(A) Let $x$ be the asset at time $t$. $\frac{dx}{dt} = -k\sqrt{x}$,where $k > 0$. Separating variables,we get $\frac{dx}{\sqrt{x}} = -k dt$. Integrati

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(A) Let $x$ be the asset at time $t$. $\frac{dx}{dt} = -k\sqrt{x}$,where $k > 0$. Separating variables,we get $\frac{dx}{\sqrt{x}} = -k dt$. Integrating both sides,$2\sqrt{x} = -kt + c$. At $t = 0$,$x = 10,00,000$,so $2\sqrt{10,00,000} = c \implies c = 2000$. Thus,$2\sqrt{x} = -kt + 2000$. At $t = 3$,$x = 10,000$,so $2\sqrt{10,000} = -3k + 2000 \implies 200 = -3k + 2000 \implies 3k = 1800 \implies k = 600$. For bankruptcy,$x = 0$. $0 = -600T + 2000 \implies 600T = 2000 \implies T = \frac{2000}{600} = \frac{10}{3}$ years.

Class 12 Mathematics Differential Equations Application of differential equations

Medium

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

MHT CET 2024 All MHT CET

A
$\frac{10}{3}$
B
$\frac{10}{9}$
C
$\frac{20}{9}$
D
$\frac{20}{3}$

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