Question

the number of air - safety inspectors for selected years is shown in the table.
year inspectors
1992 3319
1993 3178
1994 3087
(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 1999. compare your answer to the actual value of 3444. did your estimate involve interpolation or extrapolation?
(a) a linear function f that models these data is f(x)= - 116(x - 1992)+3319 (round to the nearest integer as needed.)
is f exact or approximate? choose the correct answer below.
exact approximate
(b) the estimate of the number of inspectors in 1999 is (simplify your answer. type an integer or a decimal.)

Explanation:

Step1: Find the linear - function formula

We have two data points \((x_1,y_1)=(1992,3319)\) and \((x_2,y_2)=(1994,3087)\). The slope \(m\) of the linear function \(y = mx + b\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3087 - 3319}{1994 - 1992}=\frac{- 232}{2}=-116\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(1992,3319)\), we get \(y-3319=-116(x - 1992)\), which simplifies to \(y=-116x+116\times1992 + 3319=-116x+231072+3319=-116x + 234391\).

Step2: Estimate the number of inspectors in 1999

Substitute \(x = 1999\) into the linear function \(y=-116x + 234391\). Then \(y=-116\times1999+234391=-116\times(2000 - 1)+234391=(-116\times2000+116)+234391=(-232000 + 116)+234391=-231884+234391 = 2507\).

Step3: Compare with the actual value

The problem mentions an actual value of 3444. The linear - function estimate is an extrapolation since 1999 is outside the range of the given data years (1992 - 1994).

Answer:

The estimated number of air - safety inspectors in 1999 is 2507.