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 ax ‘ 10 lakhs and they dwindle down to ‘ 10,000 after 2 years, show that the person will be bankrupt in `2 2/9` years from the start.

Answer 1

Let x be the assets of the person at time t years. Then the rate of reduction is `"dx"/"dt"` which is proportional to `sqrt"x"`.

∴ `"dx"/"dt" prop sqrt"x"`

∴ `"dx"/"dt" = - "k" sqrt "x"`, where k > 0

∴ `"dx"/sqrt"x"` = - k dt

Integrating both sides, we get

`int "x"^(-1/2)"dx" = - "k" int "dt"`

∴ `"x"^(1/2)/(1/2) = - "kt" + "c"` 

∴ `2sqrt"x"` = - kt + c

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

∴ `2sqrt1000000` = - k (0) + c

∴ c = 2000

∴ `2sqrt"x" = - "kt"` + 2000    ...(1)

Also, when t = 2, x = 10,000

∴ `2sqrt10000 = - "k" xx 2 + 2000`

∴ 200 = - 2k + 2000

∴ 2k = 1800

∴ k = 900

∴ (1) becomes,

∴ `2sqrt"x" = - 900"t" + 2000`

When the person will be bankrupt, x = 0

∴ 0 = - 900 t + 2000

∴ 900 t = 2000

∴ t = `20/9 = 2 2/9`

Hence, the person will be bankrupt in `2 2/9` years.