CBSE 2024 · Region 3 · Set 1 · Q37 · 4 marks
A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated.
The differential equation representing the growth of bacteria is given as : $\displaystyle \frac{d P}{d \mathrm{t}}=k P$, where $\displaystyle P$ is the population of bacteria at any time ' $\displaystyle \mathrm{t}$ '. Based on the above information, answer the following questions :(i)Obtain the general solution of the given differential equation and express it as an exponential function of 't'.(ii)If population of bacteria is $\displaystyle 1000$ at $\displaystyle \mathrm{t}=0$, and $\displaystyle 2000$ at $\displaystyle \mathrm{t}=1$, find the value of k . Case Study - $\displaystyle 3$
A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated.
The differential equation representing the growth of bacteria is given as : $\displaystyle \frac{d P}{d \mathrm{t}}=k P$, where $\displaystyle P$ is the population of bacteria at any time ' $\displaystyle \mathrm{t}$ '. Based on the above information, answer the following questions :
(i)
Obtain the general solution of the given differential equation and express it as an exponential function of 't'.
(ii)
If population of bacteria is $\displaystyle 1000$ at $\displaystyle \mathrm{t}=0$, and $\displaystyle 2000$ at $\displaystyle \mathrm{t}=1$, find the value of k . Case Study - $\displaystyle 3$
Marking-scheme solution
$$\begin{aligned}
& \frac{d P}{d \mathrm{t}}=k P \Rightarrow \int \frac{d P}{P}=\int k d \mathrm{t} \\
& \Rightarrow \log P=k \mathrm{t}+C \text { or } P=e^{k \mathrm{t}+C}
\end{aligned}
\begin{aligned}
& \log P=k \mathrm{t}+C \\
& \text { when } \mathrm{t}=0, P=1000 \Rightarrow C=\log 1000 \\
& \text { when } \mathrm{t}=1, P=2000 \Rightarrow \log 2000=k+\log 1000 \\
& \Rightarrow k=\log 2
\end{aligned}
$$
Differential EquationsMethods of Solving First Order, First Degree Differential EquationsApplycase_studymedium
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CBSE Class 12 Mathematics past-paper question from the 2024board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.