Question

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Answer 1

Answer:

The average rate of change for the first five weeks of population growth is;

$3100\text{ bacteria per week}$

Explanation:

Given that the growth of a population can be modeled by the exponential function;

$P(t)=500.2^t$

The average rate of change for the first five weeks can be calculated using the formula;

$m=\frac{P(b)-P(a)}{b-a}$

For the first five weeks;

$\begin{gathered} a=0 \\ b=5 \end{gathered}$

substituting to get the value of the function at this points;

$\begin{gathered} P(t)=500\cdot2^t \\ P(0)=500\cdot2^0=500\cdot1 \\ P(0)=500 \end{gathered}$$\begin{gathered} P(t)=500\cdot2^t \\ P(5)=500\cdot2^5=500\cdot32 \\ P(5)=16000 \end{gathered}$

So, the average rate of change is;

$\begin{gathered} m=\frac{16000-500}{5-0} \\ m=\frac{15500}{5} \\ m=3100 \end{gathered}$

Therefore, the average rate of change for the first five weeks of population growth is;

$3100\text{ bacteria per week}$