Question

the average monthly amount isabella has spent on gasoline since 1990 is shown in the table.
use the data in the table to complete the statements.
let x be the number of years since 1990. the year 2005 corresponds to an x - value of.
the function that best models the data, with numerical values rounded to the nearest hundredth, is f(x) = x + 22.08.
models have their limitations. for which year would this model not make sense to use?

Explanation:

Step1: Calculate the x - value for 2005

Since \(x\) is the number of years since 1990, for the year 2005, we use the formula \(x = 2005 - 1990\).
\(x=15\)

Step2: Calculate the slope of the linear function

We have two points \((x_1,y_1)=(10,135)\) (for the year 2000, \(x = 2000 - 1990=10\)) and \((x_2,y_2)=(15,199)\) (for the year 2005).
The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
\(m=\frac{199 - 135}{15 - 10}=\frac{64}{5}=12.8\)

Step3: Determine the year for which the model is not valid

The model is based on data from 1990 - 2009. If we consider the context of the data provided (historical data up to 2009), a year like 1985 (\(x=1985 - 1990=- 5\)) would be outside the domain of the data - based model.

Answer:

The year 2005 corresponds to an \(x\) - value of \(15\). The function that best models the data is \(f(x)=12.8x + 22.08\). A year like \(1985\) (any year before 1990) would be a year for which this model does not make sense to use.