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In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria increased in 12 hours by using the concept of application of differ…

Q.3 (B) (ii) In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria increased in 12 hours by using the concept of application of differential equations.

Doubtly Asked Board exam · June 2026 hsc 12th Save WhatsApp Report
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Answer: 8 times

Let \(x\) be the number of bacteria at time \(t\) hours. Then \(\frac{dx}{dt}=kx\Rightarrow\int\frac{dx}{x}=k\int dt\Rightarrow\log x=kt+c\).

At \(t=0\), \(x=x_0\): \(c=\log x_0\). So \(\log\frac{x}{x_0}=kt\), i.e. \(x=x_0e^{kt}\).

At \(t=4\), \(x=2x_0\): \(e^{4k}=2\).

At \(t=12\): \(x=x_0e^{12k}=x_0\left(e^{4k}\right)^3=x_0\cdot 2^3=8x_0\).

The bacteria increase 8 times in 12 hours.

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report
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