Question

1. the consumer price index (cpi), which measures the cost of a typical package of consumer goods, was 245.8 in 2011 and 279.2 in 2016. let x = 11 correspond to the year 2011. 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 to 2 decimal places.

a. (8) let y represent the cpi. state the linear equation that best models the cpi.
b. (4) state the cpi in 2012.
c. (4) state the cpi in 2020.

Explanation:

Step1: Define data points

We have two points: when $x=11$ (2011), $y=245.8$; when $x=16$ (2016), $y=279.2$.

Step2: Calculate slope $m$

Slope formula is $m=\frac{y_2-y_1}{x_2-x_1}$

$$ m=\frac{279.2-245.8}{16-11}=\frac{33.4}{5}=6.68 $$

Step3: Find y-intercept $b$

Use point-slope form $y-y_1=m(x-x_1)$, substitute $x_1=11,y_1=245.8,m=6.68$:

$$ 245.8=6.68\times11 + b \\ 245.8=73.48 + b \\ b=245.8-73.48=172.32 $$

Step4: Write linear equation

Combine slope and intercept into $y=mx+b$

$$ y=6.68x + 172.32 $$

Step5: Find 2012 CPI ($x=12$)

Substitute $x=12$ into the linear equation:

$$ y=6.68\times12 + 172.32=80.16+172.32=252.48 $$

Step6: Find 2020 CPI ($x=20$)

Substitute $x=20$ into the linear equation:

$$ y=6.68\times20 + 172.32=133.6+172.32=305.92 $$

Answer:

a. $y=6.68x + 172.32$
b. 252.48
c. 305.92