Answer:
0 meters/minute
Explanation:
To find the rate at which the water level in the cylindrical tank is increasing when the plane passes directly over a radar station, we can use related rates. We'll need to find a relationship between the plane's position and the water level in the tank.
First, let's establish some variables:
Let h be the height of the water in the tank (in meters).
Let r be the radius of the tank, which is 5 meters.
The plane is flying horizontally at an altitude of 1 mile, which is equivalent to 1 mile = 1609.34 meters.
The plane's speed is increasing at a rate of 3 m/min.
The plane passes directly over a radar station, which is located directly above the center of the cylindrical tank.
Now, we need to find a relationship between the position of the plane and the water level in the tank. The plane's horizontal position (x) is related to its speed:
x = 3t,
where t is the time in minutes. The plane is moving horizontally, so its horizontal position increases linearly with time.
Next, let's find the relationship between the water level (h) in the tank and the volume of water in the tank. The volume of water in a cylindrical tank is given by the formula:
V = πr²h.
Now, differentiate both sides of this equation with respect to time (t):
dV/dt = π(2rh)(dh/dt).
We can rearrange this to find dh/dt:
dh/dt = (1 / (2πrh)) * dV/dt.
Now, we need to find dV/dt, the rate at which the volume of water is increasing. Since the tank is being filled with water, dV/dt is the rate of water inflow, and it's given as 500 m³/hour.
Convert this rate to m³/min (since we're using minutes for time):
500 m³/hour = (500/60) m³/min = 25/3 m³/min.
Now, we can calculate dh/dt when the plane passes directly over the radar station, i.e., when x = 0 (plane's horizontal position is 0).
At that moment, h = 0 because the water level hasn't risen yet.
So, dh/dt = (1 / (2πrh)) * dV/dt = (1 / (2π * 5 * 0)) * (25/3) = 0.
Therefore, the rate at which the height of the water is increasing when the plane passes over the radar station is 0 meters per minute. The plane's presence doesn't affect the water level in the tank at that instant.