Calculating The Enclosed Area Of Parabolic Fences A Mathematical Approach
Farmer Jones and his wife, Dr. Jones, a mathematician, faced a challenge: how to build a fence to keep their sheep safe in the field. Dr. Jones, leveraging her mathematical expertise, proposed an intriguing solution using parabolic equations. This article delves into the mathematical problem they encountered and explores the steps to determine the area enclosed by the fences described by the equations y = 11x² and y = x² + 7. We will explore the concepts of parabolas, intersections, and integral calculus to find the solution. This scenario provides a practical application of mathematical principles, demonstrating how abstract concepts can be used to solve real-world problems. We will explore the importance of mathematical precision in agricultural planning, especially when dealing with resources and the safety of livestock.
Understanding the Problem
To begin, it's crucial to visualize the problem. Dr. Jones suggested fences that follow the shapes of parabolas, which are U-shaped curves defined by quadratic equations. The equations she proposed were y = 11x² and y = x² + 7. The first equation, y = 11x², represents a parabola that opens upwards, with its vertex (the lowest point) at the origin (0,0). The coefficient 11 stretches the parabola vertically, making it narrower. The second equation, y = x² + 7, also represents an upward-opening parabola, but its vertex is at the point (0,7). The addition of 7 shifts the parabola upwards by 7 units. Farmer Jones needs to know the precise area enclosed by these two parabolic fences to plan effectively for his sheep. This involves calculating the region where the two parabolas intersect and understanding the space contained within their boundaries. Understanding these parabolic equations and their graphical representation is the first step towards calculating the fenced area.
Dr. Jones’s mathematical approach ensures that the fenced area is both functional and efficient. By using parabolic shapes, she optimizes the use of fencing materials while maximizing the enclosed space for the sheep. This method also provides a visually appealing design, blending mathematical precision with the natural landscape. The problem highlights the intersection of mathematics and practical agriculture, showcasing how abstract concepts can lead to tangible solutions. This section will further examine the significance of accurately determining the enclosed area, which directly impacts the number of sheep the field can safely accommodate and the overall management of the farm.
Finding the Points of Intersection
The next step is to determine where these two parabolas intersect. These intersection points define the boundaries of the fenced area. To find them, we need to solve the two equations simultaneously. This means finding the x and y values that satisfy both equations. We can set the two equations equal to each other: 11x² = x² + 7. Now, we solve for x. Subtract x² from both sides: 10x² = 7. Divide both sides by 10: x² = 7/10. Take the square root of both sides: x = ±√[7/10]. This gives us two x-values, one positive and one negative, representing the two points where the parabolas intersect. To find the corresponding y-values, we can substitute these x-values back into either of the original equations. Let’s use y = x² + 7. For x = √[7/10], y = (√[7/10])² + 7 = 7/10 + 7 = 77/10. Similarly, for x = -√[7/10], y = (-√[7/10])² + 7 = 7/10 + 7 = 77/10. So, the points of intersection are (√[7/10], 77/10) and (-√[7/10], 77/10). These points are crucial because they define the limits of integration when we calculate the area between the curves.
These points of intersection are not just mathematical solutions; they represent the physical locations where the two fences meet. Accurately calculating these points ensures that the fence is built correctly and that the enclosed area is precisely what Farmer Jones needs. This step in the process underscores the importance of algebraic manipulation and solving equations in real-world applications. The precision achieved in finding these intersection points reflects Dr. Jones’s meticulous approach to problem-solving, blending theoretical mathematics with practical construction needs. This careful calculation will ultimately contribute to the safety and security of the sheep, making the mathematical effort a worthwhile endeavor.
Calculating the Area Enclosed
Now that we have the points of intersection, we can calculate the area enclosed by the two parabolas. This requires the use of integral calculus. The area between two curves, f(x) and g(x), from x = a to x = b, is given by the integral ∫[a, b] |f(x) - g(x)| dx. In our case, f(x) = x² + 7 and g(x) = 11x². The limits of integration, a and b, are the x-coordinates of the intersection points, which are -√[7/10] and √[7/10]. Since x² + 7 is greater than 11x² within the interval, we subtract 11x² from x² + 7. Thus, the integral becomes ∫[-√[7/10], √[7/10]] (x² + 7 - 11x²) dx. Simplify the integrand: ∫[-√[7/10], √[7/10]] (7 - 10x²) dx. Now, we evaluate the integral. The antiderivative of 7 is 7x, and the antiderivative of -10x² is -10/3 x³. So, the integral evaluates to [7x - 10/3 x³] from -√[7/10] to √[7/10].
Substituting the limits of integration, we get (7√[7/10] - 10/3 (√[7/10])³) - (7(-√[7/10]) - 10/3 (-√[7/10])³). This simplifies to 2 * (7√[7/10] - 10/3 (7/10)√[7/10]). Further simplification yields 2 * (7√[7/10] - 7/3 √[7/10]) = 2 * (14/3 √[7/10]). So, the area is 28/3 √[7/10] square units. This final calculation provides Farmer Jones with the precise area enclosed by the fence, allowing him to make informed decisions about the number of sheep he can safely accommodate. The use of integral calculus in this context highlights its power in solving practical problems involving areas and volumes.
Practical Implications and Conclusion
The calculated area, 28/3 √[7/10] square units, is a crucial piece of information for Farmer Jones. It allows him to determine the number of sheep he can safely graze within the fenced area. Overcrowding can lead to overgrazing, soil erosion, and health issues for the sheep. Therefore, knowing the precise area helps Farmer Jones manage his resources sustainably and ensure the well-being of his livestock. Dr. Jones’s mathematical approach not only provides an accurate measurement but also demonstrates a proactive and responsible approach to farming. This collaboration between Farmer Jones and Dr. Jones exemplifies how mathematical expertise can be applied to practical agricultural challenges, leading to more efficient and sustainable farming practices.
In conclusion, the problem of building a fence for the sheep is an excellent example of how mathematics, specifically parabolic equations and integral calculus, can be used to solve real-world problems. By finding the points of intersection and calculating the area enclosed by the parabolic fences, Farmer Jones and Dr. Jones ensured the safety and well-being of their sheep while optimizing the use of their land. This scenario underscores the importance of interdisciplinary thinking and the value of mathematical precision in practical applications. The detailed calculations and the strategic planning reflect a commitment to both mathematical rigor and agricultural sustainability, highlighting the profound impact of combining theoretical knowledge with practical needs. The success of this project serves as a testament to the power of mathematics in shaping and improving our world, one fence, and one flock of sheep at a time.
