A garden against the wall of a house is to be surrounded on three sides by a total of 336m of fencing. What dimensions of the garden will result in an area of 14,112m^2?
To solve this problem, we need to find the dimensions of the garden that will result in an area of 14,112 square meters, while being surrounded on three sides by 336 meters of fencing.
Let's assume the length of the garden is L meters and the width is W meters.
Since the garden is surrounded on three sides by fencing, we can calculate the perimeter using the formula:
Perimeter = 2(L + W) + L
Given that the total amount of fencing is 336 meters, we can write the equation:
336 = 2(L + W) + L
Simplifying the equation, we have:
336 = 2L + 2W + L
Combining like terms:
336 = 3L + 2W
Now, we need to find the area of the garden. The area of a rectangle can be calculated by multiplying its length by its width:
Area = L * W
Given that the area is 14,112 square meters, we can write the equation:
14,112 = L * W
Now we have a system of equations:
1) 336 = 3L + 2W
2) 14,112 = L * W
To solve this system, we can use substitution or elimination method.
Method 1: Substitution
From the first equation, solve for W:
W = (336 - 3L) / 2
Substitute this expression for W in the second equation:
14,112 = L * [(336 - 3L) / 2]
Multiply both sides by 2 to eliminate the fraction:
28,224 = L * (336 - 3L)
Expand the expression:
28,224 = 336L - 3L^2
Rearrange it in the standard quadratic form:
3L^2 - 336L + 28,224 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula.
After solving the equation, we find two possible solutions for L: L = 112 or L = 56.
For L = 112:
Plug this value back into the equation for W:
W = (336 - 3(112)) / 2 = 40
So, one possible dimension for the garden is 112 meters by 40 meters.
For L = 56:
Plug this value back into the equation for W:
W = (336 - 3(56)) / 2 = 80
So, another possible dimension for the garden is 56 meters by 80 meters.
Therefore, the dimensions of the garden that will result in an area of 14,112 square meters while being surrounded on three sides by 336 meters of fencing are either 112 meters by 40 meters or 56 meters by 80 meters.