SOLUTION: A small square pool is surrounded by a concrete walk two meters wide. If the area of the pool is 2/3 of the total area of the pool and the walk, find the outer dimensions of the wa

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: A small square pool is surrounded by a concrete walk two meters wide. If the area of the pool is 2/3 of the total area of the pool and the walk, find the outer dimensions of the wa      Log On


   

Question 785392: A small square pool is surrounded by a concrete walk two meters wide. If the area of the pool is 2/3 of the total area of the pool and the walk, find the outer dimensions of the walk.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
x= outer side of the concrete walk (in meters)
Since there are red%282%29 meters of concrete walk to either side of the pool, the length of one side of the pool,
x-2%2Ared%282%29=x-4= length of the side of the pool (in meters).
x%2Ax=x%5E2= total area of the pool plus concrete walk (in square meters)
%28x-4%29%5E2 area of the pool (in square meters).
What the problem says in words translates as
%28x-4%29%5E2=%282%2F3%29%2Ax%5E2
x%5E2-8x%2B16=%282%2F3%29x%5E2
Multiplying times 3 both sides of the equation, we get
3%28x%5E2-8x%2B16%29=2x%5E2
3x%5E2-24x%2B48=2x%5E2
3x%5E2-24x%2B48-2x%5E2=0%7D%7D%0D%0A%7B%7B%7Bx%5E2-24x%2B48=0
SOlving that quadratic equation, we get two solutions.
One is negative, which does not make sense.
The other one is
x=12%2B4sqrt%286%29=%22approx.%2221.8
The outside dimensions of the walk are 21.8 meters by 21.8 meters.
(The pool sides measure 21.8 meters - 4 meters = 17.8 meters).

SOLVING THE QUADRATIC EQUATION:
Using the quadratic formula:
x+=+%28-%28-24%29+%2B-+sqrt%28%28-24%29%5E2-4%2A1%2A48+%29%29%2F%282%2A1%29+
x+=+%2824+%2B-+sqrt%28576-192%29%29%2F2
x+=+%2824+%2B-+sqrt%28384%29%29%2F2
x+=+%2824+%2B-+sqrt%2864%2A6%29%29%2F2
x+=+%2824+%2B-+sqrt%2864%29%2Asqrt%286%29%29%2F2
x+=+%2824+%2B-+8%2Asqrt%286%29%29%2F2
x+=+12+%2B-+4%2Asqrt%286%29
Completing the square:
x%5E2-24x%2B48=0
x%5E2-24x=-48
x%5E2-24x%2B144=-48%2B144
%28x-12%29%5E2=96
x=12+%2B-+sqrt%2896%29
x=12+%2B-+sqrt%2816%2A6%29
x=12+%2B-+sqrt%2816%29%2Asqrt%286%29
x=12+%2B-+4sqrt%286%29