Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 12 (b)

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MATH100 B December 2024

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

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  •  December 2023  •  December 2024  •  

Question 12 (b)

Suppose a chemical is being converted to a product in tank. The amount of the chemical is $S$, which is nonnegative and changes with respect to time. If the chemical is being added to the tank at a constant nonnegative rate $R$, then the amount of the chemical in the tank satisfies the differential equation $$\frac{dS}{dt}=R-\frac{S}{2+S}.$$ Describe in words what happens to the amount of the chemical in the tank over a long period of time if $R=\frac{3}{2}$.

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Hint

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Solution
For all $S\ge 0$, $\frac{S}{2+S}<1$. Since $\frac{3}{2}>1$, we have that $$\frac{3}{2}-\frac{S}{2+S}>\frac{3}{2}-1>0,$$ so $\frac{dS}{dt}>0$ for all nonnegative $S$. That is, over time the substrate builds up without bound.

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