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Use the formula for continuous compounding to compute the balance in the account after 1, 5, and 20 years. Also, find the APY for the account. A $2000 deposit in an account with an APR of 4.5%

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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.

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