a farmer has available 1032 feet of fencing and wishes to enclose a rectangular area. If x represents the width of the rectangle for what value of x is the area the largest
A) 256.5 feet b) 258 feet
c) 256 feet d) 257 feet
To find the width (x) that will result in the largest area, we need to use the given information that the farmer has 1032 feet of fencing and the perimeter of a rectangle is equal to twice the width plus twice the length.
Let's denote the length of the rectangle as y.
Based on the given information, we can formulate the equation:
2(x + y) = 1032
Simplifying the equation, we get:
x + y = 516
To find the value of x that maximizes the area, we need to express the area (A) in terms of x and y.
The area of a rectangle is given by A = x * y.
Since we have the equation x + y = 516, we can write y = 516 - x.
Substituting this expression for y into the area equation, we get:
A = x * (516 - x)
Expanding the equation, we have:
A = 516x - x^2
To find the maximum value of the area, we need to find the vertex of the parabola represented by this equation. The x-coordinate of the vertex gives us the value of x that maximizes the area.
The x-coordinate of the vertex for a quadratic equation in the form -ax^2 + bx + c can be found using the formula:
x = -b / (2a)
In our case, a = -1, b = 516, and c = 0.
Plugging in these values, we have:
x = -516 / (2(-1))
x = 258
Therefore, the width (x) that maximizes the area is 258 feet.
So, the correct answer is (b) 258 feet.