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Home » But in math olympiad, maybe exact. But solution format says final answer boxed. - Leaselab

But in math olympiad, maybe exact. But solution format says final answer boxed. - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 The predicted technical score is $\boxed{81}$.

📌 Question: In a biotechnology experiment, a bacterial culture grows such that the population $ P(t) $ after $ t $ hours is modeled by $ P(t) = P_0 \cdot e^{kt} $. If the population triples every 5 hours, how long will it take to grow from 200 to 16200 bacteria?

📌 We are given the exponential growth model:

📌 P(t) = P_0 \cdot e^{kt}

📌 We know that $ P(5) = 3P_0 $, so:

📌 3P_0 = P_0 \cdot e^{5k} \Rightarrow 3 = e^{5k}

📌 Take the natural logarithm of both sides:

📌 \ln(3) = 5k \Rightarrow k = \frac{\ln(3)}{5}

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r^3 - (r - 2)^3 = 96
Expand \( (r - 2)^3 \):
r^3 - (r^3 - 6r^2 + 12r - 8) = 96
6r^2 - 12r + 8 = 96
6r^2 - 12r - 88 = 0
3r^2 - 6r - 44 = 0
r = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot (-44)}}{2 \cdot 3} = \f...
\sqrt{564} = \sqrt{4 \cdot 141} = 2\sqrt{141}
r = \frac{6 \pm 2\sqrt{141}}{6} = \frac{3 \pm \sqrt{141}}{3}
Since radius must be positive and greater than 2, we take the positive...

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