Exponential and Logarithmic Functions
Finding the initial amount in a word problem on continuous...
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How much should be invested now at an interest rate of $6.5 \%$ per year, compounded continuously, to have $\$ 2000$ in two years?

Do not round any intermediate computations, and round your answer to the nearest cent.
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Explanation
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Step 1 ：The problem is asking for the initial amount to be invested to reach a certain amount in the future, given a continuous interest rate. This is a problem of continuous compound interest, which can be solved using the formula for continuous compounding: \(A = P \cdot e^{rt}\) where: \(A\) is the amount of money accumulated after \(n\) years, including interest. \(P\) is the principal amount (the initial amount of money). \(r\) is the annual interest rate (in decimal). \(t\) is the time the money is invested for in years.

Step 2 ：In this case, we know \(A = \$2000\), \(r = 6.5\%\) or \(0.065\) in decimal form, and \(t = 2\) years. We need to solve for \(P\).

Step 3 ：We can rearrange the formula to solve for \(P\): \(P = A / e^{rt}\)

Step 4 ：We can then substitute the known values into the formula and solve for \(P\).

Step 5 ：Substituting the values we get \(P = 2000 / e^{(0.065 \cdot 2)}\)

Step 6 ：Solving the above expression we get \(P = 1756.1908618411228\)

Step 7 ：Rounding to the nearest cent, we get \(P = \$1756.19\)

Step 8 ：Final Answer: The initial amount that should be invested now at an interest rate of \(6.5 \%\) per year, compounded continuously, to have \$2000 in two years is approximately \(\boxed{\$1756.19}\)