Following the birth of a child, a parent wants to make an initial investment P_{0} that will grow to 40,000 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 40,000 due 18 years from now.

Question

Following the birth of a child, a parent wants to make an initial investment P_{0} that will grow to  40,000 for the child&#x27;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  40,000 due 18 years from now.

Accepted Answer

Step:1 ：We are given that the future value of the investment (A) is 40,000, the annual interest rate (r) is 5% or 0.05 in decimal form, and the time period (t) is 18 years. We need to find the present value of the investment (P_{0}).Step:2 ：The formula for continuous compounding is given by: (A = P_{0}e^{rt})Step:3 ：We can rearrange the formula to solve for (P_{0}): (P_{0} = frac{A}{e^{rt}})Step:4 ：Substituting the given values into the formula, we get: (P_{0} = frac{40000}{e^{0.05*18}})Step:5 ：Calculating the above expression, we find that (P_{0} approx 16262.79)Step:6 ：Final Answer: The initial investment should be approximately (boxed{16262.79})

Answer

Step: 1 ：We are given that the future value of the investment (A) is 40,000, the annual interest rate (r) is 5% or 0.05 in decimal form, and the time period (t) is 18 years. We need to find the present value of the investment (P_{0}).Step: 2 ：The formula for continuous compounding is given by: (A = P_{0}e^{rt})Step: 3 ：We can rearrange the formula to solve for (P_{0}): (P_{0} = frac{A}{e^{rt}})Step: 4 ：Substituting the given values into the formula, we get: (P_{0} = frac{40000}{e^{0.05*18}})Step: 5 ：Calculating the above expression, we find that (P_{0} approx 16262.79)Step: 6 ：Final Answer: The initial investment should be approximately (boxed{16262.79})

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