Respuesta :
Answer:
[tex]\displaystyle \frac{121}{8}\; {\rm ft^{2}}[/tex] (Approximately [tex]15.1\; {\rm ft^{2}}[/tex].)
Step-by-step explanation:
In the rectangular pen in this question, three of the sides would be made of the fence while the other would be the barn. Assume that the length of the side parallel to the barn is [tex]x\; {\rm ft}[/tex]. The total length of the other two sides would [tex](11 - x)\; {\rm ft}[/tex], and the length of each side would be [tex]((11 - x) / 2)\; {\rm ft}[/tex].
The total area of this rectangular pen would be:
[tex]\displaystyle \left(\frac{11 - x}{2}\right)\, (x)\; {\rm ft^{2}}[/tex].
Simplify to obtain an expression for the area of the pen (in [tex]{\rm ft^{2}}[/tex]):
[tex]\displaystyle -\frac{1}{2}\, x\, (x - 11)[/tex].
This expression represents a quadratic function (a parabola) of [tex]x[/tex]. Because the leading coefficient [tex](-1/2)[/tex] is negative, this parabola would open downward. The maximum value of this expression would be achieved at the vertex of this parabola.
By the factor theorem, the two roots of this parabola would be [tex]x = 0[/tex] and [tex]x = 11[/tex]. The vertex of the parabola is at the middle of the two roots: [tex]x = (0 + 11) / 2 = (11/2)[/tex].
In other words, the area of this rectangular pen is maximized when the length of the side opposite to the barn is [tex](11/2)\; {\rm ft}[/tex]. The area of this pen would be:
[tex]\displaystyle -\frac{1}{2}\, x\, (x - 11) = -\frac{1}{2}\, \left(\frac{11}{2}\right)\, \left(\frac{11}{2} - 11\right) = \frac{121}{8}[/tex].
In other words, the maximum possible area of this pen would be [tex](121/8)\; {\rm ft^{2}}[/tex].
Final answer:
To find the maximum area of the rectangular pen, use the given information about the fence length and the barn as one of the sides. The maximum area is 15.125 square feet.
Explanation:
To find the maximum area of the rectangular pen, we can use the given information that the farmer has 11 feet of fence and wants to use the barn as one of the sides. Let's assume the length of the rectangular pen is x feet. Since the barn will be one side, the perimeter of the rectangular pen will be x + 2w, where w is the width of the pen. The equation representing the perimeter is: x + 2w = 11.
Solving this equation for w, we get w = (11 - x)/2. The area of the rectangle is given by A = xw. Substituting the value of w, we have [tex]A = x((11 - x)/2) = (11x - x^2)/2.[/tex]
To find the maximum area, we need to find the value of x that maximizes the expression for A. Taking the derivative of A with respect to x and setting it equal to 0, we can solve for x. The maximum area occurs at x = 11/2, which is the length of the rectangular pen. Substituting x = 11/2 back into the equation for w, we find w = (11 - 11/2)/2 = 11/4.
Therefore, the maximum area possible for the pen is A = (11/2)(11/4) = 121/8 = 15.125 square feet.
