Question

The population P = P(t) at time 't' of a certain species follows the differential equation `("dp")/("dt")` = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is ______.

Options

• `1/2log_e18`

• 2log_e18

• log_e9

• log_e18

Answer 1

The population P = P(t) at time 't' of a certain species follows the differential equation `("dp")/("dt")` = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is `underlinebb(2log_e18)`.

Explanation:

Given: Population (P) = P(t) at time 't' of a certain species follows the differential equation.

`("dP")/("dt")` = 0.5P – 450 = `"P"/2 - 900/2`

⇒ `("dP")/("dt") = ("P" - 900)/2`

It is in variable-separable form

∴ `("dP")/("P" - 900) = 1/2"dt"`

Integrate both sides

P = 850 at t = 0

P = 0 at t = t

`int_850^0 ("dP")/("P" - 900) = 1/2int_0^1"dt"`

⇒ ln `|"P" - 900|_850^0 = 1/2["t"]_0^"t"`  ...`((∵ int("d"x)/x = ln  |x| + c),(int"d"x = x + "c"))`

⇒ ln 900 – In 50 = `1/2"t"`

⇒ ln `(900/50) = "t"/2`  ......`["Using"  ln  x - ln  "y" = ln  x/"y"]`

⇒ ln 18 = `"t"/2`

⇒ t = 2 ln 18