You want to be able to withdraw $$ \$ 30,000 $$ each year for 30 years. Your account earns $$ 10 \% $$ interest. a) How much do you need in your account at the beginning? b) How much total money will you pull out of the account? $$ \$ 900,000 $$

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UpStudy AI · Accepted Answer

We start with the present value formula for an annuity:

$$
PV = \frac{W \left( 1 - (1+r)^{-n} \right)}{r}
$$

where  
$$ W = 30000 $$ (annual withdrawal),  
$$ r = 0.1 $$ (annual interest rate), and  
$$ n = 30 $$ (number of years).

**Step 1. Plug in the values:**

$$
PV = \frac{30000 \left( 1 - (1.1)^{-30} \right)}{0.1}
$$

**Step 2. Compute $$(1.1)^{-30}$$:**

$$
(1.1)^{-30} = \frac{1}{(1.1)^{30}}
$$

Using the logarithmic approximation:

$$
(1.1)^{30} \approx e^{30 \ln(1.1)} \approx e^{30 \times 0.09531} \approx e^{2.8593} \approx 17.45
$$

Thus,

$$
(1.1)^{-30} \approx \frac{1}{17.45} \approx 0.0573
$$

**Step 3. Evaluate the term in the parentheses:**

$$
1 - (1.1)^{-30} \approx 1 - 0.0573 = 0.9427
$$

**Step 4. Calculate the present value:**

$$
PV \approx \frac{30000 \times 0.9427}{0.1} = 30000 \times 9.427 \approx 282810
$$

Rounding to the nearest dollar, you need approximately $$\$282,810$$ in your account at the beginning.

**Step 5. Compute the total amount withdrawn over 30 years:**

$$
\text{Total Withdrawn} = 30000 \times 30 = 900000
$$

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**Answers:**

a) Approximately $$\$282,810$$ required at the beginning.  
b) A total of $$\$900,000$$ will be withdrawn over the 30 years.
a) You need approximately \$282,810 in your account at the beginning. b) You will withdraw a total of \$900,000 over 30 years.

You want to be able to withdraw \( \$ 30,000 \) each year for 30 years. Your account earns \( 10 \% \) interest. a) How much do you need in your account at the beginning? b) How much total money will you pull out of the account? \( \$ 900,000 \)

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