Question

following the birth of a child, a parent wants to make an initial investment $p_{0}$ that will grow to $60,000 for the childs education at age 19. interest is compounded continuously at 5%. what should the initial investment be? such an amount is called the present value of $60,000 due 19 years from now. the present value is about $\square$. (do not round until the final answer. then round to two decimal places as needed.)

Explanation:

Step1: Write the formula for continuous compounding

The formula for continuous - compounding is \(A = P_0e^{rt}\), where \(A\) is the final amount, \(P_0\) is the initial investment, \(r\) is the annual interest rate (in decimal form), and \(t\) is the time in years. We need to solve for \(P_0\), so we can rewrite the formula as \(P_0=\frac{A}{e^{rt}}\).

Step2: Identify the values of \(A\), \(r\), and \(t\)

Given that \(A = 60000\), \(r=0.05\) (since \(5\%=0.05\)), and \(t = 19\).

Step3: Substitute the values into the formula

Substitute \(A = 60000\), \(r = 0.05\), and \(t=19\) into \(P_0=\frac{A}{e^{rt}}\). Then \(P_0=\frac{60000}{e^{0.05\times19}}\).
First, calculate the exponent: \(0.05\times19 = 0.95\). So \(P_0=\frac{60000}{e^{0.95}}\).
We know that \(e^{0.95}\approx2.5857\). Then \(P_0=\frac{60000}{2.5857}\).

Step4: Calculate the value of \(P_0\)

\(P_0=\frac{60000}{2.5857}\approx23199.99\).

Answer:

\(23200.00\)