Tank-Modeling Problem: Solving for Salt Amount with Differential Equations

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In summary, the problem involves a tank initially containing 30 gallons of water and 5 pounds of salt. A salt solution is continuously pumped into the tank at a rate of 3 gallons per minute, while the well-mixed solution is pumped out at the same rate. At time t = 10, a hopper begins dumping salt into the tank at a rate of 5 pounds per minute. At time t = 20, the hopper suddenly stops and at time t = 30, it ruptures and dumps 10 pounds of salt into the tank all at once. The task is to find a function that represents the amount of salt in the tank at any given time. This can be done by using the differential equation for
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Homework Statement



A large tank initially contains 30 gallons of water and 5 pounds of salt. A salt solution consisting of 1 lb/gal is pumped into the tank at a rate of 3 gal/min, and the well-mixed solution is pumped out at the same rate. At time t = 10 an large hopper begins dumping salt into the tank at a rate of 5 lbs/min. At time t = 20, the hopper starts making an alarming noise, and the flow of salt stops. Then at time t = 30, the hopper ruptures and dumps 10 lbs of salt into the tank all at once. Find the function which gives the amount of salt in the tank at time t

Homework Equations



The differential equation for the amount of salt in tank= rate in - rate out

The Attempt at a Solution


I tried to use a heaviside function as well as the dirac delta at the end of the equation. I got stuck trying to set it up with heaviside step functions
 
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Related to Tank-Modeling Problem: Solving for Salt Amount with Differential Equations

1. What is the "Tank-modeling problem"?

The "Tank-modeling problem" is a mathematical problem that involves determining the optimal strategy for filling and draining a tank in order to minimize the amount of time it takes to reach a desired volume or level.

2. Why is the "Tank-modeling problem" important?

The "Tank-modeling problem" is important because it has real-world applications in areas such as engineering, economics, and environmental management. It can help in optimizing resource usage and minimizing costs.

3. What are the key variables in the "Tank-modeling problem"?

The key variables in the "Tank-modeling problem" include the initial and desired tank volumes, the rate of inflow and outflow, and any constraints or limitations on the system, such as maximum flow rate or minimum volume requirements.

4. What are the different approaches to solving the "Tank-modeling problem"?

There are several approaches to solving the "Tank-modeling problem", including analytical methods such as differential equations and optimization algorithms, as well as numerical methods such as numerical integration and simulation techniques.

5. What are some real-world examples of the "Tank-modeling problem"?

The "Tank-modeling problem" can be found in various real-world scenarios, such as managing water levels in reservoirs or designing fuel tanks for spacecraft. It can also be applied to manufacturing processes, such as filling and emptying tanks in chemical plants or breweries.

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