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The Consumer Price Index (CPI), which measures the cost of a typical package of consumer goods, was 202.9 in 2011 and 233.2 in 2016. Let x=11 correspond to the year 2011 and estimate the CPI in 2013 and 2014. Assume that the data can be modeled by a straight line and that the trend continues indefinitely. Use two data points to find such a line and then estimate the requested quantities. Let y represent the CPI. The linear equation that best models the CPI is____ (Simplify your answer. Use integers or decimals for any numbers in the equation. Round to the nearest hundredth as needed.)

Respuesta :

The first thing we have to identify in our problem are the variables

[tex]\begin{gathered} x\to\text{time} \\ y\to\text{CPI} \end{gathered}[/tex]

Now we see the points (x,y) that gives us the problem

[tex]\begin{gathered} 2011\to(11,202.9) \\ 2016\to(16,233.2) \end{gathered}[/tex]

Since behavior can be modeled by a straight line, we use the general equation of the straight line

[tex]y=mx+b[/tex]

Where m is the slope of the line and b is the y-intercept.

Taking this into account and with the 2 points that they give us, we proceed to calculate the equation of the line starting with the slope:

[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{233.2-202.9}{16-11} \\ m=\frac{30.3}{5} \\ m=6.06 \end{gathered}[/tex][tex]\begin{gathered} y=6.06x+b \\ 202.9=6.06(11)+b \\ b=202.9-66.66 \\ b=136.24 \end{gathered}[/tex]

The equation that models the behavior of the CPI is

[tex]y=6.06x+136.24[/tex]

Now we calculate the CPI values for the years 2013 and 2014

[tex]\begin{gathered} 2013\to x=13 \\ y=6.06(13)+136.24 \\ y=78.78+136.24 \\ y=215.02 \end{gathered}[/tex][tex]\begin{gathered} 2014\to x=14 \\ y=6.06(14)+136.24 \\ y=84.84+136.24 \\ y=221.08 \end{gathered}[/tex]

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