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 ₹ 10 lakhs and they dwindle down to ₹ 10,000 after 2 years, show that the person will be bankrupt in 2\(\frac{2}{9}\) years from the start.

Answer 1

Let x be the assets of the presort at time t years.

Then the rate of reduction is which is \(\frac{dx}{dt}\) proportional to √x.

Integrating both sides, we get

At the beginning, i.e. at t = 0, x = 10,00,000 

2√10,00,000 = -k(0) + c

∴ c = 2000 

∴ 2√x = -kt + 2000 ……..(1) 

Also, when t = 2, x = 10,000 

∴ 2√10000 = -k × 2 + 2000

∴ 2k = 1800 

∴ k = 900 

∴ (1) becomes, 

∴ 2√x = -900t + 2000

When the person will be bankrupt, x = 0

∴ 0 = -900t + 2000

∴ 900t = 2000

Hence, the person will be bankrupt in 2\(\frac{2}{9}\) years.