$8$ . The three precalculus teachers want $\$ 3,000,000$ at the end of $10$ years. How much money should they put into an account each month that pays $7 \%$ annual interest compounded monthly?

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8

. The three precalculus teachers want

\$ 3,000,000

at the end of

10

years. How much money should they put into an account each month that pays

7 \%

annual interest compounded monthly?

Identify Variables: First, let's identify the variables: $\newline$ Future Value (FV) = $\$3,000,000$ $\newline$ Annual Interest Rate (r) = $7\%$ or $0.07$ $\newline$ Number of Years (t) = $10$ $\newline$ Compounds per Year (n) = $12$ (monthly) $\newline$ We need to find the Present Value (PV), which is the amount they should deposit monthly.
Convert Interest Rate: Convert the annual interest rate to a monthly interest rate by dividing by the number of compounds per year: $\newline$ Monthly Interest Rate $i = \frac{r}{n} = \frac{0.07}{12}$
Calculate Monthly Interest Rate: Calculate the monthly interest rate: $i = \frac{0.07}{12} = 0.0058333\ldots$
Use Annuity Formula: Now, we'll use the formula for the present value of an annuity to find the monthly deposit (PMT): $PMT = \frac{FV}{((1 + i)^{(n*t)} - 1) / i}$
Plug in Values: Plug in the values to the formula: $\newline$ PMT = \frac{$\newline$\$\$3,000,000\)}{$\newline$\left(\left($1$ + $0$.$0058333$\right)^{$12$\times $10$} - $1$\right) / $0$.$0058333$\)}
Calculate Exponent: Calculate the exponent part of the formula: \(\newline $(1 + 0.0058333)^{(12\times10)} = (1 + 0.0058333)^{120}$
Calculate Value: Calculate the value of $(1 + 0.0058333)^{120}$ : $\newline$ This step involves using a calculator to find the precise value.
Substitute Value: After calculating, we get: $(1 + 0.0058333)^{120} \approx 2.0398873$
Calculate Denominator: Now, substitute this value back into the PMT formula: $\newline$ $PMT = \frac{\$3,000,000}{(2.0398873 - 1) / 0.0058333}$
Calculate Monthly Deposit: Calculate the denominator of the PMT formula: $\newline$ egin{equation} $\newline$ \frac{( $2$ . $0398873$ - $1$ )}{ $0$ . $0058333$ } \approx $178$ . $412$ $\newline$ \end{equation}
Calculate Monthly Deposit: Calculate the denominator of the PMT formula: $\newline$ $(2.0398873 - 1) / 0.0058333 \approx 178.412$ Finally, calculate the monthly deposit (PMT): $\newline$ $PMT = \$3,000,000 / 178.412$
Calculate Monthly Deposit: Calculate the denominator of the PMT formula: $\newline$ $(2.0398873 - 1) / 0.0058333 \approx 178.412$ Finally, calculate the monthly deposit (PMT): $\newline$ $PMT = \$3,000,000 / 178.412$ After the calculation, we get: $\newline$ $PMT \approx \$16,812.57$