Question

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are 27 gm of certain substance and 3 h later it is found that 8 gm are left, then the amount left after one more hour is ______.

Options

• `20/3`gm

• `16/3`gm

• `19/3`gm

• `17/3`gm

Answer 1

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are 27 gm of certain substance and 3 h later it is found that 8 gm are left, then the amount left after one more hour is `underline(16/3  "gm")`.

Explanation:

Let N_gm be amount present at any time t

According to given conditions,

`"dN"/"dt" = "kN"`

`=> "dN"/"N"` = kdt

`=> int "dN"/"N"`

= k ∫ dt

⇒ log N = kt + C

⇒ N = Ae^kt

When t = 0, N = 27 gm

∴ A = 27 gm

N = 27e^kt

When t = 3, N = 8

∴ 8 = 27e^3k 

`=> "e"^"k" = (8/27)^(1//3) = 2/3`

∴ N = `27(2/3)^"t"`

put t = 4

∴ N = 27`(2/3)^4 = (27 xx 16)/81 = 16/3`

N = `16/3`gm