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A population's instantaneous growth rate is the rate at which it grows for every instant in time. The function [tex]r[/tex] gives the instantaneous growth rate of a bacterial culture [tex]x[/tex] hours after the start of an experiment.
[tex]
r(x)=0.01(x+2)\left(x^2-9\right)
[/tex]
How many hours after the experiment began was the instantaneous growth rate equal to 0?
To determine the number of hours after the experiment began when the instantaneous growth rate was equal to 0, we need to solve the equation given by the function [tex]\( r(x) \)[/tex] for when it equals 0.
The function given for the instantaneous growth rate is: [tex]\[ r(x) = 0.01 (x + 2) (x^2 - 9) \][/tex]
To find when the growth rate is 0, we set the function [tex]\( r(x) \)[/tex] equal to 0 and solve for [tex]\( x \)[/tex]: [tex]\[ 0.01 (x + 2) (x^2 - 9) = 0 \][/tex]
First, we can ignore the constant multiplier [tex]\( 0.01 \)[/tex] because it does not affect the solutions. So we have: [tex]\[ (x + 2) (x^2 - 9) = 0 \][/tex]
Next, we solve the equation by finding the values of [tex]\( x \)[/tex] that satisfy this equality. We use the Zero Product Property, which states that if a product of two factors is zero, at least one of the factors must be zero.
So, we set each factor equal to 0: [tex]\[ x + 2 = 0 \][/tex] [tex]\[ x^2 - 9 = 0 \][/tex]
Solving for [tex]\( x \)[/tex] in each equation: 1. [tex]\( x + 2 = 0 \)[/tex]: [tex]\[ x = -2 \][/tex]
2. [tex]\( x^2 - 9 = 0 \)[/tex]: [tex]\[ x^2 = 9 \][/tex] Taking the square root of both sides, we get: [tex]\[ x = 3 \][/tex] [tex]\[ x = -3 \][/tex]
Thus, the solutions to the equation [tex]\( (x + 2) (x^2 - 9) = 0 \)[/tex] are: [tex]\[ x = -2, x = 3, x = -3 \][/tex]
Now, we look at the provided options to find the correct hours after the experiment began. The potential answers are [3, 9, 0, 2].
From our solutions, the only value corresponding to the given choices is: [tex]\[ \boxed{3} \][/tex]
Therefore, the correct answer is: [tex]\[ \boxed{3} \][/tex] This means that the instantaneous growth rate of the bacterial culture was equal to zero 3 hours after the start of the experiment.
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