Question

The area of the circle is increasing at a uniform rate of 2 cm²/sec. How fast is the circumference of the circle increasing when the radius r = 5 cm?

Answer 1

Let radius of the circle be r cm. 

Given: `(dA)/(dt)` = 2 cm²/s

since A = πr²

∴ `(dA)/(dt) = 2pir (dr)/(dt)`         ...(i)

Also, circumference, C = 2πr

∴ `(dC)/(dt) = 2pi (dr)/(dt)`         ...(ii)

from (i), `2 = 2pir (dr)/(dt)`

⇒ `(dr)/(dt) = 1/(pir)`

Now, substituting the value of `(dr)/(dt)` in eq. (ii) we get

`(dc)/(dt) = 2pi 1/(pir) = 2/r`

Now, `(dC)/(dt)|_(at  r = 5)`

= `2/5`

= 0.4 cm/s

Thus, circumference of circle increases at the rate of 0.4 cm/s.