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 \(x\) - 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: Find the slope of the linear function

We can use two - point formula for the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take the points \((0,23)\) (corresponding to 1990, where \(x = 0\)) and \((10,135)\) (corresponding to 2000, where \(x = 10\)).
\(m=\frac{135 - 23}{10-0}=\frac{112}{10}=11.2\)

Step3: Check the model's limitation

The model is based on data starting from 1990 (\(x\geq0\)). For the year 1985, \(x=1985 - 1990=- 5\). Since the data collection starts from 1990, using the model for \(x=-5\) (year 1985) does not make sense as there is no data prior to 1990 to support the linear relationship for negative \(x\) values.

Answer:

The year 2005 corresponds to an \(x\) - value of \(15\). The function that best models the data is \(f(x)=11.2x + 22.08\). The model does not make sense to use for the year 1985.