Water drains from a tank. Write a differential equation.

The volume of a cone is V = (1/3)πr²h, where r is the radius of the base and h is the height. The height is related to the depth of water y by the equation h = (1/3)r². Therefore, the volume of water in the tank can be expressed as V = (1/3)πr²((1/3)r²-y). Differentiating with respect to time gives dV/dt = (1/3)πr²(dy/dt). Since the rate of change of volume is equal to the rate at which water is draining, we can write dV/dt = -k. Substituting this into the equation, we get (1
  • #1
cp255
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2. Water drains out of an inverted conical tank at a rate proportional to the depth y of water in the tank. Write a differential equation for y as a function of time.

My answer was dy/dt = ky.

This was from a weekly homework set where there were only 5 problems. I feel like I am missing something since my answer is too simple and it didn't take very much work. Am I missing something?
 
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  • #2
cp255 said:
2. Water drains out of an inverted conical tank at a rate proportional to the depth y of water in the tank. Write a differential equation for y as a function of time.

My answer was dy/dt = ky.

This was from a weekly homework set where there were only 5 problems. I feel like I am missing something since my answer is too simple and it didn't take very much work. Am I missing something?

Yes, that's too simple. The rate at which the water drains is related to the rate of change of the volume of the tank.
 

1. What is the meaning of a "water drains from a tank" scenario?

A "water drains from a tank" scenario refers to a situation where water is being removed from a tank, typically due to gravity or a pumping mechanism.

2. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model and analyze various systems in fields such as physics, engineering, and economics.

3. How is a differential equation related to a "water drains from a tank" scenario?

In this scenario, a differential equation can be used to model the rate at which the water level in the tank decreases over time. This is done by setting up an equation that relates the volume of water in the tank to the rate of change of the water level.

4. Can you provide an example of a differential equation for a "water drains from a tank" scenario?

Yes, an example of a differential equation for this scenario is:
dV/dt = -kV, where dV/dt represents the rate of change of the volume of water in the tank, k is a constant representing the rate at which water is draining from the tank, and V is the volume of water in the tank.

5. How can a differential equation for a "water drains from a tank" scenario be solved?

The differential equation can be solved using various techniques, such as separation of variables, integrating factors, or using numerical methods. The specific method used will depend on the complexity of the equation and the desired level of accuracy.

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