Chapter 4.1, Problem 31PPS

a.

To determine

A linear equation to find the number of golden retrievers $\left(G\right)$ that will be registered in year $\left(t\right)$ when $\left(t=0\right)$ in the year 2000.

Answer to Problem 31PPS

The linear equation is $y=6.4x+49.7$ .

Explanation of Solution

Given: It is given that in 2001, there were about 56.1 thousand golden retrievers registered in United States and in 2002 this number was 62.5 thousand.

Calculation:

Considering that $\left(y\right)$ represents the number of retrievers in thousand and $\left(x\right)$ represents the number of years in thousands. Therefore, the relationship between $\left(x\right)$ and $\left(y\right)$ is expressed as

  $y=mx+c\mathrm{......}\left(1\right)$

Now, from 2001 to 2002 the number of retrievers in thousands increased from 56.1 to 62.5.

Therefore, slope is calculated as,

  $\begin{array}{c}m=\frac{62.5-56.1}{2002-2001}\\ =6.4\end{array}$

Substituting the value of slope $\left(m\right)$ and $\left(x,y\right)\to \left(1,56.1\right)$ in (1), the y-intercept is evaluated as

  $\begin{array}{c}56.1=\left(6.4\right)\left(1\right)+c\\ c=49.7\end{array}$

Again from (1) plugging for $\left(m\right)$ and $\left(c\right)$

  $y=6.4x+\mathrm{49.7......}\left(1\right)$

b.

To determine

The graph for the equation.

Explanation of Solution

Given: It is given that in 2001, there were about 56.1 thousand golden retrievers registered in United States and in 2002 this number was 62.5 thousand.Graph: The graph for the equation obtained $y=6.4x+49.7$ from (1) is drawn as

  

c.

To determine

The number of golden retrievers that will be registered in 2012

Answer to Problem 31PPS

The number of golden retrievers that will be registered in 2012 are 126.5 thousands.

Explanation of Solution

Given: It is given that in 2001, there were about 56.1 thousand golden retrievers registered in United States and in 2002 this number was 62.5 thousand.

Calculation:

Considering that $\left(y\right)$ represents the number of retrievers in thousand and $\left(x\right)$ represents the number of years in thousands. Therefore, the relationship obtained between $\left(x\right)$ and $\left(y\right)$ from (1) is $y=6.4x+49.7$ .

Also, from (1), the relationship shows the number of retrievers registered at $\left(t=0\right)$ in 2000. Therefore, the number of retrievers registered in 2012 can be calculated by plugging,

  $\begin{array}{c}x=2012-2000\\ =12\end{array}$

Hence, from (1) plugging for $x=12$

  $\begin{array}{c}y=6.4\left(12\right)+49.7\\ =126.5\text{ thousands}\end{array}$