# Question e6a1f

Sep 27, 2014

It would take about 42 h.

This involves several assumptions. Here are mine:

• The mass of a bacterium is 1 × 10⁻¹⁵ kg.
• The mass of a human is 70 kg.
• There is only exponential growth, with no lag phase or stationary phase.

The equation for exponential growth is

N_t = N_0 × 2^(t/t_"d") where

${N}_{t}$ is the number (mass) at time $t$
${N}_{0}$ is the number (mass) at time $t$ = 0
${t}_{\text{d}}$ is the division time (analogous to half-life in exponential decay)

This gives

$\ln \left({N}_{t} / {N}_{0}\right) = \left(\frac{t}{t} _ \text{d}\right) \ln 2$ and

$t = \left({t}_{\text{d}} / \ln 2\right) \ln \left({N}_{t} / {N}_{0}\right)$

t = ("45 min"/ln2) ln("70 kg"/(1× 10^-15 "kg")) = 64.9 min × ln(7 × 10^16)# =
64.9 min × 38.8 h = 2520 min = 42 h (2 significant figures)

Scary!