Final answer:
To calculate the amount of money Rick had in his retirement account, use the formula for compound interest. By solving for the interest rate r, it is found to be approximately 7.1%. Therefore, option d) $2,400,000 is the closest to the actual amount he had in his account.
Step-by-step explanation:
To calculate the amount of money Rick had in his retirement account, you can use the formula for compound interest. The formula is: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount (initial investment), r is the annual interest rate (expressed as a decimal), n is the number of times that interest is compounded per year, and t is the number of years.
- Plug in the given values into the formula:
A = 5000(1 + r/n)^(nt) - Calculate the exponent: nt = (65-23) = 42
- Solve the formula for A:
A = 5000(1 + r/n)^(nt)
2.2 × 10^6 = 5000(1 + r/n)^42 - Divide both sides of the equation by 5000:
(1 + r/n)^42 = (2.2 × 10^6) / 5000 - Raise both sides of the equation to the power of (1/42):
1 + r/n = [(2.2 × 10^6) / 5000]^(1/42) - Subtract 1 from both sides of the equation:
r/n = [(2.2 × 10^6) / 5000]^(1/42) - 1 - Multiply both sides of the equation by n:
r = n * [(2.2 × 10^6) / 5000]^(1/42) - 1 - Since it is not specified how often interest is compounded, we can assume it is compounded yearly, so n = 1:
r = 1 * [(2.2 × 10^6) / 5000]^(1/42) - 1 - Plug in the values and solve for r:
r = (2.2 × 10^6) / 5000)^(1/42) - 1 - Calculate the value of r:
r ≈ 0.071 - Convert the annual interest rate to a decimal:
0.071 * 100 = 7.1%
Therefore, the annual interest rate Rick earned on his retirement account is approximately 7.1%, which means option d) $2,400,000 is the closest to the actual amount he had in his account.