Final answer:
The height of the water in the tank drops at approximately 0.00199 cm/sec when it is being drained at a rate of 25 cm³/sec.
Step-by-step explanation:
To find out how fast the height of water in the tank drops when the water is being drained at a rate of 25 cm³/sec, we use the formula for the volume of a cylinder, which is Volume = π × radius² × height. In this case, the radius of the tank is 20 cm.
Since the rate at which the water level changes is related to the rate at which the water is being drained, we can express this with the following derivative relation: dh/dt = dV/dt ÷ (1 / (π × radius²)), where dh/dt is the change in water height over time, and dV/dt is the change in volume over time, which in this case is the drainage rate.
Plugging the given values into the equation, we get:
dh/dt = 25 cm³/sec ÷ (1 / (π × (20 cm)²))
After calculating, we find that the water level drops at a rate of approximately 0.00199 cm/sec or 1.99×10⁻³ cm/sec.