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

Let \(x\) be the asset at time t .
\(\therefore \frac{\mathrm{d} x}{\mathrm{dt}} \propto \sqrt{x} \)
\( \Rightarrow \frac{\mathrm{~d} x}{\mathrm{dt}}=-\mathrm{k} \sqrt{x}, \text { where } \mathrm{k}\gt0 \)
\( \Rightarrow \frac{\mathrm{~d} x}{\sqrt{x}}=-\mathrm{kdt}\)
Integrating on both sides, we get
\(2 \sqrt{x}=-\mathrm{kt}+\mathrm{c} \)
\( \text {When } \mathrm{t}=0, x=10,00,000 \)
\( \therefore 2 \sqrt{1000000}=-\mathrm{k}(0)+\mathrm{c} \)
\( \Rightarrow \Rightarrow \mathrm{c}=2(1000)=2000 \)
\( \therefore 2 \sqrt{x}=-\mathrm{kt}+2000...(i) \)
\(\text {When } \mathrm{t}=3, x=10,000 \)
\( \therefore 2 \sqrt{10000}=-3 \mathrm{k}+2000 \)
\( \Rightarrow 2(100)=-3 \mathrm{k}+2000 \)
\( \Rightarrow 3 \mathrm{k}=1800 \)
\( \Rightarrow \mathrm{k}=600 \)
\( \therefore 2 \sqrt{x}=-600 \mathrm{t}+2000...[From(i)]\)
Time to go bankrupt \(=\mathrm{T}\)
When \(\mathrm{t}=\mathrm{T}, x=0\)
\(\therefore 0 =-600 \mathrm{~T}+2000 \)
\( \Rightarrow \mathrm{T}=\frac{2000}{600}=\frac{10}{3} \text { years}\)