Science:Math Exam Resources/Courses/MATH103/April 2013/Question 07 (e)

MATH103 April 2013

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Question 07 (e)

Biofuel: Suppose you work at an experimental algae farm where algae is turned into biofuel. The algae grows in tanks at a rate $\displaystyle \alpha =2$ and the steady supply of nutrients allows each tank to sustain $\displaystyle K$ kilograms of algae. In scenario (d). assuming you start with $\displaystyle M=K/4$ kilograms of algae, at what rate H do you have to harvest the algae such that the tank always contains $\displaystyle K/4$ kilograms of algae?

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
Once again, we are looking for a steady state. Given that we already solved for the values of M in the previous part, we can use that to solve this problem OR we can argue directly from the modified differential equation in part d.

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. 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 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Setting the derivative to 0 in $\displaystyle {\frac {dM}{dt}}=2M{\big (}1-{\frac {M}{K}}{\big )}-H$ with $\displaystyle M=K/4$ we have that $\displaystyle 0=2\cdot {\frac {K}{4}}\left(1-{\frac {\tfrac {K}{4}}{K}}\right)-H={\frac {K}{2}}\cdot {\frac {3}{4}}-H$ Isolating for H gives $\displaystyle H={\frac {3}{8}}K$

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Using the solution from part d. and plugging in $\displaystyle M=K/4$, we have ${\begin{aligned}{\frac {K}{4}}=M&={\frac {K\mp K{\sqrt {1-{\tfrac {2H}{K}}}}}{2}}\\\end{aligned}}$ Multiplying by 2 and noticing that the original differential equation has the restriction that K cannot be 0, we may divide by K as well to reduce the problem to ${\begin{aligned}{\frac {1}{2}}=M&=1\mp {\sqrt {1-{\tfrac {2H}{K}}}}\\\end{aligned}}$ Isolating for the square root and squaring both sides gives ${\begin{aligned}{\frac {1}{4}}=M&=1-{\tfrac {2H}{K}}\\\end{aligned}}$ Solving for H gives $\displaystyle H={\frac {3}{8}}K$

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