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.
