Problem

Suppose Christian stops for coffee a given number of times a week. Use the information in the following table to calculate how much Christian would save if he deposite. the amount he would spend on coffee into an ordinary annuity instead. Assume there are four weeks in a month.
\begin{tabular}{|c|c|c|c|}
\hline \begin{tabular}{c}
Number of Coflees \\
Per Week
\end{tabular} & \begin{tabular}{c}
Price of \\
One Cup
\end{tabular} & \begin{tabular}{c}
Interest Rate \\
for Annulty
\end{tabular} & \begin{tabular}{c}
Number of \\
Years
\end{tabular} \\
\hline 8 & $\$ 4.30$ & $3.4 \%$ & 17 \\
\hline
\end{tabular}

Christian would have $\$ \square$ in the annuity.
(Round to the nearest cent as needed.)

Answer

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Answer

Round to the nearest cent to get the final answer: Christian would have \(\boxed{\$36409.13}\) in the annuity.

Steps

Step 1 :Define the variables: number of coffees per week is 8, price per coffee is $4.30, annual interest rate is 3.4%, and number of years is 17.

Step 2 :Calculate the amount deposited each period, \(P\), by multiplying the number of coffees per week by the price per coffee. So, \(P = 8 \times 4.30 = \$34.40\).

Step 3 :Calculate the number of periods, \(t\), by multiplying the number of years by 4 (since there are 4 weeks in a month). So, \(t = 17 \times 4 = 68\).

Step 4 :Calculate the future value of the annuity, \(FV\), using the formula \(FV = P \times \left( \frac{(1 + \frac{annual\_interest\_rate}{4})^{4t} - 1}{\frac{annual\_interest\_rate}{4}} \right)\).

Step 5 :Substitute the values into the formula to get \(FV = 34.40 \times \left( \frac{(1 + \frac{0.034}{4})^{4 \times 68} - 1}{\frac{0.034}{4}} \right)\).

Step 6 :Calculate the future value to get \(FV = \$36409.13\).

Step 7 :Round to the nearest cent to get the final answer: Christian would have \(\boxed{\$36409.13}\) in the annuity.

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