A rectangular field has a perimeter of 56 rods. If a strip 2 rods wide is cut from all sides of the field, the remaining part of the field has an area of 91 rods. Find the dimensions of the field.
If the field has width w, then the length is 28-w.
(w-4)(28-w-4) = 91
w^2-28w+187 = 0
w = 11
The field is 11x17
The smaller area is 7x13 = 91
Could you explain why the length is 28-w?
because the perimeter is 56,
2w + 2l = 56
l + w = 28
l = 28-w
To solve this problem, we can start by assuming the length and width of the rectangular field before the strips are cut.
Let's assume the original length of the field is L rods, and the original width is W rods.
After the strips are cut, the new length and width of the remaining part of the field would be (L - 4) rods and (W - 4) rods, respectively.
The perimeter of the remaining part can be calculated as follows:
Perimeter = 2(length + width)
Substituting the values, we get:
56 = 2((L - 4) + (W - 4))
28 = L + W - 8
L + W = 36 ---(Equation 1)
Now, we need to find the area of the remaining part of the field, which is given as 91 square rods:
Area = (length) * (width)
Substituting the values, we get:
91 = (L - 4) * (W - 4)
91 = LW - 4L - 4W + 16
LW - 4L - 4W = 75 ---(Equation 2)
To find the dimensions of the field, we need to solve these two simultaneous equations (Equation 1 and Equation 2).
One way to solve these equations is by substitution. Rearrange Equation 1 to get:
L = 36 - W
Substitute this value of L in Equation 2:
(36 - W)W - 4(36 - W) - 4W = 75
36W - W^2 - 144 + 4W - 4W = 75
W^2 - 36W + 144 = 0
Now we have a quadratic equation in terms of W. We can solve it by factoring or applying the quadratic formula.
By factoring, we find that:
(W - 12)(W - 12) = 0
W - 12 = 0
So, W = 12 rods.
Substituting this value of W in Equation 1, we can find the value of L:
L + 12 = 36
L = 24 rods
Therefore, the dimensions of the field before the strips are cut are 24 rods (length) and 12 rods (width).