Suppose you want to have $$ \$ 400,000 $$ for retirement in 35 years. Your account earns $$ 10 \% $$ interest. Round answers to the nearest cent. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

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Let the desired future amount be 
$$
FV = \$400,\!000.
$$
The account earns an annual interest rate of $$10\%$$. Since deposits are made monthly, the monthly interest rate is

$$
i = \frac{0.10}{12} \approx 0.00833333.
$$

There are

$$
n = 35 \times 12 = 420 \text{ months}.
$$

The future value of a series of monthly deposits (an annuity) is given by

$$
FV = PMT \cdot \frac{(1+i)^n - 1}{i},
$$

where $$PMT$$ is the monthly payment.

Solve for $$PMT$$:

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

Substitute the known values:

$$
PMT = \frac{400,\!000 \cdot \frac{0.10}{12}}{(1+\frac{0.10}{12})^{420} - 1}.
$$

Thus,

$$
PMT = \frac{400,\!000 \cdot 0.00833333}{(1.00833333)^{420} - 1}.
$$

Step 1. Compute $$(1.00833333)^{420}$$:

Using the properties of logarithms,

$$
(1.00833333)^{420} = \exp\Big(420 \cdot \ln(1.00833333)\Big).
$$

A good approximation is

$$
\ln(1.00833333) \approx 0.00829876,
$$

so

$$
420 \cdot 0.00829876 \approx 3.48550,
$$

and

$$
(1.00833333)^{420} \approx e^{3.48550} \approx 32.67.
$$

Step 2. Compute the denominator:

$$
(1.00833333)^{420} - 1 \approx 32.67 - 1 = 31.67.
$$

Step 3. Now compute $$PMT$$:

$$
PMT \approx \frac{400,\!000 \cdot 0.00833333}{31.67} \approx \frac{3333.33}{31.67} \approx 105.26.
$$

Thus, the monthly deposit is approximately

$$
\boxed{\$105.26}.
$$

For part (b), we need to determine the total interest earned.

Step 4. Total amount contributed by the deposits is

$$
\text{Total Contributions} = PMT \times n \approx 105.26 \times 420 \approx 44,\!209.20.
$$

Step 5. The interest earned is the difference between the final account balance and the total contributions:

$$
\text{Interest Earned} = FV - \text{Total Contributions} \approx 400,\!000 - 44,\!209.20 \approx 355,\!790.80.
$$

Thus, the interest earned is approximately

$$
\boxed{\$355,\!790.80}.
$$
a) You need to deposit approximately \$105.26 each month. b) You will earn about \$355,790.80 in interest.

Suppose you want to have \( \$ 400,000 \) for retirement in 35 years. Your account earns \( 10 \% \) interest. Round answers to the nearest cent. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

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