Question

A bacterial infection in an internal wound grows as $N^{\prime }(t)=N_0$ exp(t), where the time $t$ is in hours. A dose of antibiotic, taken orally, needs $1$ hour to reach the wound. Once it reaches there, the bacterial population goes down as $\frac{dN}{dt}=-5N^2$. What will be the plot of $\frac{N_0}{N}$ vs. $t$ after $1$ hour ?

• A.
• B.
• C.
• D.

Answer 1

Answer: (A)

From $0$ to $1$ hour, $N^{\prime }=N_0{e}^t$
From 1 hour onwards $\frac{dN}{dt}=-5N^2$
So at $t=1$ hour, $N^{\prime }$ = $e{N}_0$
$\frac{dN}{dt}=-5N^2$
${\int }_{e{N}_0}^N{N}^{-2}dN=-5{\int }_1^{t}dt$
$\frac{1}{N}-\frac{1}{e{N}_0}=5\left(t-1\right)$
$\frac{N_0}{N}-\frac{1}{e}=5N_0\left(t-1\right)$
$\frac{N_0}{N}=5N_0\left(t-1\right)+\frac{1}{e}$
$\frac{N_0}{N}=5N_{0}t+\left(\frac{1}{e}-5N_0\right)$
which is following $y=mx+C$