A rectangular garden is fenced on three sides with wall forming the fourth side. The total length of the fence is 120m. The ares of the garden is 1600 m(squared). Find the dimensions of the garden.
Let's assume the length of the garden is L and the width is W.
Since the garden is fenced on three sides and a wall forms the fourth side, we can calculate the total length of the fence using the perimeter formula of a rectangle:
Perimeter = 2(L + W)
Given that the total length of the fence is 120 meters, we have:
120 = 2(L + W)
Simplifying the equation, we get:
60 = L + W
We are also told that the area of the garden is 1600 square meters. The area (A) of a rectangle is given by:
A = L * W
Substituting the given values, we have:
1600 = L * W
We have two equations now:
60 = L + W (Equation 1)
1600 = L * W (Equation 2)
To solve these equations, we can use substitution or elimination. Let's solve by substitution:
From Equation 1, we can express L in terms of W:
L = 60 - W
Now substitute the value of L in Equation 2:
1600 = (60 - W) * W
Expanding the equation, we have:
1600 = 60W - W^2
Rearranging the equation, we get:
W^2 - 60W + 1600 = 0
Now we can solve this quadratic equation to find the possible dimensions of the garden. Factoring the quadratic equation, we have:
(W - 40)(W - 40) = 0
This gives us only one possible value for W:
W = 40
Now substitute this value of W back into Equation 1:
60 = L + 40
Subtracting 40 from both sides:
L = 20
Therefore, the dimensions of the garden are a length of 20 meters and a width of 40 meters.
To find the dimensions of the garden, we need to set up a system of equations using the given information.
Let's assume that the length of the garden is "L" and the width is "W". The total length of the fence is 120m, which means that the sum of all four sides of the garden (three sides fenced and one side formed by the wall) is equal to 120m.
Since the three sides fenced have a total length of 120m, we can write the equation:
2L + W = 120 (Equation 1)
The area of a rectangle is calculated by multiplying its length by its width. So, based on the area of the garden being 1600 m², we have:
L * W = 1600 (Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two unknowns (L and W). We can solve this system to find the dimensions of the garden.
Let's rearrange Equation 1 to express W in terms of L:
W = 120 - 2L
Now, substitute this value of W into Equation 2:
L * (120 - 2L) = 1600
Simplifying the equation:
120L - 2L^2 = 1600
Rearranging to a quadratic equation form:
2L^2 - 120L + 1600 = 0
Now, we can solve this quadratic equation for L using factoring, completing the square, or the quadratic formula. In this case, it's easier to use factoring:
2L^2 - 80L - 40L + 1600 = 0
2L(L - 40) - 40(L - 40) = 0
(2L - 40)(L - 40) = 0
Therefore, we have two possible solutions:
1) 2L - 40 = 0 or L - 40 = 0
2L = 40 or L = 40
L = 40/2 or L = 40
L = 20 or L = 40
If L = 20, substitute this value back into Equation 1 to find W:
2*20 + W = 120
40 + W = 120
W = 120 - 40
W = 80
Therefore, one set of dimensions for the garden is L = 20m and W = 80m.
If L = 40, substitute this value back into Equation 1 to find W:
2*40 + W = 120
80 + W = 120
W = 120 - 80
W = 40
Therefore, the other set of dimensions for the garden is L = 40m and W = 40m.
So, the dimensions of the garden can be either 20m by 80m or 40m by 40m.
3 * 40 = 120
40 * 40 = 1600