Final answer:
The balance of an account with continuous compounding can be calculated for various time periods using the formula A = P × e¹¹(rt) and the APY is found using APY = (e¹¹(r) - 1) × 100%. For a $2000 deposit at an annual interest rate of 4.5%, the calculations will provide the future balance after 1, 5, and 20 years, as well as the APY of the account.
Step-by-step explanation:
To answer the student's question regarding the balance of an account with continuous compounding, we use the formula:
A = P × e¹¹(rt)
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- e is the base of the natural logarithm, approximately equal to 2.71828.
- r is the annual interest rate (decimal).
- t is the time the money is invested for, in years.
Given:
- P = $2000
- r = 4.5% = 0.045 (as a decimal)
Let's calculate the balance for 1, 5, and 20 years.
After 1 year (t=1):
A = 2000 × e¹¹(0.045×1) = 2000 × e¹¹(0.045)
After 5 years (t=5):
A = 2000 × e¹¹(0.045×5) = 2000 × e¹¹(0.225)
After 20 years (t=20):
A = 2000 × e¹¹(0.045×20) = 2000 × e¹¹(0.9)
Now, for the APY (Annual Percentage Yield), we can use the formula:
APY = (e¹¹(r) - 1) × 100%
APY = (e¹¹(0.045) - 1) × 100%
The above calculations yield the future balance for different time intervals and the APY for the account.