A gardener wants to plant a uniform border of flowers within his rectangular back yard. The dimensions of the yard are 12m by 6m. The area of the border should be one quarter the area of the entire yard. What is the width of the border
To find the width of the border, we first need to calculate the area of the yard.
The area of a rectangle is calculated by multiplying its length by its width.
In this case, the length of the yard is 12m and the width is 6m, so the area of the yard is 12m * 6m = 72m².
Now we need to find one quarter of the area of the yard.
One quarter of 72m² is (1/4) * 72m² = 18m².
Let's assume the width of the border is x meters.
The length of the inner rectangle, which represents the garden area, can be determined by subtracting twice the width of the border from the original length and width of the yard.
So the length of the garden area is (12m - 2x) and the width is (6m - 2x).
The area of the garden area can be calculated by multiplying its length by its width.
The area of the garden area is (12m - 2x) * (6m - 2x).
The problem states that the area of the border should be one quarter the area of the entire yard, so we can set up the equation:
(12m - 2x) * (6m - 2x) = 18m²
We can solve this quadratic equation by multiplying out the two binomials and setting the equation equal to 18m²:
(12m - 2x) * (6m - 2x) = 18m²
72m² - 24mx - 24mx + 4x² = 18m²
72m² - 48mx + 4x² = 18m²
72m² - 18m² - 48mx + 4x² = 0
54m² - 48mx + 4x² = 0
We can simplify this equation by dividing the entire equation by 2:
27m² - 24mx + 2x² = 0
Now we have a quadratic equation in standard form: ax² + bx + c = 0
a = 2, b = -24m, c = 27m²
To find the width of the border (x), we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values, we get:
x = (-(-24m) ± √((-24m)² - 4(2)(27m²))) / (2(2))
Simplifying further:
x = (24m ± √(576m² - 216m²)) / 4
x = (24m ± √(360m²)) / 4
x = (24m ± 6√10m) / 4
Now we can simplify the expression further:
x = 6m ± 3√10m
Since the width of the border cannot be negative, we discard the negative solution:
x = 6m + 3√10m
Therefore, the width of the border is 6m + 3√10m.