What is the equation that represents Simon's rectangular garden's area in terms of its width? How does the area change with different width values?
The equation A(x) = -x(x-80) gives Simon's rectangular garden's area in terms of its width. It illustrates that the area increases until half of the fencing (80 meters) is used for its width, after which it decreases.
Understanding Simon's Rectangular Garden Area Function
Simon has 160 meters of fencing to build a rectangular garden. The area of the garden can be represented by a mathematical function A(x) = -x(x-80), where 'x' is the width of the garden.
The given function A(x) = -x(x-80) represents the area of the rectangular garden where 'x' is the width and '80-x' is the length. Simon's total fencing length is 160 meters, which means the perimeter of the garden is also 160 meters. This relationship is shown by the equation 2x + 2(80-x) = 160.
The area of a rectangle is calculated by multiplying its width and length, represented by x(80-x). In this specific function, the area is given as a negative value for some reason. To further analyze the function, it can be interpreted in the form of a quadratic equation y = ax² + bx + c. In this case, it's represented as A=0, B=160, C=-x².
The function shows that the area of the garden increases as the width grows until half of the fencing (80 meters) is utilized for the width. After reaching this point, the area starts to decrease as the width continues to increase.
For a deeper understanding of Mathematical Functions and how they can be applied in real-life scenarios like Simon's garden, you can explore more resources on the subject. Understanding the relationship between different variables and how they influence the overall outcome is essential in solving mathematical problems and optimizing solutions.
In conclusion, the equation A(x) = -x(x-80) provides valuable insights into maximizing the area of Simon's rectangular garden within the constraints of 160 meters of fencing. By analyzing the function and its implications, we can better comprehend the dynamics of area optimization in practical situations.