Final answer:
Atalia Johnson wants to calculate the total interest earned on a $7,000 investment in a CD with a 2.75% interest rate, compounded daily, after it matures twice over a total of 8 years. We use the compound interest formula to calculate the amount accumulated at the end of each 4-year term, then find the interest amounts for each term and sum them up to get the total interest earned.
Step-by-step explanation:
Atalia Johnson invested $7,000 in a 4-year CD with an interest rate of 2.75%, compounded daily. To calculate the total interest earned by Atalia on her original investment when the CD matures a second time after another 4 years with the same terms, we need to apply the formula for compound interest:
A = P(1 + \frac{r}{n})^{nt}
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
In Atalia's case:
- P = 7000
- r = 0.0275
- n = 365 (daily compounding)
- t = 4 years (for each CD term)
After calculating the total amount A after 4 years, we subtract the principal (P) from A to find the interest earned in the first term. We then use the resulting amount as the new principal (P) for the next 4 years to find the total amount A at the end of the second term.
Calculating:
- Initial investment after first 4 years: A1 = 7000(1 + \frac{0.0275}{365})^{365 * 4}
- Interest earned in first term: I1 = A1 - 7000
- Total amount at the end of 8 years: A2 = A1(1 + \frac{0.0275}{365})^{365 * 4}
- Interest earned in second term: I2 = A2 - A1
- Total interest earned after second term: Total I = I1 + I2
Calculating these values will give us the total interest earned on the original investment after 8 years.