Draining Trough Problem Solving Quadratic Equations Explained

In this article, we delve into a fascinating problem involving a draining trough, a scenario that elegantly demonstrates the application of quadratic functions in real-world contexts. We'll dissect the problem statement, explore the underlying mathematical concepts, and provide a comprehensive solution, ensuring a clear understanding for students and enthusiasts alike.

Problem Statement

Finn removes the plug from a trough to drain the water. The volume, in gallons, in the trough after it has been unplugged can be modeled by ${ f(x) = 12x^2 - 13x + 3 }$, where ${ x }$ is the time in minutes. Our objective is to determine which of the following equations will help us find the time it takes for the trough to completely drain.

This problem presents a classic application of quadratic functions, where the volume of water in the trough changes over time. The quadratic equation ${ f(x) = 12x^2 - 13x + 3 }$ represents this relationship, with ${ f(x) }$ denoting the volume of water at time ${ x }$. To find the time it takes for the trough to completely drain, we need to determine when the volume of water, ${ f(x) }$, becomes zero. This translates to solving the quadratic equation ${ 12x^2 - 13x + 3 = 0 }$.

Understanding the Quadratic Function

Before diving into the solution, it's crucial to grasp the properties of quadratic functions. A quadratic function is a polynomial function of degree two, generally expressed in the form ${ f(x) = ax^2 + bx + c }$, where ${ a }$, ${ b }$, and ${ c }$ are constants, and ${ a \neq 0 }$. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if ${ a > 0 }$ and downwards if ${ a < 0 }$.

In our problem, the quadratic function is ${ f(x) = 12x^2 - 13x + 3 }$. Here, ${ a = 12 }$, ${ b = -13 }$, and ${ c = 3 }$. Since ${ a = 12 > 0 }$, the parabola opens upwards. The roots of the quadratic equation, which are the values of ${ x }$ for which ${ f(x) = 0 }$, represent the points where the parabola intersects the x-axis. In the context of our problem, these roots correspond to the times when the trough is completely drained.

Methods for Solving Quadratic Equations

There are several methods for solving quadratic equations, each with its own advantages and suitability depending on the specific equation:

1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If we can factor the quadratic equation, we can easily find the roots by setting each factor equal to zero.
2. Quadratic Formula: The quadratic formula is a general formula that provides the solutions to any quadratic equation. It is given by: ${ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }$ This formula is particularly useful when factoring is not straightforward.
3. Completing the Square: This method involves manipulating the quadratic equation to form a perfect square trinomial, which can then be easily solved.

Solving the Draining Trough Problem

Now, let's apply these concepts to solve the draining trough problem. We need to find the equation that will help us determine the time it takes for the trough to completely drain. As we established earlier, this means finding the values of ${ x }$ for which ${ f(x) = 0 }$. Therefore, the equation we need to solve is:

${ 12x^2 - 13x + 3 = 0 }$

This equation represents the scenario where the volume of water in the trough is zero, indicating that the trough is completely drained. Solving this equation will give us the time(s) at which this occurs.

Applying the Quadratic Formula

To solve the equation ${ 12x^2 - 13x + 3 = 0 }$, we can use the quadratic formula. Plugging in the values ${ a = 12 }$, ${ b = -13 }$, and ${ c = 3 }$ into the formula, we get:

${ x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(12)(3)}}{2(12)} }$

${ x = \frac{13 \pm \sqrt{169 - 144}}{24} }$

${ x = \frac{13 \pm \sqrt{25}}{24} }$

${ x = \frac{13 \pm 5}{24} }$

This gives us two possible solutions:

${ x_1 = \frac{13 + 5}{24} = \frac{18}{24} = \frac{3}{4} }$

${ x_2 = \frac{13 - 5}{24} = \frac{8}{24} = \frac{1}{3} }$

Therefore, the solutions are ${ x = \frac{3}{4} }$ and ${ x = \frac{1}{3} }$ minutes. These values represent the times at which the trough is completely drained.

Interpreting the Solutions

In the context of the problem, both solutions, ${ x = \frac{3}{4} }$ and ${ x = \frac{1}{3} }$ minutes, are valid. This implies that the trough is completely drained at two different times. This might seem counterintuitive at first, but it's important to remember that the quadratic function models the volume of water in the trough, and the parabola's shape allows for two points where the volume is zero.

Graphical Representation

To further illustrate this, we can visualize the graph of the quadratic function ${ f(x) = 12x^2 - 13x + 3 }$. The parabola intersects the x-axis at two points, corresponding to the solutions ${ x = \frac{1}{3} }$ and ${ x = \frac{3}{4} }$. The portion of the parabola below the x-axis represents negative volume, which is not physically meaningful in this context. Therefore, we only consider the positive values of ${ x }$ where the parabola intersects the x-axis.

Conclusion

In conclusion, the equation ${ 12x^2 - 13x + 3 = 0 }$ is the key to finding the time it takes for the trough to completely drain. By solving this quadratic equation, we determined that the trough is drained at ${ x = \frac{1}{3} }$ and ${ x = \frac{3}{4} }$ minutes. This problem highlights the practical application of quadratic functions in modeling real-world scenarios. Understanding the properties of quadratic functions and mastering the techniques for solving quadratic equations are essential skills for students and anyone interested in mathematical modeling.

Key Takeaways

• Quadratic functions can model real-world scenarios, such as the volume of water in a draining trough.
• The roots of a quadratic equation represent the times when the trough is completely drained.
• The quadratic formula is a powerful tool for solving quadratic equations.
• Interpreting the solutions in the context of the problem is crucial for understanding the results.

This comprehensive analysis of the draining trough problem provides a solid foundation for understanding quadratic functions and their applications. By mastering these concepts, you'll be well-equipped to tackle similar problems and appreciate the power of mathematics in describing the world around us.

Which equation will help find the time it takes for the trough to completely drain, given the volume in gallons is modeled by ${ f(x) = 12x^2 - 13x + 3 }$, where ${ x }$ is time in minutes?