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Explanation:
x = width of the sidewalk path
The variable x is some placeholder for a positive number.
Check out the diagram below.
The green rectangle is the garden itself. The gray portion represents the sidewalk path. It's only along one vertical side and one horizontal side. So this sidewalk does not entirely encompass the garden.
If the horizontal component of the green garden rectangle is 12 meters, then it bumps up to 12+x meters when we incorporate the sidewalk.
Similarly, the vertical component of 5 meters bumps up to 5+x meters.
The entire figure is (12+x) by (5+x) which leads to an area of...
area = length*width
area = (12+x)(5+x)
area = 12(5+x) + x(5+x)
area = 60+12x+5x+x^2
area = x^2+17x+60
Set this equal to the desired area of 98 and solve for x.
x^2+17x+60 = 98
x^2+17x+60-98 = 0
x^2+17x-38 = 0
(x+19)(x-2) = 0
x+19 = 0 or x-2 = 0
x = -19 or x = 2
We stated earlier that x was positive, so we're going to ignore the first solution. Only x = 2 is practical here, so it's the final answer.
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Note that if x = 2, then,
- horizontal length = 12+x = 12+2 = 14 meters
- vertical width = 5+x = 5+2 = 7 meters
- larger area = length*width = 14*7 = 98 square meters
This helps us confirm we have the correct answer.
