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 $7000 deposit in an account with an APR of 4.5% The balance in the account after 1 year is approximately $ _ (Round to the nearest cent as needed.) The balance in the account after 5 years is approximately $_ (Round to the nearest cent as needed.) . The balance in the account after 20 years is approximately $_____ (Round to the nearest cent as needed.) The APY for the account is approximately. _% (Round to two decimal places as needed.)
The Correct Answer and Explanation is :
To solve this problem, we will use the formula for continuous compounding, which is given by:
[
A = Pe^{rt}
]
Where:
- ( A ) is the amount of money accumulated after time ( t ),
- ( P ) is the principal amount (the initial deposit),
- ( r ) is the annual interest rate (in decimal form),
- ( t ) is the time the money is invested or borrowed for, in years,
- ( e ) is the base of the natural logarithm (approximately equal to 2.71828).
Given:
- ( P = 7000 )
- ( r = 4.5\% = 0.045 )
Calculating the Balance
- After 1 Year (( t = 1 )):
[
A = 7000 \times e^{0.045 \times 1}
]
[
A \approx 7000 \times e^{0.045} \approx 7000 \times 1.04603 \approx 7322.21
]
The balance after 1 year is approximately $7322.21. - After 5 Years (( t = 5 )):
[
A = 7000 \times e^{0.045 \times 5}
]
[
A \approx 7000 \times e^{0.225} \approx 7000 \times 1.25232 \approx 8766.24
]
The balance after 5 years is approximately $8766.24. - After 20 Years (( t = 20 )):
[
A = 7000 \times e^{0.045 \times 20}
]
[
A \approx 7000 \times e^{0.9} \approx 7000 \times 2.45960 \approx 17217.22
]
The balance after 20 years is approximately $17217.22.
Calculating the APY
The Annual Percentage Yield (APY) for an account with continuous compounding is calculated as:
[
APY = e^r – 1
]
Substituting ( r = 0.045 ):
[
APY = e^{0.045} – 1 \approx 1.04603 – 1 \approx 0.04603
]
Converting to percentage:
[
APY \approx 0.04603 \times 100 \approx 4.60\%
]
Summary of Results
- The balance in the account after 1 year is approximately $7322.21.
- The balance in the account after 5 years is approximately $8766.24.
- The balance in the account after 20 years is approximately $17217.22.
- The APY for the account is approximately 4.60%.
Explanation
Using continuous compounding provides a way to calculate the growth of an investment with interest applied continuously, leading to potentially higher returns than standard compounding methods. The exponential function reflects how investments can grow significantly over time, particularly in a longer investment horizon. Understanding these concepts can help investors make informed decisions about where to place their money to achieve their financial goals.