Answer:
approximately 0.0255 meters per minute.
Step-by-step explanation:
To find the rate at which the water level is increasing, we need to determine the rate of change of the height of the water in the cylindrical tank with respect to time.
Let's denote the height of the water in the tank as h(t) at time t, where t is measured in minutes. The volume V of water in the tank at time t is given by the formula for the volume of a cylinder:
V(t) = πr^2h(t),
where r is the radius of the tank (5m in this case). The rate at which the water level is increasing is equal to the rate at which the volume is increasing with respect to time, which we can find by differentiating the volume equation with respect to time:
dV(t)/dt = 2πr^2dh(t)/dt.
We are given that the rate of water flow into the tank is 4m^3/min, which means that the volume of water is increasing at a constant rate. Therefore, we can write:
dV(t)/dt = 4m^3/min.
Substituting the values into the equation, we have:
2π(5^2)dh(t)/dt = 4.
Simplifying the equation further:
50πdh(t)/dt = 4,
dh(t)/dt = 4/(50π),
dh(t)/dt ≈ 0.0255 m/min.
Therefore, the water level is increasing at a rate of approximately 0.0255 meters per minute.