Question

A stone is dropped into a pond waved in the form of circles are generated and the radius of the outer most ripple increases at the rate 2 inches/sec. How fast is the area increasing when the (a) radius is 5 inches (b) after 5 sec.?

Answer 1

Given \(\frac{dr}{dt}\)= 2inch/sec, r = 5 inch, \(\frac{dA}{dt}\) = ? 

(a) A = πr² 

\(\frac{dA}{dt}\) = π2r. \(\frac{dr}{dt}\)= π × 2 × 5 × 2 = 20π sq. inches / sec. 

(b) After 5 sec, dr = 2dt 

r = 2t 

⇒ when t = 5, r = 10 inches 

\(\frac{dA}{dt}\) = π . 2r \(\frac{dr}{dt}\)= π × 2 × 20 × 2 = 40π square inches/sec.