SOLUTION: A rectangular pool is 14 metres long and 6 metres wide. It is surrounded by a pebble path of uniform width {{{x}}} metres. If the area of the path is {{{96m^2}}}, find the width.

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Question 207877: A rectangular pool is 14 metres long and 6 metres wide. It is surrounded by a pebble path of uniform width metres. If the area of the path is , find the width.
Answer by pjalgeb(9) (Show Source):
You can put this solution on YOUR website!
Given:
Length of Pool = 14 metres
Width of Pool = 6 metres
Area of Path =

Required:
x = Width of Path = ?

Solution:
Area of Pool = L x W
= 14m x 6m
=

Area of Path =

Total Area = Area of Pool + Area of Path
= +
=

Because the pebble path is of uniform width, then the equation for total area is:

(Length of Pool+Width of Path)* (Width of Pool +Width of Path) = Total Area
or
--> We use 2x because the path is on both sides
--> Distributive Property
--> Combine Like Terms

--> Divide both sides by 4
--> The result is a quadratic equation

--> Subtract 21 from both sides

--> The resulting equation can be solved by method of completing the square

-->Add 25 to both sides to complete the square

-->The left side is now a perfect square trinomial

-->Factor the left side to a perfect binomial square

-->We can now solve for X by finding the square root of both sides

--> Even radicals have two roots(+ and -)

-->Factor the Radicand into 4 and 11

--> The square root of 4 is 2, so this can be moved out of the radical

--> There is no such thing as negative length so we use the other answer.
--> This is the simplified answer

or by using calculator we get:

Lets check our answer:
(2x+14)(2x+6) = 160 --> From the equation for the total area
[2(1.633) + 14][2(1.633) +6] = 160 --> Substitute the value of with our answer
(17.266)(9.266) = 160
160 = 160

Answer:
The width of the path is metres.