Answer-
The radius is increasing at a rate of [tex]\dfrac{1}{16}[/tex]
Solution-
The rate of change of the area of the crater is [tex]\dfrac{\pi}{2}[/tex], i.e
[tex]\dfrac{dA}{dt}=\dfrac{\pi}{2}[/tex]
We know that,
[tex]Area =\pi r^2[/tex]
Putting this,
[tex]\Rightarrow \dfrac{d(\pi r^2)}{dt}=\dfrac{\pi}{2}[/tex]
Applying chain rule,
[tex]\Rightarrow \dfrac{d(\pi r^2)}{dr}\times \dfrac{dr}{dt}=\dfrac{\pi}{2}[/tex]
[tex]\Rightarrow (2\pi r)\times \dfrac{dr}{dt}=\dfrac{\pi}{2}[/tex]
Putting the value of radius as 4 (given),
[tex]\Rightarrow 2\pi \times 4\times \dfrac{dr}{dt}=\dfrac{\pi}{2}[/tex]
[tex]\Rightarrow 8\pi \dfrac{dr}{dt}=\dfrac{\pi}{2}[/tex]
[tex]\Rightarrow \dfrac{dr}{dt}=\dfrac{\pi}{2\times 8\pi}[/tex]
[tex]\Rightarrow \dfrac{dr}{dt}=\dfrac{1}{16}[/tex]
