Mar 182009
There is a pool with a sidewalk around it. The pool measures 6m by 10m, and the total area of the pool and walk is 96 square meters. What is the width of the sidewalk?
(6 + 2x) * (10 + 2x) = 96
<=>
60 + 12x + 20x + 4x² = 96
<=>
4x² + 32x – 36 = 0
<=>
x² + 8x – 9 = 0
Delta = 64 + 36 = 100
Sqr (delta) = 10
x = (-8 + 10) / 2 = 1
The width of the sidewalk is one meter.
Verification :
1 + 6 + 1 = 8
1 + 10 + 1 = 12
8 * 12 = 96 => ok
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2m I know it's 8 x 12, but what the hell…
length = 10 + x = 10 + 2 = 12
width = 6 + x = 6 + 2 = 8
area = 96 (since the sidewalk goes around the pool the actual width of the sidewalk is 2/2 or 1m.)
(10 + x) (6 + x ) = 96
(x + 10) ( x + 6) – 96 = 0
X^2 + 16x +60 – 96 = 0
x^2 + 16x – 36 = 0
(x – ) (x – ) = 0
x = (-b +/- sqrt(b^2 -4ac)/2a)
x= (-16 +/- sqrt(256 + 144)/2a)
x = (-16 + 20)/2= 2 take the positive x
x = (-16 -20/2 = -18
References :
(6 + 2x) * (10 + 2x) = 96
<=>
60 + 12x + 20x + 4x² = 96
<=>
4x² + 32x – 36 = 0
<=>
x² + 8x – 9 = 0
Delta = 64 + 36 = 100
Sqr (delta) = 10
x = (-8 + 10) / 2 = 1
The width of the sidewalk is one meter.
Verification :
1 + 6 + 1 = 8
1 + 10 + 1 = 12
8 * 12 = 96 => ok
References :
math sucks
References :
the assumptioin is that the sidewalk has a constant width x meters.and there is a sidewalk on both sides and both ends
the area of the pool plus sidewalk shape 96 m^2 is the length x + 10 + x or 2x + 10 times the width x + 6 + x or 2x + 6.
(2x + 10)(2x + 6) = 4x^2 + 12x + 20x + 60 = 4x^2 + 32x + 60 = 96
simplify 4x^2 + 32x + 60 – 96 = 0 4x^2 + 32x – 36 =0
divide everything by 4 since you can without changing the (value of the solution to the) equation.
x^2 + 8x – 9 = 0 the factors of -9 that add up to 8 are 9 and -1
factor x^2 + 8x – 9 = (x + 9)(x – 1) = 0 set each factor = 0 to get the two solutions
x + 9 = 0 so x = -9 and x – 1 = 0 so x = 1
first check your work
(2(-9) + 10)(2(-9) + 6) = (-18 +10)(-18 + 6) = -8(-12) = 96 check
(2(1) + 10)(2(1) + 6) = (2 + 10)(2 + 6) = 12(8) = 96 check
clearly the sidewalk can't have a negative width in real application so we are only interested in the positive solution. x = 1
References :
You know the pool is 6 x 10. If the walk is of uniform width, then the total area of the pool plus the walk (the area of the rectangle covered by the pool and the walk) will be (6 + 2n) x (10 + 2n), if the width of the walk is n meters. (The walk surrounds the pool, so the total width would be the width of the walk plus the width of the pool plus the width of the walk on the other side of the pool, or (6 + 2n). The same thought process gets you the length as (10 + 2n).)
Now, you know the total area of the pool and the walk is 96 square meters, so (6 + 2n) x (10 + 2n) = 96.
Multiply it out:
60 + 20n + 12n + 4n^2 -96 = 0
Group:
60 – 96 + 20n + 12n + 4n^2 = 0
Simplfy:
4n^2 + 32n -36 = 0
Factor:
(n + 9) (4n – 4) = 0
Solve:
n = 1, n= -9
(you get here by noticing that if either (n + 9) = 0 or (4n – 4) = 0, then the whole thing is 0, regardless of what the other is, because 0 times anything is 0)
Introduce what you know about physical reality, to interpret the mathematical result:
You know that a width of -9 meters for a physical walkway makes no sense, so the other solution must be the answer: 1 meter.
Check your result:
If the walkway is 1 meter wide, then the whole area of the pool plus the walkway is 6 + 2 by 10 + 2, or 8 x 12 meters. 8×12=96, so it works.
Now go do the rest of your homework yourself. <g>
References :