Question
Your starting annual salary of \( \$ 69,000 \) increases by \( 4 \% \) each year a) Write the exponential function. b) What will be your annual salary in 10 years? Round your answer to the nearest dollar.
Ask by Tran Fuentes. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**a)** The exponential function is \( S(t) = 69,\!000 \times (1.04)^t \).
**b)** Your annual salary in 10 years will be approximately \$102,137.
Solution
Let's address each part of your question step-by-step.
### **a) Exponential Function**
The exponential function to model your annual salary increase can be written as:
\[
S(t) = 69,\!000 \times (1.04)^t
\]
- **\( S(t) \)**: Salary after \( t \) years
- **\( 69,\!000 \)**: Starting annual salary in dollars
- **\( 1.04 \)**: Growth factor (4% increase each year)
- **\( t \)**: Number of years
### **b) Annual Salary in 10 Years**
To find your annual salary after 10 years, plug \( t = 10 \) into the exponential function:
\[
S(10) = 69,\!000 \times (1.04)^{10}
\]
First, calculate \( (1.04)^{10} \):
\[
(1.04)^{10} \approx 1.480244
\]
Now multiply this by the starting salary:
\[
S(10) = 69,\!000 \times 1.480244 \approx 102,\!137
\]
**Your estimated annual salary in 10 years will be approximately \$102,137.**
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**Final Answers:**
**a)** The exponential function is \( S(t) = 69,\!000 \times (1.04)^t \).
**b)** The annual salary in 10 years will be approximately \$102,137.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To write the exponential function for a starting salary of \( \$ 69,000 \) that increases by \( 4\% \) each year, you can use the formula: \[ S(t) = S_0 \times (1 + r)^t \] where \( S_0 \) is the initial salary, \( r \) is the rate of increase, and \( t \) is the number of years. Therefore, the function is: \[ S(t) = 69000 \times (1 + 0.04)^t \] To find the annual salary in 10 years, plug in \( t = 10 \): \[ S(10) = 69000 \times (1.04)^{10} \] Calculating this, we have \( S(10) \approx 69000 \times 1.48024 \approx 102,120.56 \). Rounding to the nearest dollar, your salary in 10 years will be approximately \( \$ 102,121 \).
