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
area= L*width
perimeter= 1032=2L+2W or
L= 516-W
Area= (516-W)W
take the derivative, set it to zero, and solve for W.
To find the value of x that will maximize the area, we need to use the formula for the perimeter of a rectangle:
Perimeter = 2 * Length + 2 * Width
Given that the total length of fencing is 1032 feet, we have:
1032 = 2 * Length + 2 * Width
Simplifying this equation, we get:
Length + Width = 516
To find the area of the rectangle, we use the formula:
Area = Length * Width
Since Length + Width = 516, we can rewrite the area formula as:
Area = (516 - Width) * Width
To find the value of x (Width) that will maximize the area, we can take the derivative of the area equation with respect to Width and set it equal to 0:
d(Area)/d(Width) = 0
d((516 - Width) * Width)/d(Width) = 0
Now we can solve for Width:
516 - 2 * Width = 0
2 * Width = 516
Width = 258 feet
Therefore, the value of x that will maximize the area is 258 feet (option b).
To find the value of x that maximizes the area of the rectangular enclosure, we need to understand the relationship between the dimensions of the rectangle and the area.
Let's denote the width of the rectangle as x and the length as y. According to the problem, the perimeter of the rectangle is 1032 feet, which means that the sum of all the sides is 1032. Therefore, we have the equation:
2x + 2y = 1032
We want to maximize the area, which is given by the formula A = x * y.
To solve for y, we can rewrite the equation above as:
2y = 1032 - 2x
y = (1032 - 2x)/2
y = 516 - x
Substituting this expression for y into the area formula gives:
A = x * (516 - x)
A = 516x - x^2
To find the value of x that maximizes the area, we can take the derivative of A with respect to x and set it equal to zero. Let's differentiate A:
dA/dx = 516 - 2x
Setting dA/dx equal to zero and solving for x:
516 - 2x = 0
2x = 516
x = 258
Therefore, the width of the rectangle that maximizes the area is x = 258 feet.
Answer: b) 258 feet