Question

the bar graph shows the percentage of country a high school seniors who applied to more than three colleges for selected years from 1980 through 2013. the data in the bar graph can be modeled by ( f(x)=x + 25 ) and ( g(x)=32.9 e^{0.0216 x} ), in which ( f(x) ) and ( g(x) ) represent the percentage of high school seniors who applied to more than three colleges ( x ) years after 1980. use these functions to complete parts (a) through (c) below.
a. according to the linear model, what percentage of high school seniors applied to more than three colleges in 2010?
55 %
b. according to the exponential model, what percentage of high school seniors applied to more than three colleges in 2010?
63 % (round to the nearest percent as needed.)
c. which function is a better model for the data shown by the bar graph in 2010?
o a. linear model
o b. exponential model

Explanation:

Step1: Calculate the value of \(x\) for 2010

Since \(x\) represents the number of years after 1980, for 2010, \(x = 2010 - 1980=30\)

Step2: Use the linear model \(f(x)=x + 25\)

Substitute \(x = 30\) into \(f(x)\):
\(f(30)=30 + 25=55\)

Step3: Use the exponential model \(g(x)=32.9e^{0.0216x}\)

Substitute \(x = 30\) into \(g(x)\):

$$
LATEXBLOCK0
$$

We know that \(e^{0.648}\approx1.91\) (using a calculator for the exponential function)
\(g(30)=32.9\times1.91 = 62.839\approx63\)

Step4: Compare the model values with the bar - graph value

From the bar - graph, the value for 2010 is 63.

Answer:

a. \(55\%\)
b. \(63\%\)
c. B. Exponential model