Final answer:
To find the rate at which water is being pumped in, we can use the formula for the volume of a cone and differentiate it with respect to time. The rate at which water is being pumped in is equal to 36 times the rate of change of the height of the water in the tank.
Step-by-step explanation:
To calculate the rate at which water is being pumped into the tank, we can use the formula for the volume of a cone. The volume of a cone is given by V = (1/3)πr^2h, where r is the radius of the cone's base and h is the height of the cone. In this case, we are given that the top diameter of the cone is 12 meters, so the radius (r) is half of that, which is 6 meters. We are also given that the height (h) is 8 meters.
Substituting these values into the formula, we get V = (1/3)π(6^2)(8) = 301.59 cubic meters. To find the rate at which water is being pumped in, we need to find how fast the volume is changing with respect to time. This is the derivative of the volume with respect to time, which is dV/dt.
Since the volume is changing as the height of the water in the tank increases, the rate at which water is being pumped in is equal to the rate of change of the volume with respect to time. So, we can differentiate the volume formula to find dV/dt.
The derivative of V = (1/3)πr^2h with respect to t is dV/dt = (1/3)π(2rh(dr/dt) + r^2(dh/dt)).
The values we know are r = 6 meters, h is the height of the water in the tank, and dh/dt is the rate at which the height of the water is changing. We need to find dr/dt, the rate at which the radius is changing. But since the diameter is given to be constant at 12 meters, the radius does not change, and thus the rate of change of the radius (dr/dt) is 0.
Therefore, the rate at which water is being pumped in when the water level is at a certain height is given by dV/dt = (1/3)π(12h)(0) + 36(dh/dt) = 36(dh/dt).