There is 1116 meters of fencing available to enclose a field. The field is to be divided into two sections by a fence in the middle. What is the maximum area of the field that can be enclosed?
A=L xW
2w+3L=1116
3L=1116-2w
l=1116-2w/3
my final answer came out to 51894
can you check my work please and thank you
correct.
To save some work in the future, know that for this kind of problem, the maximum area is when the fencing is divided equally among lengths and widths. 1116 = 558*2
so since you have two lengths, L = 558/2 = 279
there are 3 widths, so W = 559/3 = 186
279*186 = 51894
L = (1116-2w) / 3 I think
A = w * (1116-2w) / 3
3 A = 1116 w - 2 w^2
Find vertex of that parabola, I am going to cheat and use calculus
d(3A) /dw = 0 at max = 1116 - 4 w
so
4 w = 1116
w = 279
then L = (1116-2w) / 3 = 186
so
A = 51894
we agree
To solve this problem, you need to maximize the area of the field given a specific amount of fencing.
Let's denote the width of one section of the field as "w" and the length of one section as "L". We need to divide the total fencing equally between the two sections, so we'll divide the available fencing by 2: (1116 meters / 2) = 558 meters.
Now, we can express the perimeter of the field as the sum of the three sides: 2w + 3L = 558.
To isolate one of the variables, let's solve this equation for L:
3L = 558 - 2w.
Dividing both sides by 3 gives: L = (558 - 2w) / 3.
Now, we can substitute this expression for L into the area formula, A = Lw:
A = [(558 - 2w) / 3] * w.
To find the maximum area, we'll take the derivative of A with respect to w, set it equal to 0, and solve for w.
dA/dw = (558 - 2w) / 3 - w/3 = (558 - 3w) / 3.
Setting this equal to 0:
(558 - 3w) / 3 = 0.
Solving for w, we get: 558 - 3w = 0.
3w = 558.
w = 558 / 3.
w = 186 meters.
Now substitute w = 186 into the equation for L:
L = (558 - 2(186)) / 3.
L = (558 - 372) / 3.
L = 186 / 3.
L = 62 meters.
Finally, calculate the area of one section: A = Lw = 62 * 186 = 11532 square meters.
Since we have two sections, the maximum area that can be enclosed is 2 * 11532 = 23064 square meters.
Therefore, the correct answer is 23064, not 51894.