Answer:
[tex]80\pi^4 cm^3/min[/tex]
Step-by-step explanation:
We are given that
Height of cylinder=12 cm
Radius of cylinder=4 cm
Circumference of the cylinder increasing at the rate=[tex]\pi^4 cm/min[/tex]
We know that circumference of cylinder=[tex]2\pi r[/tex]
[tex]C=2\pi r[/tex]
Differentiate w.r t time
[tex]\frac{dC}{dt}=2\pi \frac{dr}{dt}[/tex]
[tex]\pi^4=2\pi \frac{dr}{dt}[/tex]
[tex]\frac{dr}{dt}=\frac{\pi^3}{2} cm/min[/tex]
[tex]\frac{dh}{dt}=4\frac{dr}{dt}=4(\frac{\pi^3}{2})=2\pi^3 cm/min[/tex]
Volume of cylinder,V=[tex]\pi r^2 h[/tex]
Differentiate w.r.t.time
[tex]\frac{dV}{dt}=2\pi rh\frac{dr}{dt}+\pi r^2\frac{dh}{dt}[/tex]
[tex]\frac{dV}{dt}=2\pi (4)(12)(\frac{\pi^3}{2})+\pi (4)^2(2\pi^3)[/tex]
[tex]\frac{dV}{dt}=48\pi^4+32\pi^4[/tex]
[tex]\frac{dV}{dt}=80 \pi^4 cm^3/min[/tex]
Hence, the volume of the cylinder changing at the rate=[tex]80\pi^4 cm^3/min[/tex]
