The perimeter of a rectangular field is 300 meters. This field is to be fenced along three sides only. The fencing along its two lengths and one width is to be done at $50 per meter. If the total cost of fencing is $12,000, what are the field’s dimensions
To find the dimensions of the rectangular field, we can start by setting up equations based on the given information.
Let's assume the length of the field is L meters and the width is W meters.
The perimeter of a rectangle is given by the formula P = 2L + 2W, where P is the perimeter, L is the length, and W is the width.
From the problem statement, we know that the perimeter is 300 meters, so we have the equation:
300 = 2L + 2W
Next, we are told that the field is fenced along three sides only: both lengths and one width. The cost of fencing along these sides is $50 per meter. The total cost of fencing is given as $12,000.
The cost of fencing along the lengths of the field is 2L * $50 = $100L.
The cost of fencing along the width of the field is 1W * $50 = $50W.
So, the total cost of fencing can be expressed as the sum of the costs for the lengths and the width:
$12,000 = $100L + $50W
Now, we have a system of two equations:
300 = 2L + 2W
$12,000 = $100L + $50W
To solve this system, we can use substitution or elimination methods.
Let's solve it using the substitution method. We can solve the first equation for L and substitute it into the second equation:
From equation 1, we have: 300 - 2W = 2L → L = (300 - 2W)/2
Substituting this value of L into the second equation:
$12,000 = $100L + $50W
$12,000 = $100((300 - 2W)/2) + $50W
Now we can simplify and solve for W:
$12,000 = $50(300 - 2W) + $50W
$12,000 = $15,000 - $100W + $50W
$12,000 - $15,000 = $-50W
$-3,000 = $-50W
W = $-3,000 / -$50
W = 60 meters
Substituting this value of W back into equation 1:
300 = 2L + 2W
300 = 2L + 2(60)
300 - 120 = 2L
180 = 2L
L = 90 meters
Therefore, the dimensions of the rectangular field are 90 meters in length and 60 meters in width.