At a park, a landscaper wishes to plant a uniform border of tulips around the outside of a 18m by 12m rectangular garden. The garden will look best if the area of the tulip border is half the area of the garden. How wide should the border be?
Guidance as to where I am going wrong?
To find the width of the border, we need to first find the area of the garden and then calculate half of that area.
Step 1: Calculate the area of the garden.
The garden is rectangular with dimensions 18m by 12m.
Area of the garden = Length x Width
Area of the garden = 18m x 12m
Area of the garden = 216 square meters
Step 2: Calculate half of the area of the garden.
Half of the area of the garden = 0.5 x Area of the garden
Half of the area of the garden = 0.5 x 216 square meters
Half of the area of the garden = 108 square meters
Step 3: Calculate the area of the tulip border.
The tulip border is uniform around the outside of the garden, which means it will increase the length and width of the garden by the same amount on all sides.
Let's assume that the width of the border is "x" meters.
So, the new length of the garden will be (18m + 2x) and the new width will be (12m + 2x).
Area of the tulip border = (18m + 2x) x (12m + 2x)
Step 4: Set up the equation to solve for the width of the border.
We know that the area of the tulip border should be half the area of the garden.
Therefore, (18m + 2x) x (12m + 2x) = 108 square meters
Step 5: Solve the equation to find the width of the border.
Expand the expression:
216 + 36x + 24x + 4x^2 = 108
4x^2 + 60x + 108 = 108
4x^2 + 60x = 0
Divide by 4:
x^2 + 15x = 0
x(x + 15) = 0
Setting the equations to zero, we get:
x = 0 (ignore this because it doesn't make sense in the context of the problem)
or
x + 15 = 0
Solving for x:
x = -15 (ignore this as well since the width cannot be negative)
or
x = 0
Since we cannot have a width of 0 for the border, there is no solution to this problem.
To find out the width of the border, we need to determine the area of the garden and then divide it by 2 to find the area of the tulip border.
First, find the area of the garden:
Area of garden = length * width = 18m * 12m = 216m²
Next, we need to find the area of the tulip border, which is half the area of the garden:
Area of tulip border = (1/2) * Area of garden = (1/2) * 216m² = 108m²
Now, let's assume the width of the border is 'x'. Since the border is uniform, the dimensions of the garden including the border will be (18m + 2x) by (12m + 2x) (adding 'x' to each side).
The area of the garden with the tulip border can be calculated as:
Area of garden with border = (18m + 2x) * (12m + 2x)
We know that the area of the garden with the tulip border is equal to the area of the tulip border:
(18m + 2x) * (12m + 2x) = 108m²
Now, expand the equation and solve for 'x':
216m² + 48m*x + 36m*x + 4x² = 108m²
4x² + 84mx + 108m² - 216m² = 0
4x² + 84mx - 108m² = 0
This is a quadratic equation. We can solve it using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In our equation, a = 4, b = 84m, and c = -108m².
Solving for 'x', we get:
x = (-84m ± √(84m)² - 4 * 4 * (-108m²)) / (2 * 4)
x = (-84m ± √(7056m² + 1728m²)) / 8
x = (-84m ± √8784m²) / 8
x = (-84m ± 2√2196m) / 8
x = -10.5m ± 3√2196m / 2
Since the width of the border cannot be negative, we take the positive solution:
x = -10.5m + 3√2196m / 2
Therefore, the width of the border should be approximately -10.5m + 3√2196m / 2.
P=2(18)+ 2(12)= 36 +24+ 60
A+18x12=216
Tulip border= 216/2=108
(18+x)(12+x)=(18x12)/2 +216
216+18x+12x+x^2=108+216
x^2+30x-108=0
cannot factor
quadratic equation = -40.82 or -20.82