A Tub Filled With 50 Quarts Of Water Empties At A Rate Of 2.5 Quarts Per Minute. Let \[$ W \$\] Represent The Quarts Of Water Left In The Tub, And \[$ T \$\] Represent Time In Minutes.$\[ \begin{tabular}{|c|c|} \hline \text{Time }

Introduction

In this article, we will explore a mathematical problem involving the rate at which a tub empties. The problem states that a tub filled with 50 quarts of water empties at a rate of 2.5 quarts per minute. We will use mathematical equations to model the situation and determine the amount of water left in the tub at any given time.

Mathematical Modeling

To model this situation, we can use the concept of a rate of change. The rate at which the tub empties is given as 2.5 quarts per minute, which we can represent as a negative rate of change. We can use the following equation to represent the situation:

$$\frac{dw}{dt} = -2.5$$

where $w$ represents the quarts of water left in the tub and $t$ represents time in minutes.

Initial Condition

We are given that the tub initially contains 50 quarts of water. This is our initial condition, which we can represent as:

$$w(0) = 50$$

Solving the Differential Equation

To solve the differential equation, we can integrate both sides with respect to time:

$$\int \frac{dw}{dt} dt = \int -2.5 dt$$

$$w(t) = -2.5t + C$$

where $C$ is the constant of integration.

Applying the Initial Condition

We can apply the initial condition to find the value of $C$:

$$w(0) = -2.5(0) + C$$

$$50 = C$$

Final Equation

Substituting the value of $C$ back into the equation, we get:

$$w(t) = -2.5t + 50$$

Graphical Representation

The equation $w(t) = -2.5t + 50$ represents a linear function, which we can graph to visualize the situation.

Discussion

The graph of the equation shows that the amount of water left in the tub decreases linearly over time. The rate at which the tub empties is constant at 2.5 quarts per minute.

Conclusion

In this article, we used mathematical equations to model the situation of a tub emptying at a rate of 2.5 quarts per minute. We solved the differential equation and applied the initial condition to find the final equation, which represents the amount of water left in the tub at any given time.

Example Problems

Problem 1

A tub contains 100 quarts of water and empties at a rate of 3 quarts per minute. Find the amount of water left in the tub after 10 minutes.

Solution

We can use the same equation $w(t) = -rt + C$ to solve this problem, where $r$ is the rate at which the tub empties and $C$ is the initial amount of water.

$$w(t) = -3t + 100$$

Substituting $t = 10$, we get:

$$w(10) = -3(10) + 100$$

$$w(10) = 70$$

Therefore, the amount of water left in the tub after 10 minutes is 70 quarts.

Problem 2

A tub contains 200 quarts of water and empties at a rate of 4 quarts per minute. Find the amount of water left in the tub after 15 minutes.

Solution

We can use the same equation $w(t) = -rt + C$ to solve this problem, where $r$ is the rate at which the tub empties and $C$ is the initial amount of water.

$$w(t) = -4t + 200$$

Substituting $t = 15$, we get:

$$w(15) = -4(15) + 200$$

$$w(15) = 140$$

Therefore, the amount of water left in the tub after 15 minutes is 140 quarts.

Applications

This problem has many real-world applications, such as:

• Calculating the amount of water left in a swimming pool after a certain amount of time
• Determining the amount of fuel left in a tank after a certain amount of time
• Calculating the amount of time it takes for a tank to empty

Limitations

This problem assumes that the rate at which the tub empties is constant, which may not always be the case in real-world situations. Additionally, this problem does not take into account any external factors that may affect the rate at which the tub empties.

Future Directions

Future research could involve exploring more complex scenarios, such as:

• A tub that empties at a rate that changes over time
• A tub that is filled at a rate that changes over time
• A tub that is affected by external factors, such as temperature or pressure changes.
  

Introduction

In our previous article, we explored a mathematical problem involving the rate at which a tub empties. We used mathematical equations to model the situation and determine the amount of water left in the tub at any given time. In this article, we will answer some frequently asked questions (FAQs) related to this problem.

Q&A

Q1: What is the rate at which the tub empties?

A1: The tub empties at a rate of 2.5 quarts per minute.

Q2: How much water is left in the tub after 10 minutes?

A2: To find the amount of water left in the tub after 10 minutes, we can use the equation $w(t) = -2.5t + 50$. Substituting $t = 10$, we get:

$$w(10) = -2.5(10) + 50$$

$$w(10) = 25$$

Therefore, the amount of water left in the tub after 10 minutes is 25 quarts.

Q3: How much water is left in the tub after 20 minutes?

A3: To find the amount of water left in the tub after 20 minutes, we can use the equation $w(t) = -2.5t + 50$. Substituting $t = 20$, we get:

$$w(20) = -2.5(20) + 50$$

$$w(20) = 10$$

Therefore, the amount of water left in the tub after 20 minutes is 10 quarts.

Q4: What is the initial amount of water in the tub?

A4: The initial amount of water in the tub is 50 quarts.

Q5: What is the rate of change of the amount of water in the tub?

A5: The rate of change of the amount of water in the tub is -2.5 quarts per minute.

Q6: How long will it take for the tub to empty completely?

A6: To find the time it takes for the tub to empty completely, we can set $w(t) = 0$ and solve for $t$:

$$0 = -2.5t + 50$$

$$2.5t = 50$$

$$t = 20$$

Therefore, it will take 20 minutes for the tub to empty completely.

Q7: What is the equation that represents the amount of water left in the tub at any given time?

A7: The equation that represents the amount of water left in the tub at any given time is $w(t) = -2.5t + 50$.

Q8: Can we use this equation to model other situations?

A8: Yes, we can use this equation to model other situations where a quantity is decreasing at a constant rate. For example, we can use this equation to model the amount of fuel left in a tank after a certain amount of time.

Conclusion

In this article, we answered some frequently asked questions related to the problem of a tub filled with 50 quarts of water emptying at a rate of 2.5 quarts per minute. We used mathematical equations to model the situation and determine the amount of water left in the tub at any given time. We hope that this article has been helpful in understanding this problem and its applications.

Additional Resources

• Mathematical Modeling
• Differential Equations
• Rate of Change

Related Articles

• A Tub Filled with 50 Quarts of Water Empties at a Rate of 2.5 Quarts per Minute
• Mathematical Modeling of a Tub Emptying at a Constant Rate
• Differential Equations and Rate of Change