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.

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?