Amelia wishes to retire at age 67 with $$ \$ 1,300,000 $$ in her retirement account. When she turns 25 , she decides to begin depositing money into an account with an APR of $$ 8 \% $$ compounded monthly. What is the monthly deposit that Amelia must make in order to reach her goal? Round your answer to the nearest cent, if necessary. Formulas

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To solve this problem, we will use the future value of an annuity formula, which is used to calculate the future value of a series of equal payments made at regular intervals. The formula is given by:

$$
FV = P \times \frac{(1 + r)^n - 1}{r}
$$

Where:
- $$ FV $$ is the future value of the annuity (the amount Amelia wants to have at retirement, which is $$ \$ 1,300,000 $$).
- $$ P $$ is the monthly deposit (the unknown we want to find).
- $$ r $$ is the monthly interest rate (annual interest rate divided by 12).
- $$ n $$ is the total number of deposits (number of years until retirement multiplied by 12).

### Step 1: Identify the known values
- Future Value, $$ FV = 1,300,000 $$
- Annual Interest Rate, $$ APR = 8\% = 0.08 $$
- Monthly Interest Rate, $$ r = \frac{0.08}{12} $$
- Age when she starts depositing = 25
- Retirement age = 67
- Total years of deposits = $$ 67 - 25 = 42 $$
- Total number of deposits, $$ n = 42 \times 12 $$

### Step 2: Calculate $$ r $$ and $$ n $$
Now, let's calculate $$ r $$ and $$ n $$:

$$
r = \frac{0.08}{12}
$$
$$
n = 42 \times 12
$$

### Step 3: Substitute into the formula and solve for $$ P $$
Rearranging the future value formula to solve for $$ P $$:

$$
P = \frac{FV \times r}{(1 + r)^n - 1}
$$

Now, let's calculate $$ r $$, $$ n $$, and then $$ P $$.
Calculate the value by following steps:
- step0: Calculate:
  $$42\times 12$$
- step1: Multiply the numbers:
  $$504$$
Calculate or simplify the expression $$ 0.08/12 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.08}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{2}{25}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{2}{25}\times \frac{1}{12}$$
- step3: Reduce the numbers:
  $$\frac{1}{25}\times \frac{1}{6}$$
- step4: Multiply the fractions:
  $$\frac{1}{25\times 6}$$
- step5: Multiply:
  $$\frac{1}{150}$$
### Step 2: Calculate $$ r $$ and $$ n $$

1. **Monthly Interest Rate $$ r $$**:
   $$
   r = \frac{0.08}{12} \approx 0.00666667
   $$

2. **Total Number of Deposits $$ n $$**:
   $$
   n = 42 \times 12 = 504
   $$

### Step 3: Substitute into the formula and solve for $$ P $$

Now we can substitute $$ FV $$, $$ r $$, and $$ n $$ into the rearranged formula for $$ P $$:

$$
P = \frac{1,300,000 \times 0.00666667}{(1 + 0.00666667)^{504} - 1}
$$

Let's calculate $$ (1 + r)^n $$ and then find $$ P $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.00666667\right)^{504}$$
- step1: Add the numbers:
  $$1.00666667^{504}$$
- step2: Convert the expressions:
  $$\left(\frac{100666667}{100000000}\right)^{504}$$
- step3: Use the properties of exponents:
  $$\frac{100666667^{504}}{100000000^{504}}$$
Calculate or simplify the expression $$ 1300000 * 0.00666667 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1300000\times 0.00666667$$
- step1: Multiply the numbers:
  $$8666.671$$
Calculate or simplify the expression $$ 28.470008 - 1 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$28.470008-1$$
- step1: Subtract the numbers:
  $$27.470008$$
Calculate or simplify the expression $$ 8666.671 / 27.470008 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{8666.671}{27.470008}$$
- step1: Convert the expressions:
  $$\frac{\frac{8666671}{1000}}{\frac{3433751}{125000}}$$
- step2: Multiply by the reciprocal:
  $$\frac{8666671}{1000}\times \frac{125000}{3433751}$$
- step3: Reduce the numbers:
  $$8666671\times \frac{125}{3433751}$$
- step4: Multiply:
  $$\frac{8666671\times 125}{3433751}$$
- step5: Multiply:
  $$\frac{1083333875}{3433751}$$
### Step 3: Calculate $$ P $$

1. **Calculate $$ (1 + r)^n $$**:
   $$
   (1 + 0.00666667)^{504} \approx 28.470008
   $$

2. **Calculate $$ (1 + r)^n - 1 $$**:
   $$
   28.470008 - 1 \approx 27.470008
   $$

3. **Calculate $$ 1,300,000 \times r $$**:
   $$
   1,300,000 \times 0.00666667 \approx 8666.671
   $$

4. **Now substitute into the formula for $$ P $$**:
   $$
   P = \frac{8666.671}{27.470008} \approx 315.495758
   $$

### Final Result
Rounding to the nearest cent, Amelia must make a monthly deposit of approximately:

$$
\boxed{315.50}
$$

This is the amount she needs to deposit each month to reach her retirement goal of $$ \$ 1,300,000 $$ by age 67.
Amelia needs to deposit approximately \$315.50 each month to reach her retirement goal of \$1,300,000 by age 67.

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