The rate of reduction of person's assets is proportional to square-root of his exising assets. In 2 years, his assets reduce from 25 lakhs to 6.25 lakhs, then the person will be bankrupt in another

Options

A. 4 years
B. 3 years
C. 2 years
D. 1 years

Answer

To solve the problem, we need to model the situation using a differential equation based on the information provided. Let's break this down step by step.

### Step 1: Set up the differential equation
The problem states that the rate of reduction of a person's assets is proportional to the square root of his existing assets. We can express this mathematically as:
\[ -\frac{dx}{dt} = k \sqrt{x} \]
where \( x \) is the amount of assets and \( k \) is a proportionality constant.

### Step 2: Rearranging the equation
Rearranging the equation gives us:
\[ \frac{dx}{\sqrt{x}} = -k \, dt \]

### Step 3: Integrate both sides
Now, we will integrate both sides. The left side requires the integral of \( x^{-1/2} \):
\[ \int \frac{dx}{\sqrt{x}} = -k \int dt \]
The integral of \( \frac{1}{\sqrt{x}} \) is \( 2\sqrt{x} \), so we have:
\[ 2\sqrt{x} = -kt + C \]
where \( C \) is the constant of integration.

### Step 4: Solve for the constant of integration
We need to find \( C \) using the initial condition. Initially, at \( t = 0 \), the assets \( x = 25 \) lakhs:
\[ 2\sqrt{25} = -k(0) + C \implies C = 10 \]
Thus, our equation becomes:
\[ 2\sqrt{x} = -kt + 10 \]

### Step 5: Use the second condition to find \( k \)
We know that after 2 years (when \( t = 2 \)), the assets reduce to \( x = 6.25 \) lakhs:
\[ 2\sqrt{6.25} = -k(2) + 10 \]
Calculating \( \sqrt{6.25} = 2.5 \):
\[ 2(2.5) = -2k + 10 \implies 5 = -2k + 10 \implies -2k = 5 - 10 \implies -2k = -5 \implies k = \frac{5}{2} = 2.5 \]

### Step 6: Substitute \( k \) back into the equation
Now substituting \( k \) back into our equation gives:
\[ 2\sqrt{x} = -2.5t + 10 \]

### Step 7: Find when the assets become zero
To find when the assets become zero, set \( x = 0 \):
\[ 2\sqrt{0} = -2.5t + 10 \implies 0 = -2.5t + 10 \implies 2.5t = 10 \implies t = \frac{10}{2.5} = 4 \]

### Step 8: Determine the time until bankruptcy
Since we started counting from \( t = 0 \), and we found that \( t = 4 \) years is when the assets will be zero, and we already know that 2 years have passed, the person will be bankrupt in another:
\[ 4 - 2 = 2 \text{ years} \]

### Final Answer
The person will be bankrupt in another **2 years**.

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