Chapter 8.6, Problem 42E

To determine

To find: The function that models the given technology, and predict amount US residents will spend on pets or pet- related products and services in Year 7.

Answer to Problem 42E

The function for the model is $a\left(x\right)=3.8x+49.3$ that is chosen because maximum number of data passes through a straight line, which predicts thatin the 7^th year amount a spent is 75.9 billion dollars.

Explanation of Solution

Given information:

The given table shows the amount a (in billions dollars) United State residents spent on pets or pet-related products and services.

Year, x	1	2	3	4	5
Amount, a	53.1	56.9	61.8	65.7	67.1

Formula used:

The data is graphed using graphing calculator.

Calculations:

The graph of given data is shown in Figure-1 here.

The points (1, 53.1), (2, 56.9), and (3, 61.8) are on a straight line whose equation is given by formula $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$ . We determine equation of line through first two points

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

  $\Rightarrow a-53.1=\frac{56.9-53.1}{2-1}\left(x-x_1\right)$

  $\Rightarrow a-53.1=3.8\left(x-1\right)$

  $\Rightarrow a=3.8x-3.8+53.1$

  $\Rightarrow a=3.8x+49.3$

From the graph we observe that maximum number of data points $\left(x,a\right)$ pass through a straight line $a=3.8x+49.3$ , therefore, describe the data with maximum accuracy we adopt a linear model whose function is $a\left(x\right)=3.8x+49.3$

  

Prediction:

To predict the amount “ a ” the US residents will spend on pets or pet- related products and services in Year 7, we substitute $x=7$ in the function for the model $a=3.8x+49.3$ .

  $a=3.8\times 7+49.3$ Or $a=75.9$ Billion dollars.

Also, from the graph if we sketch line $x=7$ it intersects line $a=3.8x+49.3$ at point $\left(7,75.9\right)$ that predicts that in the 7^th year amount a spent is 75.9 billion dollars.

We can also draw line through last two points but it will not be so accurate it will give amount spent in 7^th years 69.9 billion dollars.

Conclusion:

The function for the model is $a\left(x\right)=3.8x+49.3$ that is chosen because maximum number of data pass through a straight line, which predicts thatin the 7^th year amount a spent is 75.9 billion dollars.