Ster the birth of the first ofild, the Bartons plem to set up an acoount to pav for her callege education. The goal is to save $$ \$ 70,000 $$ over the next 11 a.id their financial planner suggests a bond fund that historically pays $$ 6.5 \% $$ interest compounded monthly. How much should they put into the fing now? Round your ansvier to the mearest cent. The Bartons should put \$ $$ \square $$ inta the fund now

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We need to find the amount that must be invested today (the present value) so that in 11 years the account will grow to \$70,000. The annual nominal interest rate is $$ 6.5\% $$ compounded monthly.

The formula relating present value $$ P $$ and future value $$ F $$ for compound interest is

$$
F = P\left(1+\frac{r}{m}\right)^{mt},
$$

where
- $$ r = 0.065 $$ is the annual interest rate,
- $$ m = 12 $$ is the number of compounding periods per year,
- $$ t = 11 $$ is the number of years.

We solve for $$ P $$:

$$
P = \frac{F}{\left(1+\frac{r}{m}\right)^{mt}}.
$$

Substitute in the given values:

$$
P = \frac{70000}{\left(1+\frac{0.065}{12}\right)^{12 \times 11}}.
$$

Simplify the expression inside the parentheses:

$$
1+\frac{0.065}{12} = 1+0.00541667 \approx 1.00541667.
$$

The total number of compounding periods is:

$$
12 \times 11 = 132.
$$

Thus, the expression becomes:

$$
P = \frac{70000}{\left(1.00541667\right)^{132}}.
$$

Now, compute the exponent:

$$
\left(1.00541667\right)^{132} \approx 2.040.
$$

Finally, calculate $$ P $$:

$$
P \approx \frac{70000}{2.040} \approx 34313.73.
$$

Hence, the Bartons should put approximately \$$$ 34313.73 $$ into the fund now.
The Bartons should put \$34,313.73 into the fund now.

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