the width of a rectangular garden is 2 m less than its length. Determine the dimensions of the garden if the area is 36 m2.
thanks!
w(w+2) = 36
w^2 + 2w = 36
w^2+2w+1 = 36+1
(w+1)^2 = 37
w = -1±√37
we want a positive width, so w = -1+√37 = 5.09
check: 5.09*7.09 = 36.1
close enough
Let's start by assigning variables to the dimensions of the garden. Let's say the length of the garden is L meters and the width of the garden is W meters.
Given the information from the problem, we can form two equations:
1. The width of the garden is 2 meters less than its length:
W = L - 2
2. The area of the garden is 36 square meters:
Area = Length × Width
36 = L × W
Now, substitute the value of W from equation 1 into equation 2:
36 = L × (L - 2)
Expand the equation:
36 = L^2 - 2L
Rearrange the equation to form a quadratic equation:
L^2 - 2L - 36 = 0
Now, we can solve this quadratic equation to find the values of L.
This quadratic equation can be factored as:
(L - 6)(L + 6) = 0
Setting each factor to zero and solving for L:
L - 6 = 0 or L + 6 = 0
L = 6 or L = -6
Since the length of a garden cannot be negative, we discard the solution L = -6.
Therefore, the length of the garden is L = 6 meters.
Now, substitute this value back into equation 1 to find the width of the garden:
W = L - 2
W = 6 - 2
W = 4 meters
So, the dimensions of the rectangular garden are 6 meters by 4 meters.
To find the dimensions of the rectangular garden, we can use the information given. Let's assign a variable to one of the dimensions, such as "x", for the length of the garden.
Given that the width is 2m less than the length, we can express the width as (x - 2).
The area of a rectangle is calculated by multiplying its length by its width. In this case, the area is given as 36m². So we can set up the equation:
x * (x - 2) = 36
Now, we can solve this quadratic equation. Expanding the equation, we get:
x^2 - 2x = 36
Rearranging the equation, we have:
x^2 - 2x - 36 = 0
Now, we can solve this quadratic equation either by factoring or using the quadratic formula. Since the equation can't be factored easily, let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = 1, b = -2, and c = -36. Plugging these values into the quadratic formula, we have:
x = (-(-2) ± √((-2)² - 4(1)(-36))) / (2(1))
Simplifying further:
x = (2 ± √(4 + 144)) / 2
x = (2 ± √148) / 2
x = (2 ± 2√37) / 2
Now, we can simplify this expression:
x = 1 ± √37
So, the length (x) of the garden can be expressed as either 1 + √37 or 1 - √37.
Since we are dealing with dimensions, length cannot be negative. Therefore, x = 1 + √37.
Now, we can find the width by substituting the value of x into the expression for the width:
Width = x - 2
Width = (1 + √37) - 2
Width = √37 - 1
Hence, the dimensions of the garden are approximately:
Length = 1 + √37 meters
Width = √37 - 1 meters