To model the situation where an investment grows by 5% each year, we can use the exponential growth formula.
### Exponential Growth Formula:
The formula to calculate the future value of an investment growing at a fixed annual percentage rate is given by:
[tex]\[ A = P(1 + r)^t \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (initial investment).
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( t \)[/tex] is the number of years the money is invested for.
### Given Data:
- Original (principal) amount, [tex]\( P \)[/tex] = \[tex]$15,000
- Growth rate per year, \( r \) = 5% = 0.05 (expressed as a decimal)
- Number of years, \( t \) = 3
### Step-by-Step Solution:
1. Substitute the given values into the exponential growth formula:
\[ A = 15000 \times (1 + 0.05)^3 \]
2. Simplify inside the parentheses:
\[ 1 + 0.05 = 1.05 \]
3. Raise the base (1.05) to the power of 3:
\[ 1.05^3 = 1.05 \times 1.05 \times 1.05 \]
Let's calculate it step-by-step:
\[ 1.05 \times 1.05 = 1.1025 \]
\[ 1.1025 \times 1.05 = 1.157625 \]
4. Now, multiply this result by the principal amount:
\[ A = 15000 \times 1.157625 \]
5. Perform the multiplication:
\[ A = 15000 \times 1.157625 = 17364.375 \]
So, the value of the investment after 3 years is:
\[ A = \$[/tex]17,364.38 \]
### Final Answer:
The value of the investment after 3 years is \$17,364.38.
