2
advanced-math

An investment of $15,000 is compounded continuously at 6.5% annual rate. How long until it reaches $30,000?

A

About 10.7 years

B

About 15 years

C

About 8 years

D

About 12 years

Correct Answer: A

Choice A is the correct answer. Solve for tt using continuous compounding.

  1. Formula: 30000=15000e0.065t30000 = 15000e^{0.065t}.
  2. Divide: 2=e0.065t2 = e^{0.065t}.
  3. Natural log: ln(2)=0.065t\ln(2) = 0.065t.
  4. Solve: t=ln(2)0.065=0.6930.06510.6610.7t = \frac{\ln(2)}{0.065} = \frac{0.693}{0.065} \approx 10.66 \approx 10.7 years.

💡 Strategic Tip: Doubling time for continuous growth: t=ln(2)rt = \frac{\ln(2)}{r}.

Choice B is incorrect because this is too long. Choice C is incorrect because this is too short. Choice D is incorrect because this doesn't match the calculation.