The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.

Question

Philoid · Accepted Answer

Let the surface area of the balloon be S.

∴ S = 4πr²

According to the question,

⇒

⇒ 8πr

⇒ 8πrdr = ktdt

Integrating both sides, we have

⇒ 8π∫rdr = k∫tdt

⇒

⇒ ……(1)

Given, we have r = 1 unit when the t = 0 sec

Putting the value in equation (1)

∴

⇒ 4π (1)² = k × 0 + c

⇒ c = 4π ……(2)

Putting the value of c in equation (1) we have,

……(3)

Given, we have r = 2 units when t = 3 sec

∴

⇒

⇒ ……(4)

Now, putting the value of k in equation (2),

We have,

⇒

∴