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

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 $k$.