Following the birth of a child, a parent wants to make an initial investment $P_{0}$ that will grow to $\$ 50,000$ for the child's education at age 18. Interest is compounded continuously at $5 \%$. What should the initial investment be? Such an amount is called the present value of $\$ 50,000$ due 18 years from now.
Solution
Step 1 ：A parent wants to make an initial investment $P_{0}$ that will grow to $50,000 for their child's education at age 18. The interest is compounded continuously at a rate of 5%.
Step 2 ：The formula for continuous compounding is given by: \[A = P_{0}e^{rt}\] where: \(A\) is the amount of money accumulated after n years, including interest, \(P_{0}\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (in decimal), and \(t\) is the time the money is invested for, in years.
Step 3 ：In this case, we know that \(A = \$ 50,000\), \(r = 5 \% = 0.05\), and \(t = 18\) years. We need to find \(P_{0}\).
Step 4 ：We can rearrange the formula to solve for \(P_{0}\): \[P_{0} = \frac{A}{e^{rt}}\]
Step 5 ：Substituting the given values into the formula, we get: \[P_{0} = \frac{50000}{e^{0.05*18}}\]
Step 6 ：Solving the above expression, we find that the initial investment should be approximately \(\boxed{20328.48}\).
