Growth of a bacterial culture

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A bacterial culture doubles its number of cells every $4$ hours and starts from $500$ cells.

• (a) How many cells are there after $24$ hours?
• (b) After how long does the culture reach $10\,000$ cells?

Solution

The model is $N(t)=500\cdot 2^{\,t/4}$, with $t$ in hours. (a) $N(24)=500\cdot 2^{24/4}=500\cdot 2^{6}=500\cdot 64=32\,000$ cells. (b) Impose $500\cdot 2^{\,t/4}=10\,000$, i.e. $2^{\,t/4}=20$. Taking base-$2$ logarithms: $\frac{t}{4}=\log_2 20=\frac{\ln 20}{\ln 2}\approx 4.322\;\Rightarrow\; t\approx 17.29\ \text{h}.$ $\boxed{\,N(24)=32\,000\ \text{cells},\quad t\approx 17.29\ \text{h}\,}$