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
please show work!
i need help fast
To find the value of width (x) that maximizes the area, we need to express the area as a function of x and then find the value of x that corresponds to the maximum point.
Let's start by defining the perimeter (P) of the rectangle. The perimeter is given as 1032 feet, and for a rectangle, the perimeter is calculated as:
P = 2*(length + width)
Since we're given the total amount of fencing available, we can express the length (L) in terms of x:
L = 1032 - 2x
Next, we'll define the area (A) of the rectangle. The area of a rectangle is calculated as:
A = length * width = L * x
Substituting the expression for L, we have:
A = (1032 - 2x) * x = 1032x - 2x^2
To find the value of x that maximizes the area, we need to take the derivative of the area function with respect to x and set it equal to zero. Then, solve the resulting equation for x.
dA/dx = 1032 - 4x = 0
Solving this equation:
1032 - 4x = 0
4x = 1032
x = 1032/4
x = 258
So, the value of x that maximizes the area is x = 258.
Therefore, the correct answer is b) 258 feet.