Answer: the dimensions of the field can be either 96 yards by 100 yards or 122 yards by 48 yards.
Explanation:
Let's assume the length of the soccer field is L yards and the width is W yards.
We know that the area of the field is given by:
Area = Length x Width
Given that the area is 4,800 square yards, we have:
4,800 = L x W ............(1)
We also know that the perimeter of the field is given by:
Perimeter = 2 x (Length + Width)
Given that the perimeter is 292 yards, we have:
292 = 2 x (L + W) ............(2)
We have two equations (equation 1 and equation 2) with two unknowns (L and W). We can solve this system of equations to find the dimensions of the field.
Let's solve equation 2 for L:
292 = 2L + 2W
2L = 292 - 2W
L = (292 - 2W)/2
L = 146 - W/2
Substitute this value of L into equation 1:
4,800 = L x W
4,800 = (146 - W/2) x W
4,800 = 146W - W^2/2
Rearrange the equation:
W^2/2 - 146W + 4,800 = 0
Now we have a quadratic equation. We can solve it to find the value of W. Once we have W, we can substitute it back into equation 1 or equation 2 to find the corresponding value of L.
Using a quadratic solver or factoring, we find that W = 100 yards or W = 48 yards.
If W = 100 yards, then L = 146 - W/2 = 146 - 100/2 = 96 yards.
If W = 48 yards, then L = 146 - W/2 = 146 - 48/2 = 122 yards.