Question

the number of bacteria in a culture is given by the function, ( n(t)=965 e^{0.2 t} ), where ( t ) is measured in hours.
a) what is the relative rate of growth of this bacterium population?
( % )
b) what is the initial population of the culture?
( square ) bacteria
c) how many bacteria will the culture contain at time ( t = 5 )? round to the nearest whole bacteria.
( square ) bacteria

Explanation:

Step1: Find the relative rate of growth

The general form of an exponential growth function is \(n(t)=n_0e^{rt}\), where \(r\) is the relative rate of growth.
Comparing \(n(t) = 965e^{0.2t}\) with \(n(t)=n_0e^{rt}\), we get \(r = 0.2\).
To convert to a percentage, we multiply by \(100\), so \(0.2\times100 = 20\%\).

Step2: Find the initial population

The initial population is when \(t = 0\).
Substitute \(t = 0\) into \(n(t)=965e^{0.2t}\), we get \(n(0)=965e^{0.2\times0}=965\times1 = 965\).

Step3: Find the population at \(t = 5\)

Substitute \(t = 5\) into \(n(t)=965e^{0.2t}\), we have \(n(5)=965e^{0.2\times5}=965e^{1}\).
Since \(e^{1}\approx2.71828\), then \(n(5)=965\times2.71828\approx2623\).

Answer:

a) \(20\%\)
b) \(965\)
c) \(2623\)