According to a model, the mass of bacteria, ${m \textrm{ g}}$ in a test culture at time ${t \textrm{ s}}$ grows at a rate proportional to its current mass and dies at a constant rate of ${8 \textrm{ g/s}.}$

It is given that the mass of bacteria stays constant if ${m=16 \textrm{ g}.}$

(a) Show that the mass of bacteria can be described by the differential equation ${\displaystyle \frac{\mathrm{d}m}{\mathrm{d}t}=\frac{1}{2}m-8.}$

(b) Find ${m}$ in terms of ${t}$, given that the initial mass of bacteria is ${\frac{24}{5}\textrm{ g}.}$

(c) Find the time when there is no bacteria left in the test culture.

(d) Sketch the graph of ${m}$ against ${t}$ for the parts relevant to the context of the question.

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