Question

If r is the radius of spherical balloon at time t and the surface area of balloon changes at a constant rate K, then ______.

Options

• `4pi"r"^2 = "Kt"^2/2 + "c"`

• 8πr² = Kt + c

• `pi"r"^2 = "Kt"^2/2 + "c"`

• 4πr² = Kt + c

Answer 1

If r is the radius of spherical balloon at time t and the surface area of balloon changes at a constant rate K, then 4πr² = Kt + c.

Explanation:

According to question,

`"d"/"dx" (4pi"r"^2)` = K   (∵ surface area of spherical ballon with radius r is 4πr²)

`=> 4pi (2"r") "dr"/"dt" = "K" => 8pi"r"  "dr" = "Kdt"` 

On Integrating both sides, we get

`8pi int "r"  "dr" = "K" int "dt"`

`=> 8pi "r"^2/2 = "Kt" + "C"`

`= 4pi"r"^2 = "Kt" + "C"`