A bank account pays interest at an annual rate of 4%. If the initial deposit on the account is $1,250 and no other deposits or withdrawals are made to the account, which of the following functions A models the account of money in the bank account after y years?

Options

A. `A(y)=0.04(1,250)^(y)`
B. `A(y)=1.04(1,250)^(y)`
C. `A(y)=1,250(0.04)^(y)`
D. `A(y)=1,250(1.04)^(y)`

Answer

To model the amount of money in a bank account after y years with an initial deposit of $1,250 and an annual interest rate of 4%, we can use the formula for compound interest. Here’s how to derive the function step by step:

### Step 1: Identify the formula for compound interest
The general formula for compound interest is given by:
\[ A(y) = A_0 \times (1 + r/n)^{nt} \]
where:
- \( A(y) \) is the amount of money in the account after \( y \) years,
- \( A_0 \) is the initial deposit,
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of times interest is compounded per year,
- \( t \) is the number of years.

### Step 2: Assign values to the variables
In this case:
- The initial deposit \( A_0 = 1250 \),
- The annual interest rate \( r = 4\% = 0.04 \),
- Since the problem states that no other deposits or withdrawals are made, we assume the interest is compounded annually, so \( n = 1 \),
- The time period is \( t = y \) years.

### Step 3: Substitute the values into the formula
Since the interest is compounded annually, we substitute \( n = 1 \) into the formula:
\[ A(y) = 1250 \times (1 + 0.04/1)^{1 \cdot y} \]

### Step 4: Simplify the expression
Now simplify the expression:
\[ A(y) = 1250 \times (1 + 0.04)^{y} \]
\[ A(y) = 1250 \times (1.04)^{y} \]

### Final Function
Thus, the function that models the amount of money in the bank account after \( y \) years is:
\[ A(y) = 1250 \times (1.04)^{y} \]