Science:Math Exam Resources/Courses/MATH101/April 2007/Question 06 (a)
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 (a) • Q4 (b) • Q5 • Q6 (a) • Q6 (b) •
Question 06 (a) 

FullSolution Problems. Justify your answers and show all your work. Simplification of answers is not required. The population of fish in a lake is m million, where varies with time t measured in years. The number of fish is currently 2 million. Suppose m satisfies the logisticgrowth differential equation
When will the number of fish equal 3 million? You may use the fact that the general solution to the logisticgrowth differential equation is where A is constant. 
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 

First solve for the constant A using the initial condition 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. With and , we know from the problem that
Solving for A, we use the fact that and see that
Cross multiplying, we see that
and solving gives . Thus, we have . We want to solve for t when . Plugging this in yields
Cross multiplying, we see that
Bringing the 3 over and dividing by 3 yields
Taking logarithms and then dividing by 16 gives
and this completes the problem. 