a rectangular chicken yard was built against an existing she wall. 30m of fencing was used to enclose 108m^2, Find the dimesion of the Yard
what are the factors of 108?
1 108
2 54
3 36
4 27
6 18
9 12
So, how can those be used to add up to 30? (Remember that one has to be used twice)
Let's assume the length of the yard is L meters and the width is W meters.
According to the given information, the perimeter of the yard is 30 meters, which means the sum of all four sides is equal to 30 meters. Since opposite sides of a rectangle are equal in length, we can write the following equation:
2L + 2W = 30
Now, we know that the area of a rectangle is given by the equation:
Area = Length × Width
In this case, the area is given as 108 square meters. So, we can write the second equation:
L × W = 108
Now, we have two equations:
1) 2L + 2W = 30
2) L × W = 108
Let's solve this system of equations using substitution.
From equation 1, we can express L in terms of W:
2L = 30 - 2W
L = (30 - 2W) / 2
L = 15 - W
Substituting this value of L in equation 2, we get:
(15 - W) × W = 108
Expanding the equation, we have:
15W - W^2 = 108
Rearranging this equation, we get:
W^2 - 15W + 108 = 0
Now, we can factorize this quadratic equation:
(W - 9)(W - 12) = 0
This gives us two possible values for the width: W = 9 or W = 12.
If W = 9, then using equation 1:
L = 15 - W
L = 15 - 9
L = 6
If W = 12, then using equation 1:
L = 15 - W
L = 15 - 12
L = 3
So, the dimensions of the rectangular yard can be either 9m x 12m or 12m x 9m.
To find the dimensions of the rectangular chicken yard, you can start by setting up equations based on the given information.
Let's assume that the length of the rectangular yard is L meters, and the width is W meters.
The perimeter of a rectangle is given by the formula:
Perimeter = 2(L + W)
In this case, the perimeter of the chicken yard is given as 30 meters. So, we have:
Perimeter = 2(L + W) = 30
The area of a rectangle is given by the formula:
Area = Length × Width
In this case, the area of the chicken yard is given as 108 square meters. So, we have:
Area = L × W = 108
Now, using these two equations, we can solve for the dimensions of the yard.
Step 1: Solve the perimeter equation for one variable.
2(L + W) = 30
L + W = 15
L = 15 - W
Step 2: Substitute the value of L in terms of W into the area equation.
L × W = 108
(15 - W) × W = 108
Step 3: Simplify the equation and solve for W.
15W - W^2 = 108
W^2 - 15W + 108 = 0
Step 4: Factor or use the quadratic formula to solve for W.
(W - 9)(W - 12) = 0
W = 9 or W = 12
Step 5: Substitute the values of W into the equation L = 15 - W to find the corresponding values of L.
If W = 9, then L = 15 - 9 = 6
If W = 12, then L = 15 - 12 = 3
Therefore, the possible dimensions of the rectangular yard are:
Length = 6 meters and Width = 9 meters
or
Length = 3 meters and Width = 12 meters