Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 10

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MATH100 B December 2024

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Question 10

Consider a cell that is shaped like a triangular prism. Its triangular sides have base length $x$ and height $x$ (which may change over time, and are both measured in micrometers, μm) and its depth is fixed at $10\mu m$ (not changing). Its volume is $5x^2\mu m^3$, and its surface area is $x^2+10(1+\sqrt{5})x\mu m^2$. The cell absorbs nutrients at a rate $A(x)$, and consumes nutrients at a rate of $C(x)$, where $$A(x)=k\cdot \text{Area}(x)\text{and}C(x)=\text{Volume}(x)$$ for a positive constant $k$. Here, $\text{Area}(x)$ and $\text{Volume}(x)$ are the surface area and volume of the triangular prism respectively, as functions of $x$. Such cells exhibit the following behaviours: When $A(x)>C(x)$, the cell grows. When $A(x)<C(x)$, the cell shrinks. When $A(x)=C(x)$, the cell remains the same size. Initially, $x$ is a very small, positive number. Determine what will happen to the cell after a really long time has passed. Your answer may depend on $k$.

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
Since $A(x)=k(x^2+10(1+\sqrt{5})x$ and $C(x)=5x^2$, the leading terms agree when $k=5$. Consider the behaviour of the cell in the three separate cases $k<5$, $k=5$, and $k>5$.

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

We are given that the surface area of the prism is $x^2+10(1+\sqrt{5})x$ and and the volume is $5x^2$, so $$A(x)=k(x^2+10(1+\sqrt{5})x\text{ and }C(x)=5x^2.$$ The leading terms in the above expressions are $k{x}^2$ and $5x^2$, so we consider the cases $k<5$, $k=5$, and $k>5$. First suppose $k<5$. Since $x$ starts out small, linear terms dominate over quadratic terms, so $A(x)$ dominates over $C(x)$ and the cell grows. It cannot grow arbitrarily large, however, since if it did the quadratic terms would start to dominate and $C(x)$ would become larger than $A(x)$. Thus, the cell will grow but approach an upper bound given by the expression $A(x)=C(x)$, which happens when $$x=\frac{10k(1+\sqrt{5})}{5-k}.$$ Alternatively, if $k=5$ or $k>5$, then $A(x)>C(x)$ regardless of $x$, so the cell will grow forever without bound.

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