7FegGrowth


 * Exponential Growth**

For a poplutation experiencing exponential growth, given t = 0, P = P 0 math . \qquad P = P_0e^{kt}, \quad k>0 \; \; \textit{k is constant} math

Example

Bacteria is being grown in a culture. The number of bacteria is observed to be growing exponentially such that the population doubles in five hours. If the initial number of bacteria is 2000, find the equation for the population after **t** hours. Hence find the number of hours for the population to reach 28000 bacteria.

From the information math \\ . \qquad P(0)=2000 \\. \\ . \qquad P(5)=4000 math

Using the rule for exponential population growth: math . \qquad P = 2000e^{kt} math

Use condition P(5) = 4000 math \\ . \qquad 4000 = 2000e^{5k} \\. \\ . \qquad e^{5k} = 2 math Use log laws math \\ . \qquad 5k = log_e 2 \\ .\\ . \qquad k = \frac{1}{5}log_e 2 = 0.139 math

hence equation is math . \qquad P=2000e^{0.139t} math

Part b) Hence find the number of hours for the population to reach 28000 bacteria.

math \\ . \qquad 28000 = 2000e^{0.139t} \\. \\ . \qquad e^{0.139t} = 14 math

Use log laws math \\ . \qquad 0.139t = log_e 14 \\. \\ . \qquad t = \dfrac {log_e 14}{0.139} \\ .\\ . \qquad t = 19 \, \text{ hours} math

Notice on the graph that t > 0 so the graph is restricted to 1st Quadrant only.

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