A small garden measures 14 ft by 6 ft. A uniform border around the garden increases the total area to 209 ft2. What is the width of the border?
You have set up the first rectangular area = 84...then realize that the sides of new rect. are x+6;and x+14.
Set up the equation..
(X+6)(x+14) = 209 be careful when mult. Yu should get....
X=5 Now plug this value into both binomials and the sides are 11 and 19..
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To find the width of the border, we need to first find the original area of the garden and then subtract it from the total area including the border.
The original area of the garden can be found by multiplying the length and width of the garden: 14 ft * 6 ft = 84 ft².
Now, let's subtract the original area from the total area to find the area of the border. We have:
Total area - Original area = Area of the border
209 ft² - 84 ft² = 125 ft²
Since the border is uniform, it surrounds the entire garden. The border will have the same width on all sides.
To find the width of the border, we need to calculate the additional length and width that were added to the garden.
Let's say the width of the border is x ft. In that case, the new length and width of the garden would be:
New length = original length + 2x
New width = original width + 2x
In this case, the original length is 14 ft and the original width is 6 ft. Replacing these values, we have:
New length = 14 ft + 2x
New width = 6 ft + 2x
We can find the area of the border using the new length and new width:
Area of the border = New length * New width
Since we know the area of the border is 125 ft², we can set up the following equation:
125 ft² = (14 ft + 2x) * (6 ft + 2x)
To solve this equation, we can distribute and simplify:
125 ft² = (84 ft² + 28 ft*x + 12 ft*x + 4x²)
Now, let's collect like terms:
125 ft² = (84 ft² + (28 ft*x + 12 ft*x) + 4x²)
125 ft² = (84 ft² + 40 ft*x + 4x²)
Rearranging the equation:
4x² + 40 ft*x - 41 ft² = 0
At this point, to solve for x, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation, the coefficients are:
a = 4
b = 40 ft
c = -41 ft²
Plugging these values into the quadratic formula:
x = (-40 ft ± √((40 ft)² - 4 * 4 * -41 ft²)) / (2 * 4)
Simplifying:
x = (-40 ft ± √(1600 ft² + 664 ft²)) / 8
x = (-40 ft ± √(2264 ft²)) / 8
Calculating the square root:
x = (-40 ft ± 47.66 ft) / 8
Now, let's solve for x by using both the positive and negative square root:
x₁ = (-40 ft + 47.66 ft) / 8
x₁ = 7.66 ft / 8
x₁ = 0.9583 ft
x₂ = (-40 ft - 47.66 ft) / 8
x₂ = -87.66 ft / 8
x₂ = -10.9575 ft
Since we are dealing with a physical garden, the width of the border cannot be negative. Therefore, the width of the border is approximately 0.9583 ft (rounded to the nearest hundredth).