A retail lumberyard plans to store lumber in a rectangular region adjoining the sales office. The region will be fenced on three sides and the fourth side will be bounded by the wall of the office. Find the dimensions of the region if 350 feet of fencing is available and the area of the region is 12,500 square feet.
If the region is LxW then
L+2W = 350
LW = 12500
L = 350-2W
(350-2W)W = 12500
350W - 2W^2 = 12500
W = 50,125
L = 250,100
So, the area could be 50x250 or 125x100
To find the dimensions of the rectangular region, we can use a system of equations based on the given information.
Let's assume the length of the region is L feet and the width is W feet.
According to the problem, we know that the region will be fenced on three sides and the remaining side will be bounded by the wall of the office. This implies that the perimeter of the region will be equal to the sum of the lengths of the three sides, which is 350 feet.
Therefore, the perimeter equation is:
2L + W = 350 ---(Equation 1)
We're also given that the area of the region is 12,500 square feet.
The area of a rectangle is given by the formula: Area = Length * Width.
So, the area equation is:
L * W = 12,500 ---(Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of L and W.
We can rearrange Equation 1 to solve for W:
W = 350 - 2L
Substitute this value of W in Equation 2:
L * (350 - 2L) = 12,500
Expand and rearrange the equation:
350L - 2L^2 = 12,500
Rearrange the equation by bringing all terms to one side:
2L^2 - 350L + 12,500 = 0
Now we have a quadratic equation. We can solve it by factoring or by using the quadratic formula.
In this case, the equation can be factored as:
(L - 50)(2L - 250) = 0
Setting each factor to zero gives us two possible values for L:
L - 50 = 0 -> L = 50
OR
2L - 250 = 0 -> L = 125
Since the length cannot be shorter than the width, we can ignore the value L = 125.
Therefore, the length of the region is L = 50 feet.
Now we can substitute this value of L into Equation 1 to find the width:
2(50) + W = 350
100 + W = 350
W = 350 - 100
W = 250
So, the width of the region is W = 250 feet.
Therefore, the dimensions of the rectangular region are 50 feet by 250 feet.