Question

the number of air safety inspectors for selected years is shown in the table.
year inspectors
2005 3341
2006 3163
2007 3085
(a) find a linear function f that models these data. is f exact or approximate?
(b) use f to estimate the number of inspectors in 2012. compare your answer to the actual value of 3465. did your estimate involve interpolation or extrapolation?
(a) a linear function f that models these data is f(x)= (round to the nearest integer as needed.)

Explanation:

Step1: Assume linear - function form

Let the linear function be \(y = mx + b\), where \(x\) is the year and \(y\) is the number of inspectors. Let \(x = 0\) correspond to the year 2005. Then for the year 2005 (\(x_1=0\)), \(y_1 = 3341\); for the year 2006 (\(x_2 = 1\)), \(y_2=3163\).
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3163 - 3341}{1-0}=\frac{- 178}{1}=-178\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with the point \((x_1,y_1)=(0,3341)\), the linear function is \(y=-178x + 3341\).

Step2: Adjust for 2012

For the year 2012, \(x = 2012 - 2005=7\).
Substitute \(x = 7\) into the linear function \(y=-178\times7 + 3341\).
First, calculate \(-178\times7=-1246\).
Then \(y=-1246 + 3341=2095\).

Answer:

2095