Question

the bar graph on the right shows the average cost of room and board at four - year public and private colleges for four selected years from 2011 through 2017. the data can be modeled by $f(x)=941x + 3175$ and $g(x)=3851e^{0.133x}$ in which $f(x)$ and $g(x)$ represent the average cost of room and board at public four - year colleges in the school year ending $x$ years after 2010. use these functions to complete parts (a) through (c) below. c. which function is a better model for the data for the school year ending in 2017? a. the linear model is better b. the exponential model is better

Explanation:

Step1: Calculate the value of the linear model \(f(x)\) for \(x = 7\) (since \(2017 - 2010=7\))

Substitute \(x = 7\) into \(f(x)=941x + 3175\):

$$
LATEXBLOCK0
$$

Step2: Calculate the value of the exponential model \(g(x)\) for \(x = 7\)

Substitute \(x = 7\) into \(g(x)=3851e^{0.133x}\):

$$
LATEXBLOCK1
$$

Step3: Compare the model - predicted values with the actual value for public colleges in 2017

The actual value for public colleges in 2017 (from the bar - graph) is \(10430\).
The error for the linear model: \(|10430 - 9762|=668\)
The error for the exponential model: \(|10430 - 9761| = 669\)

Since the errors are approximately the same, but if we consider the general trend of the bar - graph (which is more in line with a linear increase for public colleges as the differences between consecutive years for public colleges in the bar - graph seem relatively constant compared to the nature of an exponential function), the linear model is a better fit.

Answer:

A. The linear model is better