# Biophysics Problem 37

A cell's oxygen requirement is proportional to its mass, but its oxygen intake is proportional to its surface area. Show that a cell cannot grow indefinitely and survive.

#### First Step

Since the oxygen required is proportional to mass and mass itself is proportional to volume $(M = \rho V),$ the oxygen required scales as $L^3.$
Thus the oxygen required will equal some constant multiplied by $L^3,$ i.e.,

$O_{required} = K_1 L^3,$

where $K_1$ is this constant.

Since the oxygen supplied is proportional to \emph{surface area}, it similarly scales as some constant times $L^2,$ i.e.,

$O_{supplied} = K_2 L^2,$

where $K_2$ is a different constant than $K_1.$

#### Calculations

In the limiting case, the required and supplied oxygen will be equal.  Therefore,

$K_1 L^3 = K_2 L^2\\ L = \frac{K_1}{K_2}$

When $(K_1/K_2) < L,$ the above equation will not be balanced and the cell's oxygen requirements cannot be satisfied.