The Gray family wants to save money to travel the world. They plan to invest in an ordinary annuity that earns $$ 2.4 \% $$ interest, compounded quarterly. Payments will be made at the end of each quarten. How much money do they need to pay into the annuity each quarter for the annulty to have a total value of $$ \$ 13,000 $$ after 13 years? Do not round intermediate computations, and round your final answer to the nearest cent. If necessary, refer to the list of financial formulas. Hill

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To solve the problem, we will use the formula for the future value of an ordinary annuity, which is given by:

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

Where:
- $$ FV $$ is the future value of the annuity.
- $$ P $$ is the payment made at the end of each period.
- $$ r $$ is the interest rate per period.
- $$ n $$ is the total number of payments.

### Step 1: Identify the known values
- Future Value ($$ FV $$) = $$ 13,000 $$
- Annual interest rate = $$ 2.4\% $$ or $$ 0.024 $$
- Compounding frequency = quarterly, so the interest rate per quarter ($$ r $$) is:

$$
r = \frac{0.024}{4} = 0.006
$$

- Total number of years = $$ 13 $$
- Total number of payments ($$ n $$) = $$ 13 \times 4 = 52 $$

### Step 2: Rearrange the formula to solve for $$ P $$
We need to isolate $$ P $$:

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

### Step 3: Substitute the known values into the formula
Now we can substitute the known values into the rearranged formula:

$$
P = \frac{13000 \times 0.006}{(1 + 0.006)^{52} - 1}
$$

### Step 4: Calculate the value of $$ P $$
Now, we will perform the calculations step by step. First, we will calculate $$ (1 + 0.006)^{52} $$ and then use it to find $$ P $$.

Let's calculate $$ (1 + 0.006)^{52} $$ and then find $$ P $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.006\right)^{52}$$
- step1: Add the numbers:
  $$1.006^{52}$$
- step2: Convert the expressions:
  $$\left(\frac{503}{500}\right)^{52}$$
- step3: Use the properties of exponents:
  $$\frac{503^{52}}{500^{52}}$$
Calculate or simplify the expression $$ 13000 * 0.006 / (1.364882 - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{13000\times 0.006}{\left(1.364882-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{13000\times 0.006}{1.364882-1}$$
- step2: Multiply the numbers:
  $$\frac{78}{1.364882-1}$$
- step3: Subtract the numbers:
  $$\frac{78}{0.364882}$$
- step4: Convert the expressions:
  $$\frac{78}{\frac{182441}{500000}}$$
- step5: Multiply by the reciprocal:
  $$78\times \frac{500000}{182441}$$
- step6: Multiply:
  $$\frac{78\times 500000}{182441}$$
- step7: Multiply:
  $$\frac{39000000}{182441}$$
### Step 5: Final Calculation
From our calculations, we found:

$$
(1 + 0.006)^{52} \approx 1.364882
$$

Now, substituting this value into the formula for $$ P $$:

$$
P \approx \frac{13000 \times 0.006}{1.364882 - 1}
$$

Calculating the denominator:

$$
1.364882 - 1 \approx 0.364882
$$

Now substituting this back into the equation for $$ P $$:

$$
P \approx \frac{78}{0.364882} \approx 213.767739
$$

### Step 6: Round the final answer
Rounding $$ P $$ to the nearest cent:

$$
P \approx 213.77
$$

### Conclusion
The Gray family needs to pay approximately **$213.77** into the annuity each quarter to have a total value of $13,000 after 13 years.
The Gray family needs to pay approximately $213.77 into the annuity each quarter.

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