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ishika20sam
a)The total area of the flower bed is (24 + 24 = 48) square feet.
b)The area equation is: [ A = (6 + 2x)(4 + 2x) ]
c)Possible solutions: (x = 3) or (x = -6).
d)The extraneous solution is (x = -6).
e)The extraneous solution is (x = 3).
Let’s break down each part of the problem step by step:
1.Total Area of the Flower Bed (Part 1):
We have a rectangular flower bed with tulips in the middle and daisies surrounding them.
The tulip area has a length of 6 feet and a width of 4 feet.
The daisy border around the tulips has a uniform width of x feet on all sides.
To find the total area, we’ll add the area of the tulips and the area of the daisies.
The area of the tulips is (6 \times 4 = 24) square feet.
Since the daisies occupy an equal area, their combined area is also (24) square feet.
Therefore, the total area of the flower bed is (24 + 24 = 48) square feet.
2.Area of the Flower Bed as an Equation (Part 2):
Let’s express the area of the flower bed using binomials.
The length of the flower bed (including the daisy border) is (6 + 2x).
The width of the flower bed (including the daisy border) is (4 + 2x).
So, the area equation is: [ A = (6 + 2x)(4 + 2x) ]
3.Solving the Equation (Part 3):
Expanding the binomials: [ A = 24 + 20x + 4x^2 ]
Since the tulips and daisies occupy equal areas, we divide the total area by 2:[ (24 + 20x + 4x^2)/(2) = 12 + 10x + 2x^2 ]
Now we solve for (x): [ 12 + 10x + 2x^2 = 48 ] [ 2x^2 + 10x - 36 = 0 ] [ x^2 + 5x - 18 = 0 ] [ (x - 3)(x + 6) = 0 ]
Possible solutions: (x = 3) or (x = -6).
4.Identifying the Extraneous Solution (Part 4):
The extraneous solution occurs when it doesn’t make sense in the context of the problem.
A border width of (x = -6) would be negative, which is not practical.
Therefore, the extraneous solution is (x = -6).
5.Width of the Daisy Border (Part 5):
The correct value of (x) after excluding the extraneous solution is (x = 3).
So, each side of the daisy border is (3) feet wide.
