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 start by setting up some equations based on the given information.
Let's assume the width of the rectangular paddock is "w" and the length is "l". Since the total enclosed area is 225 ft^2, we have the equation:
l * w = 225 (Equation 1)
The rancher wishes to subdivide the region into three smaller areas. This means there are two additional fences, each dividing the width into three equal parts. Since there are three equal parts, each part will have a width of w/3.
Considering the two additional fences, the total length of the three smaller regions is (w/3) + (w/3) + (w/3) = 3w/3 = w.
Now, let's consider the perimeter of the rectangular paddock. We are given that the total fencing available is 86 ft, so we can write:
2w + l + w = 86 (Equation 2)
Simplifying Equation 2, we get:
3w + l = 86 (Equation 3)
Now, we have two equations: Equation 1 and Equation 3. We can solve these equations simultaneously to find the values of "w" and "l".
Let's substitute the value of "l" from Equation 3 into Equation 1:
w * (86 - 3w) = 225
Expanding and rearranging the equation, we get:
86w - 3w^2 = 225
Rearranging further, we get:
3w^2 - 86w + 225 = 0
Solving this quadratic equation, we find two possible values for "w". Once we find the values of "w", we can substitute them back into Equation 3 to get the corresponding values for "l".
I can help you solve this equation if you'd like, or you can use a graphing calculator or a quadratic formula to find the values of "w".
Let's find the dimensions of the rectangular paddock.
Let's assume the length of the rectangular paddock is L and the width is W.
The perimeter of the rectangular paddock is equal to the sum of all the sides, which is given as 86 ft.
Therefore, we have the equation: 2L + 2W = 86.
Now, let's find the area of the rectangular paddock.
The area of a rectangle is given by the formula A = L x W.
We are given that the total enclosed area is 225 ft2.
Therefore, we have another equation: A = L x W = 225.
Now, we have a system of two equations with two variables:
Equation 1: 2L + 2W = 86
Equation 2: L x W = 225
We can solve this system of equations simultaneously to find the dimensions of the enclosed region.
Let's solve the first equation for L:
2L + 2W = 86
2L = 86 - 2W
L = 43 - W
Substitute this value of L into Equation 2:
(43 - W) x W = 225
43W - W^2 = 225
W^2 - 43W + 225 = 0
Now, we can solve this quadratic equation. Factoring or using the quadratic formula, we find that W = 15 or W = 28.
If we substitute W = 15 back into Equation 1, we find that L = 28.
If we substitute W = 28 back into Equation 1, we find that L = 15.
Therefore, the dimensions of the enclosed region can be either 28 ft by 15 ft or 15 ft by 28 ft.