Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Determine the amount of money required to set up a charitable endowment that pays the amount \(P\) each year indefinitely for the annual interest rate \(r\) compounded continuously. $$ P=\$ 12,000, r=6 \% $$

Step-by-Step Solution

Answer

To calculate the specific value of \(A\), use a scientific calculator or software. When the calculation is done, it will provide the exact amount of money needed to set the charitable endowment that pays \$12,000 each year indefinitely at an annual interest rate of 6% compounded continuously.

Step 1: Convert the Interest Rate to Decimal Form

First, let's start by converting the interest rate from percent to decimal form. This can be done by dividing the interest rate by 100. Thus, 6% can be converted to decimal form as \( r = 0.06\).

Step 2: Substitute Values into the Equation

Now, let's substitute the given values into the equation \( A = Pe^{rt}\). Recalling that \(P\) is the amount to pay each year (\$12,000), \(r\) is the interest rate (0.06), and \(t\) is the time (1 year), we substitute these values into the equation to find the value of \(A\). The equation to solve for \(A\) then becomes \( A = \$12,000 \cdot e^{0.06 \times 1}\).

Step 3: Solve the equation for A

Now that we have the equation with the values substituted in, the next step is to calculate the value for \(A\). After performing the calculation, one gets the value of \(A\), which represents the amount of money required to set up the endowment.