Answer:
a) 2500π cm/s
b) 300√π cm/s
Step-by-step explanation:
Given:
Rate of growth of radius, [tex]\frac{dr}{dt}[/tex] = 25 cm/s
Area of circle is given as:
A = πr²
a)Rate of growth of area, [tex]\frac{dA}{dt}=\frac{d(\pi r^2)}{dt}[/tex]
or
⇒ [tex]\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}[/tex] ............(1)
at r = 50 cm
on substituting the respective values, we get
⇒ [tex]\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}[/tex]
or
⇒ [tex]\frac{dA}{dt}[/tex] = 2π(50)25 = 2500π cm/s
b) when area , A = 36 cm²
36 = πr²
r = [tex]\frac{6}{\sqrt{\pi}}[/tex]
thus, using (1)
⇒ [tex]\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}[/tex]
on substituting the respective values, we get
⇒ [tex]\frac{dA}{dt}=(2)\pi (\frac{6}{\sqrt{\pi}})25[/tex]
or
⇒ [tex]\frac{dA}{dt}[/tex] = 300√π cm/s
