$$ 10 .(7 \mathrm{pts}) $$ Joy wants to be able to take out $$ \$ 2600 $$ for a total of 25 years from her retirement account. If the account earns $$ 6 \% $$ compounded monthly. how much money would Joy need in her account?

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First, we note that Joy plans to withdraw a fixed amount $$ R = \$2600 $$ every month for 25 years. Since the interest rate is $$ 6\% $$ compounded monthly, the monthly interest rate is

$$
i = \frac{0.06}{12} = 0.005.
$$

The total number of monthly withdrawals is

$$
n = 25 \times 12 = 300.
$$

The present value $$ PV $$ of an annuity (which is the amount required in the account now to fund these withdrawals) is given by

$$
PV = R \cdot \frac{1 - (1+i)^{-n}}{i}.
$$

Substitute the known values:

$$
PV = 2600 \cdot \frac{1 - (1.005)^{-300}}{0.005}.
$$

Now, calculate step by step:

1. Compute $$(1.005)^{300}$$. Using the exponential approximation,

$$
(1.005)^{300} \approx e^{300 \ln(1.005)}.
$$

Since

$$
\ln(1.005) \approx 0.0049875,
$$

then

$$
300 \ln(1.005) \approx 1.49625,
$$

and so

$$
(1.005)^{300} \approx e^{1.49625} \approx 4.467.
$$

2. Compute $$(1.005)^{-300}$$:

$$
(1.005)^{-300} \approx \frac{1}{4.467} \approx 0.2239.
$$

3. Now, the numerator in the present value formula is

$$
1 - 0.2239 = 0.7761.
$$

4. Divide by $$ i = 0.005 $$:

$$
\frac{0.7761}{0.005} = 155.22.
$$

5. Finally, multiply by $$ R = 2600 $$:

$$
PV \approx 2600 \times 155.22 \approx 403,572.
$$

Thus, Joy would need approximately

$$
\boxed{\$403,572}
$$

in her retirement account.
Joy needs approximately \$403,572 in her retirement account.

Joy wants to be able to take out for a total of 25 years from her retirement account. If the account earns compounded monthly.
how much money would Joy need in her account?

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Joy wants to be able to take out for a total of 25 years from her retirement account. If the account earns compounded monthly. how much money would Joy need in her account?