A Tank Initially Contains 50 Gallons Of Water At Time T = 0 T=0 T = 0 Minutes. Water Starts Being Pumped Into The Tank At The Same Time A Different Pipe Begins Draining It.The Rate At Which Water Is Being Pumped Into The Tank Is Modeled By The Function

Introduction

In this article, we will delve into a classic problem in mathematics involving a tank that is being filled and drained simultaneously. The problem is a great example of how mathematical modeling can be used to understand and analyze real-world situations. We will explore the mathematical concepts involved in this problem, including rates of change, accumulation, and differential equations.

The Problem

A tank initially contains 50 gallons of water at time $t=0$ minutes. Water starts being pumped into the tank at the same time a different pipe begins draining it. The rate at which water is being pumped into the tank is modeled by the function $f(t) = 2t + 1$ gallons per minute, where $t$ is the time in minutes. The rate at which water is being drained from the tank is modeled by the function $g(t) = t^2 + 1$ gallons per minute.

Mathematical Modeling

To understand the situation, we need to model the rate at which the water level in the tank is changing. Let $V(t)$ be the volume of water in the tank at time $t$. The rate at which the water level is changing is given by the derivative of $V(t)$ with respect to time, which we denote as $\frac{dV}{dt}$.

The rate at which water is being pumped into the tank is given by the function $f(t) = 2t + 1$ gallons per minute. This means that the rate at which the water level is increasing due to the pumping is given by $f(t)$.

On the other hand, the rate at which water is being drained from the tank is given by the function $g(t) = t^2 + 1$ gallons per minute. This means that the rate at which the water level is decreasing due to the draining is given by $-g(t)$, since the draining is reducing the water level.

The Differential Equation

The rate at which the water level is changing is given by the derivative of $V(t)$ with respect to time, which we denote as $\frac{dV}{dt}$. We can write an equation for this rate as follows:

$$\frac{dV}{dt} = f(t) - g(t)$$

Substituting the expressions for $f(t)$ and $g(t)$, we get:

$$\frac{dV}{dt} = (2t + 1) - (t^2 + 1)$$

Simplifying the expression, we get:

$$\frac{dV}{dt} = -t^2 + 2t$$

Solving the Differential Equation

To solve the differential equation, we need to find the function $V(t)$ that satisfies the equation. We can do this by integrating both sides of the equation with respect to time.

$$V(t) = \int (-t^2 + 2t) dt$$

Evaluating the integral, we get:

$$V(t) = -\frac{t^3}{3} + t^2 + C$$

where $C$ is the constant of integration.

Initial Condition

We are given that the tank initially contains 50 gallons of water at time $t=0$ minutes. This means that the initial condition is:

$$V(0) = 50$$

Substituting this into the equation for $V(t)$, we get:

$$50 = -\frac{0^3}{3} + 0^2 + C$$

Simplifying the equation, we get:

$$C = 50$$

The Final Solution

Substituting the value of $C$ into the equation for $V(t)$, we get:

$$V(t) = -\frac{t^3}{3} + t^2 + 50$$

This is the final solution to the problem.

Conclusion

In this article, we explored a classic problem in mathematics involving a tank that is being filled and drained simultaneously. We used mathematical modeling to understand the situation and derived a differential equation that describes the rate at which the water level is changing. We then solved the differential equation to find the function $V(t)$ that describes the volume of water in the tank at time $t$. The final solution is a quadratic function that describes the volume of water in the tank at any time $t$.

Applications

This problem has many real-world applications, including:

• Hydraulics: The problem can be used to model the flow of water in pipes and tanks.
• Chemical Engineering: The problem can be used to model the flow of chemicals in reactors and tanks.
• Environmental Engineering: The problem can be used to model the flow of pollutants in rivers and lakes.

Future Work

There are many possible extensions to this problem, including:

• Multiple pipes: Adding multiple pipes that fill and drain the tank at different rates.
• Non-linear rates: Using non-linear functions to model the rates at which the water is being pumped in and drained.
• Time-dependent rates: Using time-dependent functions to model the rates at which the water is being pumped in and drained.

Q&A: A Tank Filling and Draining Problem

In this article, we will continue to explore the tank filling and draining problem by answering some common questions that may arise.

Q: What is the initial condition of the tank?

A: The initial condition of the tank is that it contains 50 gallons of water at time $t=0$ minutes.

Q: What are the rates at which water is being pumped into and drained from the tank?

A: The rate at which water is being pumped into the tank is given by the function $f(t) = 2t + 1$ gallons per minute. The rate at which water is being drained from the tank is given by the function $g(t) = t^2 + 1$ gallons per minute.

Q: How do we model the rate at which the water level in the tank is changing?

A: We model the rate at which the water level in the tank is changing by using the derivative of the volume of water in the tank with respect to time, which we denote as $\frac{dV}{dt}$.

Q: What is the differential equation that describes the rate at which the water level is changing?

A: The differential equation that describes the rate at which the water level is changing is given by:

$$\frac{dV}{dt} = f(t) - g(t)$$

where $f(t)$ is the rate at which water is being pumped into the tank and $g(t)$ is the rate at which water is being drained from the tank.

Q: How do we solve the differential equation?

A: We solve the differential equation by integrating both sides of the equation with respect to time. This gives us the function $V(t)$ that describes the volume of water in the tank at time $t$.

Q: What is the final solution to the problem?

A: The final solution to the problem is the function $V(t) = -\frac{t^3}{3} + t^2 + 50$ that describes the volume of water in the tank at time $t$.

Q: What are some real-world applications of this problem?

A: Some real-world applications of this problem include:

• Hydraulics: The problem can be used to model the flow of water in pipes and tanks.
• Chemical Engineering: The problem can be used to model the flow of chemicals in reactors and tanks.
• Environmental Engineering: The problem can be used to model the flow of pollutants in rivers and lakes.

Q: What are some possible extensions to this problem?

A: Some possible extensions to this problem include:

• Multiple pipes: Adding multiple pipes that fill and drain the tank at different rates.
• Non-linear rates: Using non-linear functions to model the rates at which the water is being pumped in and drained.
• Time-dependent rates: Using time-dependent functions to model the rates at which the water is being pumped in and drained.

Conclusion

In this article, we have explored the tank filling and draining problem by answering some common questions that may arise. We have also discussed some real-world applications of this problem and possible extensions to this problem. By exploring these extensions, we can gain a deeper understanding of the mathematical concepts involved in this problem and develop new mathematical models that can be used to analyze real-world situations.

Glossary

• Differential equation: An equation that involves the derivative of a function.
• Derivative: A measure of how a function changes as its input changes.
• Integral: A measure of the accumulation of a function over a given interval.
• Volume: The amount of space occupied by a three-dimensional object.
• Rate: A measure of how quickly something is happening.

References

• [1]: "Differential Equations and Dynamical Systems" by Lawrence Perko.
• [2]: "Mathematical Modeling: A Case Study Approach" by James T. Sandefur.
• [3]: "Hydraulics and Fluid Mechanics" by John F. Kennedy.

Note: The references provided are for informational purposes only and are not necessarily related to the specific problem being discussed.