Question
An investor deposited money into an investment account that earns interest compounded annually. The function shown models the amount of money in the account in dollars after `t` years.
`A\left(t\right)=1,550\left(1.02\right)^{t}`
Which statement best interprets one value of the function?
`A\left(t\right)=1,550\left(1.02\right)^{t}`
Which statement best interprets one value of the function?
Answers
GPT-4o mini
The function \( A(t) = 1550(1.02)^t \) describes the amount of money in an investment account after \( t \) years, where:
- \( 1550 \) is the initial investment amount (the principal).
- \( 1.02 \) is the growth factor, indicating that the investment earns an interest rate of 2% compounded annually.
To interpret one value of the function, we can consider specific values of \( t \):
For example:
- If \( t = 0 \):
- \( A(0) = 1550(1.02)^0 = 1550(1) = 1550 \)
- This means that at the start (0 years), the account has $1550, which reflects the initial deposit.
- If \( t = 1 \):
- \( A(1) = 1550(1.02)^1 = 1550(1.02) = 1581 \)
- This means that after 1 year, the account has grown to $1581 due to the interest accrued.
- If \( t = 5 \):
- \( A(5) = 1550(1.02)^5 \approx 1550(1.10408) \approx 1711.16 \)
- This means that after 5 years, the amount in the account is approximately $1711.16.
Each of these values reflects the total amount of money in the account after a certain number of years, taking into account the initial investment and the compounded interest over that time period.
### Summary Statement:
One interpretation of a value of the function, such as \( A(1) \), is that it represents the total amount of money in the account after 1 year, which in this case would be $1581, indicating the initial investment of $1550 has grown due to compounded interest.
- \( 1550 \) is the initial investment amount (the principal).
- \( 1.02 \) is the growth factor, indicating that the investment earns an interest rate of 2% compounded annually.
To interpret one value of the function, we can consider specific values of \( t \):
For example:
- If \( t = 0 \):
- \( A(0) = 1550(1.02)^0 = 1550(1) = 1550 \)
- This means that at the start (0 years), the account has $1550, which reflects the initial deposit.
- If \( t = 1 \):
- \( A(1) = 1550(1.02)^1 = 1550(1.02) = 1581 \)
- This means that after 1 year, the account has grown to $1581 due to the interest accrued.
- If \( t = 5 \):
- \( A(5) = 1550(1.02)^5 \approx 1550(1.10408) \approx 1711.16 \)
- This means that after 5 years, the amount in the account is approximately $1711.16.
Each of these values reflects the total amount of money in the account after a certain number of years, taking into account the initial investment and the compounded interest over that time period.
### Summary Statement:
One interpretation of a value of the function, such as \( A(1) \), is that it represents the total amount of money in the account after 1 year, which in this case would be $1581, indicating the initial investment of $1550 has grown due to compounded interest.