The Parkhursts used 160 yd of fencing to enclose a rectangular corral and to divide it into two parts by a fence parrallel to one of the shorter sides. Find the dimensions of the corral if its area is 1000 yd^2
Thank you!
Call the width W
Call the length L
Then you know tow things:
W * L = 1000
and
3 * W + 2 * L = 160
then L = W/1000
so
3*W + 2000/W = 160
or
3 * W^2 - 160 * W + 2000 = 0
Solve that quadratic with the quadratic equation. You get two results. One of them has a short side, W, longer than the long side, L. Throw that solution out because long is longer than short. The other solution, W = 20 and then L = 50 works.
To find the dimensions of the corral, we need to set up some equations based on the given information.
Let's assume that the length of the corral is L and the width is W.
Given that the area of the corral is 1000 yd^2, we have the equation:
L * W = 1000 ---(Equation 1)
We are also told that 160 yards of fencing is used to enclose the corral and divide it into two parts. Taking into account the fence parallel to one of the shorter sides, we can break down the total amount of fencing used into three parts:
2L + W = 160 ---(Equation 2) (the two lengths and the width)
Now, we can solve this system of equations to find the values of L and W.
From Equation 2, we can express W in terms of L:
W = 160 - 2L
Substituting this value of W in Equation 1, we get:
L * (160 - 2L) = 1000
Simplifying the equation further:
160L - 2L^2 = 1000
Rearranging to put the equation in standard quadratic form:
2L^2 - 160L + 1000 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
L = (-b ± √(b^2 - 4ac)) / (2a)
Using the values from our equation:
a = 2, b = -160, c = 1000
L = (-(-160) ± √((-160)^2 - 4(2)(1000))) / (2(2))
Simplifying further:
L = (160 ± √(25600 - 8000)) / 4
L = (160 ± √17600) / 4
L = (160 ± 132.87) / 4
This gives us two potential values for L:
L₁ ≈ 72.87
L₂ ≈ 47.13
Now, substituting these values back into Equation 2 to find the corresponding width, W:
For L = 72.87:
W₁ = 160 - 2(72.87)
W₁ ≈ 14.26
For L = 47.13:
W₂ = 160 - 2(47.13)
W₂ ≈ 65.73
Therefore, the dimensions of the corral can be either approximately 72.87 yards by 14.26 yards or approximately 47.13 yards by 65.73 yards.