The bacteria has an exponential growth rate and the population of the

bacteria increases rapidly with time.

(a) The relative rate of growth is

.

(b) The initial size of the culture is 75 bacteria.

(c) The function that models the number of bacteria n(t) is;

(d) The number of bacteria after 4.5 hours is approximately 8,100 bacteria.

(e) The number of hours after which the bacteria will reach 75,000 is approximately 6.64 hours.

Reasons:

The count in the culture of bacteria after 2 hours = 600

The count after 6 hours = 38,400

(a) The relative rate of growth, k is given by the formula;

Therefore, we get;

Which gives;

4·k = ㏑(64)

(b) The initial size of the culture, C, is given by the relation;

Therefore, we get;

The initial size of the culture, C = 75

(c) The function is

Where:

y = n(t)

C = n₀

k = r

We get;

n₀ = C = 75

Which gives the function as follows;

(d) The number of bacteria after 4.5 hours is ≈ 8,100 bacteria

(e) At n(t) = 75,000, we have;

The time at which the bacteria population will reach 75,000, t ≈ 6.64 hours.

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