Mathematics

Mathematics

Problem

Culture P begins with 200 bacteria and grows continuously at a rate proportional to its size with growth rate 0.03 h−10.03\text{ h}^{-1}. Culture Q begins with 120 bacteria and grows continuously with growth rate 0.05 h−10.05\text{ h}^{-1}. Approximately how many hours after t=0t=0 will the two cultures contain the same number of bacteria? a) 17.0 h17.0\text{ h} b) 25.5 h25.5\text{ h} c) 12.8 h12.8\text{ h} d) 34.0 h34.0\text{ h}
Intuitive explanation

Both cultures follow exponential growth, so you'll write each as N(t)=N0ektN(t) = N_0 e^{kt} using the given initial populations and growth rates. The key insight is that "same number of bacteria" means setting the two expressions equal and solving for tt using logarithms—this is where the exponential exponent comes down to become a linear equation you can handle.

Intuitive explanation

Both cultures follow exponential growth, so you'll write each as N(t)=N0ektN(t) = N_0 e^{kt} using the given initial populations and growth rates. The key insight is that "same number of bacteria" means setting the two expressions equal and solving for tt using logarithms—this is where the exponential exponent comes down to become a linear equation you can handle.

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