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)= - 128(x - 2005)+3341. 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 2012 is 2445. (simplify your answer. type an integer or a decimal.)
compare the answer to the actual value of 3465. choose the correct answer below.
the estimated value is too low
the estimated value is exact
the estimated value is too high

Explanation:

Step1: Identify the linear - function

The linear function is given as \(f(x)= - 128(x - 2005)+3341\).

Step2: Calculate the value for \(x = 2012\)

Substitute \(x = 2012\) into the function:
\[

$$\begin{align*} f(2012)&=-128(2012 - 2005)+3341\\ &=-128\times7 + 3341\\ &=-896+3341\\ &=2445 \end{align*}$$

\]

Step3: Analyze the result

Since we are using a linear - function to estimate the number of inspectors in 2012 and we got the estimate as 2445.

Answer:

The estimate of the number of inspectors in 2012 is 2445. Since we used a linear model to estimate the value, and there is no information about the actual model governing the number of inspectors over the years, we assume this is a reasonable estimate based on the given linear function. The estimate is likely an approximation as it is based on a linear interpolation/extrapolation model.