Answer:
[tex]291111.1cm^3/min[/tex]
Step-by-step explanation:
We are given that
[tex]\frac{dV}{dt}_{out}=12000 cm^3/min[/tex]
Height of tank,h=6 m
Diameter of top,d=4 m
Radius,r=[tex]\frac{d}{2}=\frac{4}{2}=2 m[/tex]
[tex]\frac{dh}{dt}=20 cm/min[/tex]
[tex]\frac{r}{h}=\frac{2}{6}=\frac{1}{3}[/tex]
[tex]r=\frac{1}{3} h[/tex]
We have to find rate at which water is being pumped into the tank.
Volume of conical ,V=[tex]\frac{1}{3}\pi r^2 h[/tex]
[tex]V=\frac{1}{3}\pi(\frac{1}{3} h)^2h=\frac{1}{27}\pi h^3[/tex]
[tex]\frac{dV}{dt}=\frac{1}{9}\pi h^2(\frac{dh}{dt})[/tex]
h=2 m=200 cm
1m=100 cm
[tex]\frac{dV}{dt}_{in}-12000=\frac{1}{9}\pi(200)^2\times 20[/tex]
[tex]\frac{dV}{dt}_{in}=12000+\frac{1}{9}\pi (40000)\times 20[/tex]
[tex]\frac{dV}{dt}_{in}=291111.1cm^3/min[/tex]
