A) \[400-{{300}^{\text{et/2}}}\]
B) \[300-{{200}^{\text{et/2}}}\]
C) \[600-{{500}^{\text{et/2}}}\]
D) \[400-{{300}^{e\,-\,\text{t/2}\,}}\]
Correct Answer: A
Solution :
\[\frac{dp}{dt}=\frac{P-400}{2}\] |
\[\frac{dP}{P-400}=\frac{1}{2}dt\] |
\[In\left| P-400 \right|=\frac{1}{2}t+c\] |
at \[t=0,P=100\] |
In \[300=c\] |
\[In\left| \frac{P-400}{300}=\frac{t}{2} \right|\] |
\[\Rightarrow \] \[\left| P-400 \right|=300\,{{e}^{\text{t/2}}}\] |
\[\Rightarrow \] \[400-P=300\,{{e}^{\text{t/2}}}\,(as\,P\,<400)\] |
\[\Rightarrow \] \[P=400-300\,{{e}^{\text{t/2}}}.\] |
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