Answer :
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.
### 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.
