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
Let L=total length of fencing.
x=width
(L/2-x)=length
A(x)=Area=x(L/2-x)
Differentiate A with respect to x, and equate A'(x) to zero and solve for x=x0 which gives the maximum or minimum area.
Differentiate A'(x) again to get A"(x).
Confirm that A"(x0) is negative for a maximum (and positive for a minimum).
You should find that x0=width=length.
Post your answer for a check if you wish.
is it 258
Yes, that's correct.
To find the value of x that maximizes the area of the rectangular enclosure, we need to use the given information about the amount of fencing available.
Let's assume that the length of the rectangle is L and the width is x. Since we're dealing with a rectangular enclosure, we need to account for both the length and width when calculating the perimeter.
The perimeter of the rectangle can be calculated by adding up all the sides:
Perimeter = 2 * length + 2 * width
Given that the total available fencing is 1032 feet, we can write the equation:
1032 = 2L + 2x
Simplifying the equation, we get:
L = 516 - x
Now, we can calculate the area of the rectangle, which is equal to the length multiplied by the width:
Area = length * width
Substituting the expression for length (L) we found earlier:
Area = (516 - x) * x
To find the value of x that maximizes the area, we can differentiate the equation with respect to x and set it equal to zero:
d(Area)/dx = 0
Differentiating the equation:
d(Area)/dx = (516 - x) * 1 + (-1) * x
Setting it equal to zero:
0 = 516 - 2x
2x = 516
x = 516/2
x = 258
Therefore, the value of x that maximizes the area is 258 feet.
So, the correct answer is b) 258 feet.