ACT (Enhanced)MathematicsHard
A financial planner is evaluating a client's investment portfolio. One investment follows a continuous compounding model. If an initial investment of $10,000 grows to $12,214 after 2 years at an annual interest rate, compounded continuously, which of the following expressions represents the annual interest rate (r)?
- Ar = (ln(1.2214)) / 2
- Br = e^(1.2214 / 2)
- Cr = (1.2214 - 1) / 2
- Dr = 2 * ln(1.2214)
Show answer & explanationAnswer & explanation
Correct answer: A. r = (ln(1.2214)) / 2
The formula for continuous compounding is A = Pe^(rt). Here, A = 12214, P = 10000, t = 2. So, 12214 = 10000e^(2r). Divide by 10000: 1.2214 = e^(2r). To solve for r, take the natural logarithm (ln) of both sides: ln(1.2214) = ln(e^(2r)) = 2r. Finally, r = ln(1.2214) / 2.
Why the other options are wrong
- B. This attempts to use 'e' incorrectly and does not isolate 'r'.
- C. This represents simple interest or a misapplication of the exponential growth formula.
- D. This incorrectly multiplies by 't' instead of dividing after taking the logarithm.
Continuous Compounding Formula
The formula A = Pe^(rt) calculates the future value (A) of an investment with principal (P), annual interest rate (r), and time (t) in years, compounded continuously.
- Uses the mathematical constant 'e' (Euler's number).
- Represents the theoretical upper limit of compounding frequency.
- Often used for modeling natural growth or decay processes.
Memory trick: Pert for continuous growth, like P-E-R-T.