Question

the bar gra the right shows the average cost of room and board at four-ye c and private colleges for four selected years from 2011 throu data can be modeled by $f(x)=946x + 3134$ and $g(x)=17e^{0.133x}$ 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. (round to the nearest dollar as needed) c. which function is a better model for the data for the school year ending in 2011? a. the exponential model is better b. the linear model is better

Explanation:

Step1: Find the value of \(x\) for 2011

Since \(x\) represents the number of years after 2010, for 2011, \(x = 2011 - 2010=1\)

Step2: Calculate \(f(1)\)

Substitute \(x = 1\) into the linear function \(f(x)=946x + 3134\)
\(f(1)=946\times1+3134=946 + 3134=4080\)

Step3: Calculate \(g(1)\)

Substitute \(x = 1\) into the exponential function \(g(x)=5517e^{0.133x}\)
\(g(1)=5517e^{0.133\times1}\)
Using the formula \(y = e^{a}\approx2.718^{a}\), then \(g(1)=5517\times e^{0.133}\approx5517\times1.142=6300.314\)

Step4: Compare with the data from the bar - graph

From the bar - graph, for public colleges in 2011, the cost is \(4912\). The value of \(f(1) = 4080\) has an error of \(|4912 - 4080|=832\), and the value of \(g(1)\approx6300\) has an error of \(|4912 - 6300| = 1388\)

Answer:

B. The linear model is better