Answer :
To solve the problem of determining how much a person makes after working for 15 years with an annual salary increase, we'll use an exponential model.
1. Initial Parameters:
- The starting annual salary is [tex]$29,000.
- The annual increase rate is 6.2%.
2. Exponential Model:
We'll use the formula for exponential growth:
\[
y = a \cdot (b)^t
\]
Where:
- \( y \) is the salary after \( t \) years.
- \( a \) is the initial amount, which is the starting salary, $[/tex]29,000.
- [tex]\( b \)[/tex] is the base of the exponential function, which is [tex]\( 1 \)[/tex] plus the annual increase rate in decimal form: [tex]\( 1 + 0.062 = 1.062 \)[/tex].
- [tex]\( t \)[/tex] is the number of years, which is 15 in this case.
3. Calculating the Salary After 15 Years:
Substitute the known values into the model:
[tex]\[
y = 29000 \cdot (1.062)^{15}
\][/tex]
After evaluating this expression, you find:
- The calculated salary is approximately [tex]$71,493.36672.
4. Rounding:
Since we need to round the salary to the nearest dollar, the amount is rounded to $[/tex]71,493.
Thus, after 15 years, the person will earn approximately $71,493 annually.
1. Initial Parameters:
- The starting annual salary is [tex]$29,000.
- The annual increase rate is 6.2%.
2. Exponential Model:
We'll use the formula for exponential growth:
\[
y = a \cdot (b)^t
\]
Where:
- \( y \) is the salary after \( t \) years.
- \( a \) is the initial amount, which is the starting salary, $[/tex]29,000.
- [tex]\( b \)[/tex] is the base of the exponential function, which is [tex]\( 1 \)[/tex] plus the annual increase rate in decimal form: [tex]\( 1 + 0.062 = 1.062 \)[/tex].
- [tex]\( t \)[/tex] is the number of years, which is 15 in this case.
3. Calculating the Salary After 15 Years:
Substitute the known values into the model:
[tex]\[
y = 29000 \cdot (1.062)^{15}
\][/tex]
After evaluating this expression, you find:
- The calculated salary is approximately [tex]$71,493.36672.
4. Rounding:
Since we need to round the salary to the nearest dollar, the amount is rounded to $[/tex]71,493.
Thus, after 15 years, the person will earn approximately $71,493 annually.
