A farmer has 12000m roll of fencing. He wants to make 2 paddocks by splitting a rectangle enclosure in half. What are the dimensions of the enclosure with the largest area that he can make with the fencing he has?
I got 6,000,000 m^2, is it right?
PLZ help
split the fence equally between length and width. So, two pens of 2000x3000. I make that 12,000,000.
To be sure, use algebra.
If the pens are x long ad y wide, then
2x+3y = 12000
area = 2xy = 2x(12000-2x)/3
This is a parabola with vertex at (3000,12000000)
To find the dimensions of the enclosure with the largest area, we can start by setting up an equation using the information given. Let's assume the length of the enclosure is "l" and the width is "w".
Since the farmer wants to split the enclosure in half to create two paddocks, the total length of fencing required for the paddocks would be:
2l + w
According to the information given, the farmer has 12000m of fencing. Therefore, we can write the equation:
2l + w = 12000
Now, we need to find the dimensions that would maximize the area, which is given by:
Area = length * width
In this case, the area can be expressed as:
Area = lw
To find the maximum area, we need to eliminate one variable from the equation. We can solve the equation 2l + w = 12000 for w:
w = 12000 - 2l
Substituting this value of w into the area equation, we get:
Area = l(12000 - 2l)
Now, we have an equation that represents the area in terms of one variable, l. We can find the maximum area by finding the value of l that maximizes this equation.
To do this, we can start by expanding the equation:
Area = 12000l - 2l^2
Now, to find the maximum area, we need to find the vertex of the quadratic equation. The x-coordinate of the vertex can be found using the formula:
x = -b / 2a,
where a = -2, b = 12000, and c = 0.
Substituting the values, we get:
l = -12000 / (2 * -2)
l = 6000 / 2
l = 3000
So, the length of the enclosure that maximizes the area is 3000m. To find the width, we can substitute this length value into the earlier equation:
w = 12000 - 2l
w = 12000 - 2(3000)
w = 12000 - 6000
w = 6000
Therefore, the dimensions of the enclosure with the largest area that the farmer can make with the 12000m of fencing are 3000m (length) and 6000m (width).
To check the maximum area, we can calculate:
Area = length * width
Area = 3000 * 6000
Area = 18,000,000 m^2
So, the maximum area the farmer can achieve is 18,000,000 m^2, not 6,000,000 m^2.