Question

the number of air safety inspectors for selected years is shown in the table.
year 1992 1993 1994
inspectors 3319 3178 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 linear function f that models these data is f(x)= - 116(x - 1992)+3319
(a) 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 2507
(simplify your answer. type an integer or a decimal.)
compare the answer to the actual value of 3444. choose the correct answer below.
the estimated value is too low
the estimated value is exact
the estimated value is too high

Explanation:

Step1: Find the linear - function formula

The general form of a linear function is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. Given two points \((x_1,y_1)=(1992,3319)\) and \((x_2,y_2)=(1993,3178)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3178 - 3319}{1993 - 1992}=\frac{- 141}{1}=-141\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with the point \((1992,3319)\), we have \(y-3319=-141(x - 1992)\), then \(y=-141x+141\times1992 + 3319=-141x+280872+3319=-141x + 284191\).

Step2: Estimate the number of inspectors in 1999

Substitute \(x = 1999\) into the function \(y=-141x + 284191\). Then \(y=-141\times1999+284191=-141\times(2000 - 1)+284191=(-141\times2000+141)+284191=(-282000 + 141)+284191=-281859+284191 = 2332\).

Step3: Compare with the actual value

The actual value in 1999 is 2507. The estimated value \(2332\) is less than the actual value \(2507\), so the estimated value is too low.

Answer:

(a) \(y=-141x + 284191\)
(b) 2332, the estimated value is too low.