Question

A person's assets start reducing in such a way that the rate of reduction of assets is proportional to the square root of the assets existing at that moment. If the assets at the beginning are \(\text{₹}40\) lakhs and they dwindle down to \(\text{₹}40000\) after \(2\) years. Then the person will be bankrupt in k years from the start. Then \(k=\)

• A. \(2\frac{2}{9}\)
• B. \(2\frac{2}{7}\)
• C. \(2\frac{2}{5}\)
• D. \(2\frac{2}{3}\)

Hint:

Set up dA/dt = -k times the square root of A, integrate, and use the two given values to find when A becomes zero.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Let \(A\) be the assets. Then \(\dfrac{dA}{dt}=-k\sqrt{A}\).

Step 2: Integrate
\(\dfrac{dA}{\sqrt{A}}=-k\,dt\Rightarrow 2\sqrt{A}=-kt+C\).

Step 3: Use the Given Data
At \(t=0\): \(A=4{,}000{,}000\), so \(2\sqrt{A}=4000\) and \(C=4000\).
At \(t=2\): \(A=40{,}000\), so \(2\sqrt{A}=400\), giving \(400=-2k+4000\) and \(k=1800\).

Step 4: Find the Bankruptcy Time
Set \(A=0\): \(0=-1800t+4000\), so \(t=\dfrac{4000}{1800}=\dfrac{20}{9}=2\dfrac{2}{9}\) years.

Final Answer:
The person is bankrupt after \(2\dfrac{2}{9}\) years. \[ \boxed{\text{(A) }2\dfrac{2}{9}} \]