Mrs. Varner deposited q dollars in a bank account that has been earning annual interest. The total value of the account is based on the function f(x) = q • 1.025x, where x represents the number of years the money has been in the account. If no deposits or withdrawals are made after the initial deposit, which equation represents the total value of the account 5 years from now?
The correct Answer and Explanation is:
The total value of Mrs. Varner’s bank account after 5 years is represented by the function ( f(x) = q \cdot 1.025^x ), where:
- ( f(x) ) represents the total value of the account at year ( x ),
- ( q ) is the initial amount of money deposited (the principal),
- 1.025 represents the growth factor due to the annual interest rate of 2.5% (expressed as a decimal),
- ( x ) is the number of years the money has been earning interest.
Since you are looking for the total value of the account after 5 years, substitute ( x = 5 ) into the equation:
[
f(5) = q \cdot 1.025^5
]
This expression gives the total value of the account 5 years from now. The equation that represents the total value of the account 5 years from now is:
[
f(5) = q \cdot 1.1314
]
Here, ( 1.025^5 ) is approximately 1.1314. This value represents how much the initial deposit has grown after 5 years, based on the 2.5% annual interest compounded annually.
Explanation
The function ( f(x) = q \cdot 1.025^x ) models exponential growth, where money grows at a constant percentage each year. The base of the exponent, 1.025, is derived from the annual interest rate of 2.5%, expressed as a decimal (2.5% = 0.025). Adding 1 to this gives the growth factor of 1.025, which represents the total value increase each year, including both the original amount and the interest earned.
In exponential growth, each year, the account’s value increases by a percentage of the previous year’s total. Over 5 years, this cumulative effect causes the initial deposit to grow by a factor of approximately 1.1314, meaning that Mrs. Varner’s deposit would grow by about 13.14% over the 5-year period.
Thus, the equation ( f(5) = q \cdot 1.1314 ) accurately reflects the total value of the account after 5 years.