Final answer:
To calculate the rate at which water is being pumped into the tank, we use the volume formula for a cone and apply the concept of related rates in calculus to set up an equation that accounts for the given rates of change and the tank's dimensions.
Step-by-step explanation:
To find the rate at which water is being pumped into the tank, we need to use the concept of related rates in calculus. The volume of water in the cone-shaped tank can be described by the formula V = (1/3)πr^2h, where V is the volume, r is the radius of the water surface at a given height h, and π is Pi, approximately 3.14159.
Given that the diameter of the tank is 3 meters, the radius r is 1.5 meters. However, when the water level is at 2.5 meters (which is our h in this case), the radius r will be in proportion to the height since it's a cone. We can set up a ratio using similar triangles: (radius at 2.5 meters) / 2.5 = 1.5 / 12. From this, we can find the current radius of the water surface.
Next, we differentiate the volume with respect to time to get dV/dt = πr^2(dh/dt) + 2πrh(dr/dt), taking into account the change in height dh/dt and the change in radius dr/dt. Since water is leaking and being pumped at the same time, the rate of change of the volume (dV/dt) will be the sum of the rates of water being added and water leaking, and we also know dh/dt (the rate at which water level is rising).
The final step is to substitute the known values and solve for the rate of water being pumped in. We have the rate of leakage (6000 cm^3/min), the rate the water level is rising (22 cm/min), and the tank's dimensions to find the rate at which water is being pumped in as dV/dt.