Problem 45 - 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. $$ P=\$ 5000, r=7.5 \% $$

Step-by-Step Solution

Answer

To evaluate, calculate the value of the denominator \(e^{0.075 * 1}\) first, then divide \$5000 by this evaluated value. The result of this calculation will give the initial endowment amount necessary to yield a perpetuity of \$5000 per year with continuous compounding at an annual interest rate of 7.5%. Always remember to express the answer in currency terms, such as dollars.$.

Step 1: Express the interest rate as a decimal

The annually compounded rate, \(r\), is given as 7.5%. As a decimal, this can be expressed as 0.075 (= 7.5 / 100).

Step 2: Rearrange the equation for 'Principal'

Re-arrange the equation \(P = Principal * e^{rt}\) to solve for 'Principal'. This rearrangement gives: \[Principal = P / e^{rt}\].

Step 3: Substitute the known values into the equation

With \(P = \$5000\), \(r = 0.075\), and \(t = 1\), we can substitute these values in the equation from Step 2, which gives: \[Principal = 5000 / e^{(0.075 * 1)}\].

Step 4: Evaluate the equation

Using the accurate value of \(e\) (approximately 2.71828) and the values we substituted, the equation can now be evaluated to determine the 'Principal'.