given: It is given that,

In year 2000 the number of vacations were 337.1 million. And in year 2004 the number of vacations were 375.4 million.

Calculation: From (1) the number of vacations taken in the year 2012 can be evaluated by plugging $\left(x=12\right)$ in (1) for the year 2012. Therefore from (1),

  $\begin{array}{c}y=9.575\left(12\right)+\mathrm{337.1......}\left(\text{For }x=12\right)\\ =452\text{ million}\end{array}$