Final answer:
To find the interest rate for continuous compounding, the formula A = Pert is used. By rearranging the formula to solve for the interest rate, and using the values given (A = $3600, P = $1200, t = 21 years), the interest rate is approximately 5.23%.
Step-by-step explanation:
The student asks to find the interest rate for a savings account with continuous compounding, where a $1200 deposit grows to $3600 over 21 years with no additional deposits or withdrawals. To solve this, we can use the formula for continuous compounding interest, which is:
A = Pert
Where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial sum of money), r is the annual interest rate (in decimal), t is the time the money is invested or borrowed for, in years, and e is the base of the natural logarithms, approximately equal to 2.71828.
Rearranging the formula to solve for the interest rate:
r = (1/t) * ln(A/P)
Substituting the given values with A = $3600, P = $1200, and t = 21 years:
r = (1/21) * ln(3600/1200)
r = (1/21) * ln(3)
r ≈ (1/21) * 1.09861228866811
r ≈ 0.052315823746
The interest rate is approximately 5.23%