Question

A wound is healing in such a way that t days since Sunday the area of the wound has been decreasing at a rate of `- 6/("t" + 2)^2` cm² per day where 0 < t ≤ 8. If on Monday the area of the wound was 1.4 cm². What was the area of the wound on Sunday?

Answer 1

Let Sunday be the initial period

Given `"dA"/"dt" = - 3/("t" + 2)^2  "cm"^2/"day"`

dA = `- 3("t" + 2)^-2 "dt"`

Integrating both sides we get

`int  "dA" = int - 3("t" + 2)^-2  "dt"`

A = `- 3 (("t" + 2)^(- 2 + 1))/(- 2 + 1) + "c"`

A = `- 3 ("t" + 2)^-1/(- 1) + "c"`

A = `3/("t" + 2) + "c"`  ........(1)

When t = 1, A = 2cm² 

2 = `3/(1 + 2) + "c"`

2 = 1 + c

c = 2 – 1

⇒ c = 1

Equation (1) ⇒ A = `3/("t" + 2) + 1`  .......(2)

On Sunday (t = 0), initial period.

∴ A = `3/(0 + 2) + 1`

= `3/2 + 1`

A = `5/2`

= 2.5 sq.cm.

∴ Area of the wound on Sunday = 2.5 sq.cm