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Answer:
To find out how many minutes it took for the jar to be half filled with microbes, we can work backwards from the point where the jar was filled with germs.
Let's denote the number of microbes in the jar at any given minute \(t\) as \(N(t)\). We know that the number of microbes doubles every minute, so we can express \(N(t)\) as \(N(t) = 2^t\).
Given that the jar was filled with germs after 30 minutes, we have \(N(30) = 2^{30}\).
Now, to find the number of minutes it took for the jar to be half filled with microbes, we need to find \(t\) such that \(N(t) = \frac{1}{2} \times N(30)\).
Setting up the equation:
\[ \frac{1}{2} \times 2^{30} = 2^t \]
Solving for \(t\):
\[ 2^{30-1} = 2^t \]
\[ 2^{29} = 2^t \]
Therefore, it took \(29\) minutes for the jar to be half filled with microbes.