a garden 7m by 12m will be expanded by planting a border of flowers,the border will be of the same width around the entire garden and has an area of 92m
I assume you want the width of the border.
new length = 12+2x
new width = 7+2x
What has an area of 92 m^2 (you had 92 m) ?
if the whole area is 92 m^2
new area = (12+2x)(7+2x) = 92
84 + 38x + 4x^2 = 92
4x^2 + 38x - 8 = 0
2x^2 + 19x - 4 = 0
x = (-19 ± √393)/4
= .206 or a negative
the area of the border is 92 m^2
(12+2x)(7+2x) - (7)(12) = 92
84 + 38x + 4x^2 - 176 = 0
4x^2 + 38x - 92 = 0
(x-2)(4x+46) = 0
x = 2 or x is a negative
Hahahahahajaja
To find the width of the border, we need to find the length and width of the expanded garden first, and then subtract the original length and width.
Given:
Original garden dimensions: 7m by 12m
Area of the border: 92m²
Step 1: Find the area of the expanded garden
Let's assume the width of the border is 'x.' Since the border is of the same width around the entire garden, we need to add the width of the border to both the length and the width of the original garden.
Expanded garden length = 7m + 2x
Expanded garden width = 12m + 2x
The area of the expanded garden is:
Area of expanded garden = (Expanded garden length) * (Expanded garden width)
Step 2: Calculate the new area and solve for 'x'
Area of expanded garden = (7m + 2x) * (12m + 2x)
Area of expanded garden = 7m * 12m + 2x * 12m + 2x * 7m + 2x * 2x
Area of expanded garden = 84m² + 24xm + 14xm + 4x²
Area of expanded garden = 4x² + 38xm + 84m²
Step 3: Set up the equation
Since the area of the expanded garden is the sum of the area of the original garden and the area of the border, we can set up the following equation:
Area of expanded garden = Area of original garden + Area of border
4x² + 38xm + 84m² = 7m * 12m + 92m²
Simplifying the equation, we have:
4x² + 38xm + 84m² = 84m² + 92m²
4x² + 38xm = 92m²
Step 4: Solve for 'x'
To solve the equation, we need to isolate 'x'. Subtracting 92m² from both sides of the equation, we get:
4x² + 38xm - 92m² = 0
We can factor this quadratic equation as follows:
(x - 4)(4x + 23m) = 0
Setting each factor to zero and solving for 'x', we have two possible solutions:
x - 4 = 0 or 4x + 23m = 0
If x - 4 = 0, then x = 4.
If 4x + 23m = 0, then x = -23m/4.
Since the width of the border cannot be negative, we discard the second solution.
Therefore, the width of the border is x = 4m.
To find the width of the flower border, you need to subtract the area of the original garden from the area of the expanded garden.
The original garden has a length of 7m and a width of 12m, so its area is calculated by multiplying the length by the width:
Original garden area = 7m * 12m = 84m²
The expanded garden area, including the flower border, is equal to the original garden area plus the area of the flower border, which is given as 92m²:
Expanded garden area = Original garden area + Flower border area = 84m² + 92m² = 176m²
Next, we need to determine the dimensions of the expanded garden. The original garden dimensions remain the same, and the flower border is added around the entire garden.
Let's assume the width of the flower border is 'x'. The expanded garden's length will be increased by twice the flower border width since it is added on both sides:
Expanded garden length = Original garden length + (2 * Flower border width) = 7m + 2x
Similarly, the expanded garden's width will also be increased by twice the flower border width:
Expanded garden width = Original garden width + (2 * Flower border width) = 12m + 2x
To find the width of the flower border, we can solve the equation:
Expanded garden area = Expanded garden length * Expanded garden width
Substituting the values we have:
176m² = (7m + 2x) * (12m + 2x)
Now we can solve this quadratic equation to find the value of 'x' which represents the width of the flower border.