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)=967x + 3006 ) and ( g(x)=3944e^{0.132x} ) 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 2011? a. the linear model is better. b. the exponential model is better.

Explanation:

Step1: Calculate \( f(x) \) for \( x = 1 \) (since \( 2011 - 2010=1 \))

Substitute \( x = 1 \) into \( f(x)=967x + 3006 \).
\( f(1)=967\times1+3006=967 + 3006=3973 \)

Step2: Calculate \( g(x) \) for \( x = 1 \)

Substitute \( x = 1 \) into \( g(x)=3944e^{0.132x} \).
First, calculate \( e^{0.132\times1}\approx1.1405 \) (using the approximation \( e^a\approx1 + a+\frac{a^{2}}{2!}+\cdots\), or using a calculator). Then \( g(1)=3944\times1.1405\approx4498 \)

Step3: Compare with the actual value (public college value in 2011 is 4786)

The difference for \( f(x) \): \( |4786 - 3973|=813 \).
The difference for \( g(x) \): \( |4786 - 4498| = 288 \).

Answer:

A. The linear model is better.