SOLUTION: A rectangular flower bed 30 yards long by 20 yards wide has a walk of uniform width around it. If the area of the path is 1/4 that of the flower bed, find the width of the path.

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Question 198520: A rectangular flower bed 30 yards long by 20 yards wide has a walk of uniform width around it. If the area of the path is 1/4 that of the flower bed, find the width of the path.
Answer by nerdybill(7384) (Show Source):
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A rectangular flower bed 30 yards long by 20 yards wide has a walk of uniform width around it. If the area of the path is 1/4 that of the flower bed, find the width of the path.
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Let w = width of path
then
Area of flower bed = (30)(20) = 600 sq yards
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Area of flower bed and walk = (30+2w)(20+2w)
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Area of path = "area of flower bed and walk" - "area of flower bed"
Area of path = (30+2w)(20+2w) - 600
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(1/4)600 = (30+2w)(20+2w) - 600
150 = (30+2w)(20+2w) - 600
750 = (30+2w)(20+2w)
750 = 600+60w+40w+4w^2
750 = 4w^2+100w+600
0 = 4w^2+100w-150
0 = 2w^2+50w-75
Using the quadratic equation we get:
w = {1.42, -26.42}
Throw out the negative solution leaves us with:
w = 1.42 yards
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Details of quadratic to follow:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=3100 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 1.41941090707505, -26.4194109070751. Here's your graph: