Final answer:
The principal that must be invested at a 6% interest rate, compounded continuously for 3 years to yield $1500, is approximately $1250.85. This is found using the formula for continuous compounding, A = Pe^(rt), and solving for P.
Step-by-step explanation:
To determine the principal that must be invested at a 6% interest rate, compounded continuously for 3 years, to yield $1500, we use the formula for continuous compounding, which is A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), and t is the time in years. We know that A=$1500, r=0.06 (which is 6% as a decimal), and t=3 years.
Rearranging the formula to solve for P gives us P = A / ert. Substituting the values we have, P = 1500 / e(0.06 * 3). Calculating P gives us approximately P = $1250.85 when rounded to two decimal places. Note that compound interest has a more significant impact when it's applied continuously and over longer periods, as opposed to simple interest which is calculated only on the principal amount.