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 17 . Interest is compounded continuously at $7 \%$. What should the initial investment be? Such an amount is called the present value of $\$ 50,000$ due 17 years from now.
The present value is about $\$$
(Do not round until the final answer. Then round to two decimal places as needed.)
Answer
Rounding to two decimal places, we get the final answer: \(\boxed{15211.06}\).
Step 1 ：We are given that the final amount (A) is $50,000, the annual interest rate (r) is 7% or 0.07 in decimal form, and the time (t) is 17 years. We need to find the initial investment (P).
Step 2 ：The formula for continuous compounding is \(A = P e^{rt}\). We can rearrange this formula to solve for P: \(P = \frac{A}{e^{rt}}\).
Step 3 ：Substituting the given values into the formula, we get \(P = \frac{50000}{e^{0.07 \times 17}}\).
Step 4 ：Calculating the above expression, we find that \(P \approx 15211.0632033352\).
Step 5 ：Rounding to two decimal places, we get the final answer: \(\boxed{15211.06}\).
