You have a 1200 foot roll of fencing and a large field. You want to make two smaller fields by splitting a rectangular enclosure in half..
1. Use an algebraic solution to find the dimensions of the largest such enclosure. How big will the enclosure be?
300
To find the dimensions of the largest enclosure, we first need to set up an equation using the given information.
Let's assume the length of the entire rectangular enclosure is L and the width is W. Since we want to split the rectangular enclosure in half, we will have two smaller fields with dimensions (L/2) and W.
The perimeter of the entire rectangular enclosure is given by the equation:
Perimeter = 2L + 2W
We are told that we have 1200 feet of fencing, so the equation becomes:
1200 = 2L + 2W
Next, we need to express one of the variables, L or W, in terms of the other. Let's solve the equation for L:
2L = 1200 - 2W
L = (1200 - 2W) / 2
Now, to find the largest enclosure, we need to maximize the area, which is given by the equation:
Area = length × width
In this case, the length is L/2 and the width is W. So, the area becomes:
Area = (L/2) × W
Substituting the expression for L from earlier, the equation becomes:
Area = [(1200 - 2W) / 2] × W
To find the maximum possible area, we can take the derivative of the area equation with respect to W and set it equal to zero. However, since we are only interested in the dimensions and not the actual maximum area, we can skip this step.
To find the dimensions of the largest enclosure, we can simply substitute the value of W into the expression for L and calculate the values.
Let's arbitrarily assume W = 200 feet:
L = (1200 - 2(200)) / 2
L = (1200 - 400) / 2
L = 800 / 2
L = 400 feet
So, the dimensions of the largest enclosure would be 400 feet by 200 feet.