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 (t) =1,550 (1.02)^t
Which statement best interprets one value of the function?
A (t) =1,550 (1.02)^t
Which statement best interprets one value of the function?
Answers
GPT-4o mini
To interpret the function \( A(t) = 1,550 (1.02)^t \), let's break it down:
- **1,550**: This is the initial amount of money deposited into the investment account. It represents the principal amount.
- **1.02**: This is the growth factor, which means that the account earns an interest rate of 2% annually (as the interest is compounded).
- **\( t \)**: This represents the number of years the money is invested.
When we look at a specific value of \( t \), for example, \( t = 0 \), we can calculate \( A(0) \):
\[
A(0) = 1,550 (1.02)^0 = 1,550 \cdot 1 = 1,550
\]
This indicates that at the start (0 years), the amount in the account is $1,550, which matches the initial deposit.
Now, if we choose \( t = 1 \):
\[
A(1) = 1,550 (1.02)^1 = 1,550 \cdot 1.02 = 1,581
\]
This means that after one year, the investment grows to $1,581 due to earned interest from the initial deposit.
**Interpretation**: One value of the function can be interpreted as the amount of money in the investment account after a specific number of years (\( t \)). For instance, after 1 year, the account balance will be $1,581, indicating how much the initial investment of $1,550 will grow with a 2% annual interest rate.
In general terms, for any value of \( t \), \( A(t) \) represents the total amount in the investment account after \( t \) years, illustrating the effect of compound interest on the initial deposit.
- **1,550**: This is the initial amount of money deposited into the investment account. It represents the principal amount.
- **1.02**: This is the growth factor, which means that the account earns an interest rate of 2% annually (as the interest is compounded).
- **\( t \)**: This represents the number of years the money is invested.
When we look at a specific value of \( t \), for example, \( t = 0 \), we can calculate \( A(0) \):
\[
A(0) = 1,550 (1.02)^0 = 1,550 \cdot 1 = 1,550
\]
This indicates that at the start (0 years), the amount in the account is $1,550, which matches the initial deposit.
Now, if we choose \( t = 1 \):
\[
A(1) = 1,550 (1.02)^1 = 1,550 \cdot 1.02 = 1,581
\]
This means that after one year, the investment grows to $1,581 due to earned interest from the initial deposit.
**Interpretation**: One value of the function can be interpreted as the amount of money in the investment account after a specific number of years (\( t \)). For instance, after 1 year, the account balance will be $1,581, indicating how much the initial investment of $1,550 will grow with a 2% annual interest rate.
In general terms, for any value of \( t \), \( A(t) \) represents the total amount in the investment account after \( t \) years, illustrating the effect of compound interest on the initial deposit.