Final answer:
Account A with quarterly compounding gives a future value of approximately $4,961.87, whereas Account B with continuous compounding yields about $4,905.59. Thus, Account A provides a higher return, earning Amy approximately $56.28 more at the end of 15 years.
Step-by-step explanation:
To determine which account will yield a greater balance at the end of 15 years for Amy's $3500 investment, we need to calculate the future value of the investment in both Account A (compounded quarterly) and Account B (continuous compounding) using the given annual interest rate of 2.25%.
Account A - Quarterly Compounding
Account A uses quarterly compounding, which means the interest is calculated and added to the account four times a year. The future value (FV) can be calculated using the formula:
FV = P × (1 + r/n)nt
where:
- P is the principal amount ($3500)
- r is the annual interest rate (0.0225)
- n is the number of times interest is compounded per year (4)
- t is the number of years (15)
Plugging in the values:
FV = $3500 × (1 + 0.0225/4)4×15
Calculating the power and multiplication yields:
FV = $3500 × (1.005625)60 ≈ $3500 × 1.417676 ≈ $4961.867
- Account B - Continuous Compounding
Account B uses continuous compounding, and the future value is calculated with the formula:
FV = P × ert
where:
- e is the base of the natural logarithm (approximately 2.71828)
- P, r, and t are as previously defined
Again, plugging in the values:
FV = $3500 × e0.0225×15
Calculating the exponent and multiplication gives:
FV = $3500 × e0.3375 ≈ $3500 × 1.401597 ≈ $4905.591
Account A offers a higher future value than Account B after 15 years. To calculate how much more money Amy earns by choosing Account A: Difference = $4961.867 - $4905.591 ≈ $56.276
Amy earns approximately $56.28 more by choosing Account A over Account B after 15 years.