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You want to be able to withdraw $30,000 each year for 20 years. Your account earns 7% interest. How much do you need in your account at the beginning? Fill out the information given: {:[d=],[r=],[k=],[N=]:} Find P_(0). P_(0)=

You want to be able to withdraw $30,000 \$ 30,000 each year for 2020 years. \newlineYour account earns 7% 7 \% interest.\newlineHow much do you need in your account at the beginning?\newlineFill out the information given:\newlined=r=k=N=\begin{array}{l} d= \\ r= \\ k= \\ N= \end{array} \newlineFind P0 P_{0} .\newlineP0= P_{0}=

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Full solution

Q. You want to be able to withdraw $30,000 \$ 30,000 each year for 2020 years. \newlineYour account earns 7% 7 \% interest.\newlineHow much do you need in your account at the beginning?\newlineFill out the information given:\newlined=r=k=N=\begin{array}{l} d= \\ r= \\ k= \\ N= \end{array} \newlineFind P0 P_{0} .\newlineP0= P_{0}=
  1. Identify Given Information: Identify the given information for the annuity formula.\newlineThe problem states that you want to withdraw $30,000\$30,000 each year for 2020 years with an interest rate of 7%7\%. This information will be used to fill out the given:\newlinedd (the regular withdrawal) = $30,000\$30,000\newlinerr (the interest rate per period) = 7%7\% or 0.070.07\newlinekk (the number of withdrawals per year) = 11 (since it's each year)\newline202000 (the total number of withdrawals) = 2020 (since it's for 2020 years)
  2. Use Annuity Formula: Use the present value of an annuity formula to find P0P_0. The present value of an annuity formula is: P0=d×[1−(1+r)−Nr]P_0 = d \times \left[\frac{1 - (1 + r)^{-N}}{r}\right] We will substitute the given values into this formula to find P0P_0.
  3. Substitute Given Values: Substitute the given values into the formula.\newlineP0=$30,000×[1−(1+0.07)−200.07]P_0 = \$30,000 \times \left[\frac{1 - (1 + 0.07)^{-20}}{0.07}\right]
  4. Calculate Inside Brackets: Calculate the value inside the brackets.\newlineFirst, calculate (1+0.07)−20(1 + 0.07)^{-20}.\newline(1+0.07)−20≈0.2584(1 + 0.07)^{-20} \approx 0.2584 (using a calculator)\newlineNow, calculate 1−0.25841 - 0.2584.\newline1−0.2584=0.74161 - 0.2584 = 0.7416
  5. Complete Calculation: Complete the calculation for P0P_0. \newlineP0=($)30,000×(0.7416/0.07)P_0 = (\$)30,000 \times (0.7416 / 0.07)\newlineP0=($)30,000×10.5943P_0 = (\$)30,000 \times 10.5943 (rounded to four decimal places)\newlineP0≈($)317,829P_0 \approx (\$)317,829

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