7
advanced-math

Continuous compounding at rate rr for tt years is modeled by A=PertA = Pe^{rt}. What annual rate rr causes an investment to triple in 20 years?

A

About 5.5%

B

About 10%

C

About 15%

D

About 3%

Correct Answer: A

Choice A is the correct answer. Solve for rr when A=3PA = 3P.

  1. Equation: 3P=Pe20r3P = Pe^{20r}.
  2. Simplify: 3=e20r3 = e^{20r}.
  3. Take natural log: ln(3)=20r\ln(3) = 20r.
  4. Solve: r=ln(3)20=1.0986200.055=5.5%r = \frac{\ln(3)}{20} = \frac{1.0986}{20} \approx 0.055 = 5.5\%.

💡 Strategic Tip: Use logarithms to solve for the exponent in exponential equations.

Choice B is incorrect because this rate would more than triple the investment. Choice C is incorrect because 15% is too high. Choice D is incorrect because 3% is too low to triple in 20 years.