A rancher wishes to use 86 ft of fencing to enclose a rectangular paddock and subdivide the region into three smaller areas. If the total enclosed area is 225 ft2, find the dimensions of the enclosed region.
To find the dimensions of the enclosed region, we can use the information given.
Let's assume that the dimensions of the rectangular paddock are length (L) and width (W).
We know that the perimeter of a rectangle is given by the formula: Perimeter = 2 * (Length + Width).
In this case, the perimeter is given as 86 ft. So we can write the equation as:
86 = 2 * (L + W)
Simplifying the equation, we get:
43 = L + W
We are also given that the total enclosed area is 225 ft². The area of a rectangle is given by the formula: Area = Length * Width.
So we can write another equation:
225 = L * W
We now have two equations:
43 = L + W
225 = L * W
To solve these equations simultaneously, we can use substitution or elimination.
Let's solve them using the substitution method:
From the first equation, we can solve for L in terms of W:
L = 43 - W
Now substitute this value of L in the second equation:
225 = (43 - W) * W
Expanding the equation, we get:
225 = 43W - W²
Rearranging the equation to a standard quadratic form, we get:
W² - 43W + 225 = 0
Now we can solve this quadratic equation. We can either factorize it or use the quadratic formula.
Using the quadratic formula:
W = [-(-43) ± √((-43)² - 4 * 1 * 225)] / (2 * 1)
Calculating this, we get:
W = [43 ± √(1849 - 900)] / 2
W = [43 ± √949] / 2
W = (43 + √949) / 2 or W = (43 - √949) / 2
Since the width cannot be negative, we can ignore the second solution.
So, W = (43 + √949) / 2.
Now, substitute this value of W back into the equation L = 43 - W to find the length:
L = 43 - [(43 + √949) / 2]
L = 43 - 43/2 - √949/2
L = 86/2 - √949/2
L = (86 - √949) / 2
Therefore, the dimensions of the enclosed region are:
Length (L) = (86 - √949) / 2
Width (W) = (43 + √949) / 2