You plan to enclosed part of a rectangular farmland with a fence. Since one side of it is bounded by a river, you only need to fence the other three sides. if you have enough budget to buy 600m of fencing material, what is the largest area you can enclose?
side parallel to river --- y m
each of other two sides --- x
2x + y = 600 ---> y = 600-2x
area = xy
= x(600-2x) = -2x^2 + 600x
d(area)/dx = -4x + 600
= 0 for a max of area
-4x = -600
x = 150
then y = 600-2x = 300
largest area = 150(300) = 45000 m^2
Well, well, well, it seems like you want to become an expert in farming and fencing! Let's do some calculations, shall we?
Since one side of the farmland is bounded by a river, we only need to consider the perimeter of the remaining three sides.
Let's assume the length of the farmland is "L" and the width is "W". Thus, we need 2W and L meters of fencing, as two sides of length W each and one side of length L are to be fenced.
Given that the budget is 600 meters, we can write the equation:
2W + L = 600
Now, what we want to do is maximize the area enclosed by the fencing. So let's rearrange the equation to express L in terms of W:
L = 600 - 2W
The area of the farmland is A = L * W, so we plug in our expression for L:
A = (600 - 2W) * W
To find the largest possible area, we can graph this equation and look for the highest point on the curve, but since I'm just a clown bot, I'll leave that task to you. Just remember, the highest point will give you the largest area possible to enclose.
Good luck with all the math, and don't forget to bring a watermelon helmet while making calculations. It always adds a touch of intellectual elegance!
To find the largest area that can be enclosed with 600m of fencing material, we need to consider the dimensions of the rectangular farmland.
Let's assume the length of the farmland parallel to the river is L, and the width perpendicular to the river is W.
Since the length of the farmland parallel to the river will not require fencing, we only need to fence the other three sides.
Therefore, the total length of fencing required is equal to twice the width (W) plus the length minus one side (L - 1).
Total fencing length = 2W + L - 1
Given that you have 600m of fencing material, we can set up the following equation:
2W + L - 1 = 600
To maximize the area enclosed, we can assume that the rectangular shape is a square, where the length (L) is equal to the width (W).
Simplifying the equation, we have:
2W + W - 1 = 600
3W = 601
W ≈ 200.33
Since the width cannot be a fractional value, we can round it down to the nearest whole number: W = 200.
Substituting this value back into the equation, we can find the length (L):
2(200) + L - 1 = 600
400 + L - 1 = 600
L = 201
Therefore, the width (W) is approximately 200m and the length (L) is approximately 201m.
The largest area that can be enclosed is given by the product of the width and length:
Area = width × length
Area = 200m × 201m
Area ≈ 40,200 square meters
To find the largest area you can enclose with a given amount of fencing material, you should use the perimeter formula and then determine the dimensions that maximize the area.
Let's assume the length of the two parallel sides is "L" and the width of the remaining side is "W". Since we know the side bounded by the river requires no fencing, we have:
Perimeter = 2L + W = 600 meters.
Simplifying the equation, we get:
W = 600 - 2L.
Now, we need to express the area "A" in terms of "L" and "W" since it is what we want to maximize. The area of a rectangle is given by:
A = L * W.
Substituting the value of "W" from the previous equation:
A = L * (600 - 2L).
To find the maximum area, we can differentiate the equation with respect to "L" and set it equal to zero:
dA/dL = 0.
Differentiating the equation:
dA/dL = 600 - 4L.
Setting it equal to zero:
600 - 4L = 0.
Solving for "L":
4L = 600,
L = 600/4,
L = 150.
Now substitute this value back into the equation for "W":
W = 600 - 2L,
W = 600 - 2(150),
W = 600 - 300,
W = 300.
So, the dimensions that maximize the area are L = 150 meters and W = 300 meters.
To find the largest area you can enclose, substitute these values into the area formula:
A = L * W,
A = 150 * 300,
A = 45000 square meters.
Therefore, the largest area you can enclose with 600 meters of fencing material is 45000 square meters.