Final answer:
To find the width that gives the maximum area, we can use calculus. We find the critical points, check the second derivative, and determine that the maximum occurs at x = 5. The correct answer is option A.
Step-by-step explanation:
To find the width that gives the maximum area, we need to find the value of x that maximizes the function A = 10x - x^2. The first step is to find the critical points by taking the derivative of A with respect to x and setting it equal to zero. So, dA/dx = 10 - 2x = 0.
Solving for x, we get x = 5. This is a critical point, but we need to check if it is a maximum or minimum. To do this, we can check the second derivative. Taking the derivative of dA/dx with respect to x, we get d^2A/dx^2 = -2.
Since the second derivative is negative, this indicates that x = 5 corresponds to the maximum value of the function. Therefore, the width that gives the maximum area is 5 meters.
The question asks for the width that gives the maximum area for a rectangular dog pen with a given amount of fencing and a quadratic function to model the area. We are given a function A = 10x - x^2, where A is the area in square meters and x is the width in meters. To find the width that gives the maximum area, we need to find the vertex of the parabola represented by the quadratic function, since the vertex will give us the maximum or minimum point of the function and we know the parabola opens downwards because the coefficient of x^2 is negative.
To find the vertex, we can use the vertex formula x = -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. In this function, a = -1 and b = 10. So, x = -10/(2*(-1)) = 5. So, the width that gives the maximum area is 5 meters.
It is important to apply proper quadratic function analysis to solve for the dimensions that yield the maximum area, specially when dealing with optimization problems in mathematics.