Exponential Growth/Rate of Change of a Microbe Colony

Question

Consider a microbe (mass of 1 pg) that divides every 6 hrs. At t = 0, there is 1 cell. If we consider the growing mass to be a sphere, ignoring space between cells, where the cells have a diameter of 1 micron, how many days will it take before the sphere is expanding faster than the speed of light 'c'? ( c = 3 * 10^8 m s-1).

Answer 1

If we double every 6 hours, we double 4 times per day.

$M= 2^{4t}$ pg

With time measured in days.

Volume of a sphere of radius $\frac 12$ micron (diameter 1 micron.)

$\frac {4}{3}\pi(\frac 12)^3 = \frac {1}{6}\pi$

Density $\rho = \frac {6}{\pi}$

The volume of a massive sphere.

$V = \frac M\rho = \frac {M\pi}{6}$

Radius:

$V = \frac 43 \pi r^3\\ r= \sqrt[3]{\frac {3V}{4\pi}} = \sqrt[3]{\frac {2^{4t}}{8}} = \frac 12 {2^\frac 43 t}$

$\frac {dr}{dt} = \frac 12(\frac 43 \ln 2) 2^{\frac 43 t} = c\\ 2^{\frac 43 t} = \frac {3c}{2\ln 2}\\ \ln (2^{\frac 43 t}) = \ln(\frac {3c}{2\ln 2})\\ (\frac 43\ln 2) t = \ln 3 + \ln c - \ln 2 - \ln(\ln 2)\\ t = \frac {\ln 3 + \ln c - \ln 2 - \ln(\ln 2)}{\frac 43 \ln 2}$