SOLUTION: The length of a pool is 3 times its width, and the pool is surrounded by a grass walk 4 feet wide. If the total area enclosed by the walk (of uniform width) is 684 square feet, fi

Question 134511: The length of a pool is 3 times its width, and the pool is surrounded by a grass walk 4 feet wide. If the total area enclosed by the walk (of uniform width) is 684 square feet, find the dimensions of the pool.
Thank you for your help
Answer by nycsharkman(136) (Show Source): You can put this solution on YOUR website!
The length of a pool is 3 times its width, and the pool is surrounded by a grass walk 4 feet wide. If the total area enclosed by the walk (of uniform width) is 684 square feet, find the dimensions of the pool.
This is quadratic equation word problem using a geometric figure...our swimming pool.
Information Given:
width = x...x is the unknown.
length = 3 times x or 3x.
Area = 684ft^2...We use a square because area is measured in SQUARE UNITS.
Grass walk = 4 feet wide
We know that if the grass walk surrounds the pool, it must be 4 feet wide all AROUND the pool. Make sense?

Here is the equation:
(3x + 8) times (x + 8) = 684
We use the FOIL method on the left side to get:
3x^2 + 24x + 8x + 64, which then becomes:
3x^2 + 32x + 64
We equate 3x^2 + 32x + 64 to the given area and we get:
3x^2 + 32x + 64 = 684
Subtract 684 from BOTH sides and we are left with:
3x^2 + 32x + 64 - 684 = 0
3x^2 + 32x -620 = 0
Factor the left side using groups.
3x^2 + 62x - 30x - 620 = 0
(3x^2 + 62x) = Group A
(- 30x - 620) = Group B
Factor each group separately.
Group A becomes x(3x + 62).
Group B becomes -10(3x + 32)
We now have this equation:
(x - 10)(3x + 62) = 0
Set each factor to 0 and solve for x.
x - 10 = 0
x = 10
===========
3x + 62 = 0
3x = -62
x = -62/3....This value for x is REJECTED because we are looking for the dimensions of the pool. Dimensions are distances and DISTANCE CANNOT be negative. Is this clear?
We use x = 10 to find our dimensions.
As I said above:
width = x
length = 3 times x or 3x
I just found that x = 10, right?
Then our width is 10 feet and the length is 3x = 3(10) = 30 feet.
Width = 10 feet
Length = 30 feet
Got it?

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