Science:Math Exam Resources/Courses/MATH104/December 2016/Question 01 (b)

MATH104 December 2016

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Question 01 (b)
How long would it take an investment of $100,000 with an interest rate of 6% compounded continuously to gain $8000?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall the continuously compounded interest formula: $A=Pe^{rt},$ where $A$ is the future value after $t$ years $P$ is the principal $r$ is the annual interest rate

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Following the hint, when we gain $8000, the total amount $A$ is $108,000. Substituting this with the annual interest rate converted into decimal form, $r=0.06$ , into the continuously compounded interest formula: $A=Pe^{rt},$ we get $108000=100000e^{0.06t}.$ Then we divide by $100000$ on both sides and then take $\ln$ of both sides, to find $t$ . $1.08=e^{0.06t}\implies \ln 1.08=0.06t\implies t={\frac {\ln 1.08}{0.06}}$ Answer: $\color {blue}{\frac {\ln 1.08}{0.06}}$