The flower bed will be surrounded by a path of constant width with the same area as the flower bed. Calculate the width of the path.The parks department is planning a new flower bed outside city...

The flower bed will be surrounded by a path of constant width with the same area as the flower bed. Calculate the width of the path.

The parks department is planning a new flower bed outside city hall. It will be rectangular with dimensions 9 m by 6 m. The flower bed will be surrounded by a path of constant width with the same area as the flower bed. Calculate the width of the path.

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

We are given that the flower bed has dimensions 9m by 6m, thus an area of 54`m^2` . The walk surrounding the bed is of constant width, say `x` .

Then the area of the flower bed and walk combined is given by `A=(6+2x)(9+2x)` .(Draw a rectangle with length 9 and width 6. Then draw another rectangle around the first with the same distance from the original sides called x. Then the width of the larger enclosing rectangle is 6+2x, and the length is 9+2x)

We also know that the area of the walk is the same as the area of the flower bed which is 54, thus the area of the larger rectangle is 54+54=108.

Now `(6+2x)(9+2x)=108`

`54+30x+4x^2=108`

`4x^2+30x-54=0`

`(4x-6)(x+9)=0`

`x=3/2` or `x=-9` . From the context of the problem, x=-9 does not make sense so:

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The width of the path is `3/2` m

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Sources:
ramyaram's profile pic

ramyaram | Student, Grade 10 | (Level 1) eNoter

Posted on

Let x rep width of the path 

Width total=2x+6

Lenght total=2x+9

Total area=(area of path)(area of flower bed)

Total area=108

a=lw

108=(2x+6)(2x+9)

108=4x^2+30x+54

0=4x^2+30x-54

X=(-b+\-square root{b^2-4ac}) / 2a

X= [-30+\-square root {30^2-4(4)(-54)]\2(4)

X=1.5

Hope that helps :D

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