Science:Math Exam Resources/Courses/MATH110/April 2019/Question 01 (h)

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MATH110 April 2019

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Question 01 (h)
Suppose the number of fish in a pond $t$ months from now is $40e^{0.2t}$. At what rate (in fish per month) is the population of fish growing at $t=2$ months?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
How can you translate "rate of the population" into mathematical symbols? In class, you may have discussed how the derivative of a function gives the rate of change at a given time. See the next hint for more detail.

Hint 2
We are given the function $P(t)=40e^{0.2t}$, which describes the number of fish in a pond at time $t$. The rate of change of the number of fish in a pond at time $t$ is given by $P^{'}(t)$. Calculate the derivative of $f$!

Hint 3
Use the chain rule.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution
Let the population function be $P(t)=40e^{0.2t}$. The derivative is $P'(t)=0.2\cdot 40e^{0.2t}$ fish/month $=8e^{0.2t}$ fish/month. Therefore the rate that the population is growing in 2 months is $P'(2)=8e^{0.4}$ fish/month.

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