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
To find the dimensions of the enclosure with the largest area, we need to use the given 12000m roll of fencing to create two equal paddocks by splitting a rectangle enclosure in half.
Let's assume the enclosure has a length of 'L' and a width of 'W'. We can divide this rectangle enclosure into two equal paddocks by adding a fence lengthwise in the middle of the enclosure. So, each paddock will have a width of 'W/2'.
The total perimeter of the enclosure is equal to the sum of all four sides, which is given as 12000m. In mathematical terms:
2L + 3W = 12000
Next, we need to express the area of the enclosure 'A' in terms of 'L' and 'W'. The area of a rectangle is given by multiplying the length and width:
A = L * W
To solve the problem mathematically, we need to express the area 'A' in terms of a single variable. We can do this by substituting W = 2(W/2) in the area equation:
A = L * (2(W/2))
A = L * W
So, we have two equations:
2L + 3W = 12000 (equation 1)
A = L * W (equation 2)
Now, we need to express one variable in terms of the other from equation 1. Rearranging equation 1 gives:
2L = 12000 - 3W
L = (12000 - 3W)/2
Substituting this value of L in equation 2:
A = ((12000 - 3W)/2) * W
Now, we can maximize the area 'A' by taking the derivative with respect to W and solving for W when the derivative is equal to zero. This will give us the critical point, which represents the maximum area.
dA/dW = 0
Differentiating the equation for A with respect to W:
dA/dW = (12000 - 3W)/2 - (3W)/2
dA/dW = 12000/2 - 3W/2 - 3W/2
dA/dW = 6000 - 3W
Setting the derivative equal to zero:
6000 - 3W = 0
3W = 6000
W = 6000/3
W = 2000
Substituting the value of W back into equation 1 to find L:
2L + 3(2000) = 12000
2L + 6000 = 12000
2L = 12000 - 6000
2L = 6000
L = 6000/2
L = 3000
So, the dimensions of the enclosure that will give the largest area are L = 3000m and W = 2000m.
To calculate the maximum area, substitute these values back into equation 2:
A = L * W
A = 3000 * 2000
A = 6,000,000 m²
Therefore, your calculation of 6,000,000 m² is correct, and the maximum area is indeed 6,000,000 square meters.