Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 10/Solution 1

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.