Questions: New parents wish to save for their newborn's education and wish to have 41,000 at the end of 19 years. How much should the parents place at the end of each year into a savings account that earns an annual rate of 8.5% compounded annually? (Round your answers to two decimal places.) How much interest would they earn over the life of the account? Determine the value of the fund after 12 years. How much interest was earned during the 12th year?

Solution Steps

1. To find the annual deposit required, we can use the future value of an annuity formula.
2. To calculate the total interest earned over the life of the account, we subtract the total amount deposited from the future value.
3. To determine the value of the fund after 12 years, we use the future value of an annuity formula for 12 years.
4. To find the interest earned during the 12th year, we calculate the difference in the fund's value between the 12th and 11th years.

To determine the amount the parents should deposit at the end of each year, we use the future value of an annuity formula:

\[ A = \frac{FV}{\frac{(1 + r)^n - 1}{r}} \]

where:

• \( FV = 41000 \)
• \( r = 0.085 \)
• \( n = 19 \)

Substituting the values, we find:

\[ A = \frac{41000}{\frac{(1 + 0.085)^{19} - 1}{0.085}} \approx 938.96 \]

The total amount deposited over 19 years is:

\[ \text{Total Deposited} = A \times n = 938.96 \times 19 \approx 17840.24 \]

The total interest earned is then calculated as:

\[ \text{Total Interest} = FV - \text{Total Deposited} = 41000 - 17840.24 \approx 23159.76 \]

Using the future value of an annuity formula for 12 years:

\[ FV_{12} = A \times \frac{(1 + r)^{12} - 1}{r} \]

Substituting the values, we find:

\[ FV_{12} \approx 18355.96 \]

To find the interest earned during the 12th year, we first calculate the fund's value after 11 years:

\[ FV_{11} = A \times \frac{(1 + r)^{11} - 1}{r} \approx 16052.54 \]

The interest earned during the 12th year is:

\[ \text{Interest}_{12} = FV_{12} - FV_{11} \approx 18355.96 - 16052.54 \approx 2303.42 \]

Final Answer