Science:Math Exam Resources/Courses/MATH102/December 2017/Question 13 (c)

MATH102 December 2017

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q8  •  Q9 (a)  •  Q9 (b)  •  Q9 (c)  •  Q10 (a)  •  Q10 (b)  •  Q10 (c)  •  Q10 (d)  •  Q11 (a)  •  Q11 (b)  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  Q12 (d)  •  Q13 (a)  •  Q13 (b)  •  Q13 (c)  •  Q13 (d)  •  Q14 (a)  •  Q14 (b)  •  Q15 (a)  •  Q15 (b)  •  Q16  •  

   Other MATH102 Exams

  •  December 2014  •  December 2011  •  December 2012  •  December 2013  •  December 2016  •  December 2015  •  

Question 13 (c)

A biochemical reaction in which a substance is both produced and consumed is described by the differential equation $${\frac {dc}{dt}}={\frac {4c^{3}}{1+c^{3}}}-rc,$$ where $c(t)$ denotes the concentration of the substance, $r>0$ is a constant and the units are omitted. (c) For what positive value of $r$ is the line $y(c)=rc$ tangent to $g(c)={\frac {4c^{3}}{1+c^{3}}}$? Call this value $r_{\text{tan}}$.

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
We look for an intersection point other than the origin.

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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From $$g(c)={\dfrac {4c^{3}}{1+c^{3}}}=4{\dfrac {c^{3}+1-1}{1+c^{3}}}=4-{\dfrac {4}{1+c^{3}}}=4-4(1+c^{3})^{-1},$$ we differentiate using the power rule and chain rule to yield $$g'(c)=-4(-1)(1+c^{3})^{-2}\cdot 3c^{2}={\dfrac {12c^{2}}{(1+c^{3})^{2}}}.$$ At the point $c\neq 0$ (which is not the origin) where $g$ is tangent to $y$, we have $$g(c)=y(c),$$ $$g'(c)=y'(c),$$ that is, $${\dfrac {4c^{3}}{1+c^{3}}}=rc,$$ $${\dfrac {12c^{2}}{(1+c^{3})^{2}}}=r.$$ Let us first solve for ${\textstyle c}$. In fact, since ${\textstyle c\neq 0}$, the first equation simplifies to $${\dfrac {4c^{2}}{1+c^{3}}}=r.$$ Equating this with the second equation, we have $${\dfrac {12c^{2}}{(1+c^{3})^{2}}}={\dfrac {4c^{2}}{1+c^{3}}}\implies {\dfrac {3}{1+c^{3}}}=1\implies 1+c^{3}=3\implies c^{3}=2.$$ Therefore, $c={\sqrt[{3}]{2}}$ and $$r={\dfrac {4c^{2}}{1+c^{3}}}={\dfrac {4({\sqrt[{3}]{2}})^{2}}{3}}={\dfrac {4{\sqrt[{3}]{4}}}{3}}.$$ This is the value of ${\textstyle r_{\text{tan}}}$. Answer: ${\textstyle r_{\text{tan}}={\color {blue}{\dfrac {4{\sqrt[{3}]{4}}}{3}}}}$.