To determine the expected number of jobs in radiology [tex]\( t \)[/tex] years from the year 2014, we need to account for the annual growth rate in the number of jobs.
Here are the detailed steps to derive the function [tex]\( j(t) \)[/tex] for the expected number of jobs:
1. Identify the current number of jobs and the growth rate:
- Initial number of jobs in 2014: [tex]\( 230,600 \)[/tex]
- Annual growth rate: [tex]\( 0.9\% \)[/tex] which can be written as [tex]\( 0.009 \)[/tex] in decimal form.
2. Set up the general formula for calculating the future number of jobs:
The number of jobs in radiology, [tex]\( t \)[/tex], years after 2014 is calculated using the formula for compound growth:
[tex]\[
\text{future jobs} = \text{initial jobs} \times (1 + \text{growth rate})^t
\][/tex]
3. Substitute the given values into the formula:
[tex]\[
j(t) = 230600 \times (1 + 0.009)^t
\][/tex]
Therefore, the function that gives the expected number of jobs [tex]\( j(t) \)[/tex] in radiology [tex]\( t \)[/tex] years from 2014 is:
[tex]\[
j(t) = 230600 \times (1.009)^t
\][/tex]
This function [tex]\( j(t) \)[/tex] allows you to calculate the expected number of radiology jobs for any given year [tex]\( t \)[/tex] after 2014, assuming a consistent annual growth rate of [tex]\( 0.9\% \)[/tex].
