To solve the problem of determining how many years it will take for Susanna's deposit to grow from \[tex]$400 to \$[/tex]600 with an annual interest rate of 3%, we can follow these steps:
1. Identify the formula for compound interest:
[tex]\[ A = P(1 + r)^t \][/tex]
Where:
- [tex]\( A \)[/tex] is the target amount (in this case, \[tex]$600),
- \( P \) is the principal (initial deposit, \$[/tex]400),
- [tex]\( r \)[/tex] is the annual interest rate (3% or 0.03),
- [tex]\( t \)[/tex] is the time in years.
2. Set up the equation with the given values:
[tex]\[ 600 = 400(1 + 0.03)^t \][/tex]
3. Simplify the equation:
[tex]\[ 600 = 400(1.03)^t \][/tex]
4. Isolate the exponential term:
[tex]\[ \frac{600}{400} = (1.03)^t \][/tex]
[tex]\[ 1.5 = (1.03)^t \][/tex]
5. Take the natural logarithm (ln) of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln(1.5) = \ln((1.03)^t) \][/tex]
6. Use the property of logarithms to bring the exponent [tex]\( t \)[/tex] in front:
[tex]\[ \ln(1.5) = t \cdot \ln(1.03) \][/tex]
7. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(1.5)}{\ln(1.03)} \][/tex]
From the calculation:
- [tex]\( \ln(600) \approx 6.3969 \)[/tex]
- [tex]\( \ln(400) \approx 5.9915 \)[/tex]
- [tex]\( \ln(1.03) \approx 0.0295588 \)[/tex]
Thus:
[tex]\[ t = \frac{6.3969 - 5.9915}{0.0295588} \][/tex]
[tex]\[ t \approx 13.717 \][/tex]
Therefore, it will take approximately 13.717 years for Susanna’s deposit to grow to \$600 with an annual interest rate of 3%.
