To determine Ngozi's salary [tex]\( S(t) \)[/tex] [tex]\( t \)[/tex] years after she starts working as an interpreter, we need to consider her initial salary and the annual raise she receives.
1. Initial Salary: Ngozi starts with an initial salary of [tex]$24,000$[/tex].
2. Annual Raise: She receives an annual raise of [tex]\( 3.5\% \)[/tex] each year.
To find out the salary [tex]\( S(t) \)[/tex] after [tex]\( t \)[/tex] years, we need to account for the raise being compounded yearly. When a salary grows by a fixed percentage each year, the new salary after each year is calculated by multiplying the previous year's salary by [tex]\( 1 + \text{raise percentage} \)[/tex].
Let's break this down step-by-step:
### Step-by-step Solution
1. Initial Year (t=0):
- The salary [tex]\( S(0) \)[/tex] at the start is simply the initial salary without any raises.
[tex]\[
S(0) = \text{Initial Salary} = 24000
\][/tex]
2. First Year (t=1):
- The raise for the first year is calculated as:
[tex]\[
\text{Raise} = 24000 \times 0.035
\][/tex]
- The new salary for the first year is:
[tex]\[
S(1) = 24000 + (24000 \times 0.035) = 24000 \times (1 + 0.035)
\][/tex]
[tex]\[
S(1) = 24000 \times 1.035
\][/tex]
3. Second Year and Beyond (t=2, t=3, ...):
- For the second year, the salary will take into account the raise from the previous year:
[tex]\[
S(2) = S(1) \times 1.035
\][/tex]
- Since [tex]\( S(1) = 24000 \times 1.035 \)[/tex], substituting we get:
[tex]\[
S(2) = 24000 \times 1.035 \times 1.035 = 24000 \times (1.035)^2
\][/tex]
- This pattern continues for subsequent years. Generally, after [tex]\( t \)[/tex] years:
[tex]\[
S(t) = 24000 \times (1.035)^t
\][/tex]
### General Formula for [tex]\( S(t) \)[/tex]
Therefore, the function [tex]\( S(t) \)[/tex] that gives Ngozi's salary [tex]\( t \)[/tex] years after she starts working as an interpreter is:
[tex]\[
S(t) = 24000 \times (1.035)^t
\][/tex]
