Science:Math Exam Resources/Courses/MATH104/December 2012/Question 01 (j)
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Question 01 (j) 

You invest $100 000 now at an annual interest rate of 7%, compounded continuously. Your plan is to retire once the rate of growth of your investment is $10 000 per year. In how many years will you retire? 
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 formula for continuously compounded interest . What can we do with this equation to determine an expression for the rate of change of A, the value of the investment, in time? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall the formula for continuously compounded interest, where are the future value, principal and growth rate respectively and is the time since the investment was made. The rate of growth of the investment is simply the time derivative of the future value (i.e. ). Evaluating the time derivative of gives: By the information given in the question, we want to solve for the time when the dollars/year, given dollars and /year. Therefore, we would need to wait (100/7)ln(10/7) years for our initial investment to be growing at a rate of $10,000 per year so we can retire. 