The area covered by a certain population of bacteria increases according to a continuous exponential growth model. Suppose that a sample culture has an initial area of and an observed doubling time of hours.

The area is the region bounded by the shape of an object. The space covered by the figure or any two-dimensional geometric shape, in a plane, is the area of the shape.
Area is the term used to define the amount of space taken up by a 2D shape or surface.
We measure area in square units : cm² or m². Area is calculated by multiplying the length of a shape by its width.

As the area covered increases according to a continous growth model, we start with these relation

[tex]y (t) = C e^{kt}[/tex]

We know that the initial area y(0) is 8.6. Then

[tex]y(0) = Ce^{0} = C = 8.6[/tex]

We also know that the area duplicates after 5 days.

y( t + 5 ) = 2y(t)

[tex]\frac{y(0+5)}{y(0)} = 2 = \frac{e^{0+ 5k} }{e^{0} } = e^{5k}[/tex]

5k = In(2)

k = In(2)/5 = 0.14

Then, the model becomes

[tex]y (t) = 8.6e^{In(2).t/5}[/tex]

b) We have to calculate y(6)

[tex]y(6) = 8.6e^{In(2). t /5}[/tex] [tex]= 8.6e^{0.832} = 8.6 * 2.297 = 19.758[/tex]

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