The assets of a person are reduced in his business such that the rate of reduction is proportional to the…
The assets of a person are reduced in his business such that the rate of reduction is proportional to the square root of the existing assets. If the assets were initially ₹ $10,00,000$ and due to loss they reduce to ₹ 10,000 after 3 years, then the number of years required for the person to go bankrupt will be
$\frac{10}{3}$
$\frac{10}{9}$
$\frac{20}{9}$
$\frac{20}{3}$
Solution
Let $x$ be the asset at time t .
$\begin{aligned}
& \therefore \quad \frac{\mathrm{d} x}{\mathrm{dt}} \propto \sqrt{x} \\
& \quad \Rightarrow \frac{\mathrm{~d} x}{\mathrm{dt}}=-\mathrm{k} \sqrt{x}, \text { where } \mathrm{k}\gt0 \\
& \quad \Rightarrow \frac{\mathrm{~d} x}{\sqrt{x}}=-\mathrm{kdt}
\end{aligned}$
Integrating on both sides, we get
$\begin{array}{ll}
& 2 \sqrt{x}=-\mathrm{kt}+\mathrm{c} \\
& \text { When } \mathrm{t}=0, x=10,00,000 \\
\therefore \quad & 2 \sqrt{1000000}=-\mathrm{k}(0)+\mathrm{c} \\
\Rightarrow & \Rightarrow \mathrm{c}=2(1000)=2000 \\
\therefore \quad & 2 \sqrt{x}=-\mathrm{kt}+2000...(i)
\end{array}$
$\begin{array}{ll}
& \text { When } \mathrm{t}=3, x=10,000 \\
\therefore \quad & 2 \sqrt{10000}=-3 \mathrm{k}+2000 \\
& \Rightarrow 2(100)=-3 \mathrm{k}+2000 \\
& \Rightarrow 3 \mathrm{k}=1800 \\
& \Rightarrow \mathrm{k}=600 \\
\therefore \quad & 2 \sqrt{x}=-600 \mathrm{t}+2000...[From(i)]
\end{array}$ Time to go bankrupt $=\mathrm{T}$
When $\mathrm{t}=\mathrm{T}, x=0$
$\begin{aligned}
\therefore \quad 0 & =-600 \mathrm{~T}+2000 \\
& \Rightarrow \mathrm{~T}=\frac{2000}{600}=\frac{10}{3} \text { years }
\end{aligned}$