SOLUTION: Within a 5m by 10m area is a rectangular pool surrounded by a uniform concrete border of area 10m^2. Find the width of the border. Thank you

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Question 1151456: Within a 5m by 10m area is a rectangular pool surrounded by a uniform concrete border of area 10m^2. Find the width of the border.
Thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

A diagram is optional, but it might help visualize the problem.

The blue rectangle is the 5 m by 10 m area we have to work with.
The red rectangle is the pool itself. Its dimensions are 5-2x and 10-2x
Note how I'm subtracting off two copies of x to get the dimensions of the reduced rectangle
The green dashed lines are each x meters long to represent the uniform border width.
Since x is some length, this means that x cannot be negative, and it makes little sense to have it be 0 either. So x > 0.

Let's define some variables
A = area of blue rectangle = area of entire workspace
B = area of red rectangle = area of pool only ignoring the border
C = area of border only

We know that
A = 5*10 = 50
B = (5-2x)(10-2x) = 50 - 30x + 4x^2 using the FOIL rule
C = A - B
C = (50) - (50-30x+4x^2)
C = -4x^2+30x
in addition, C = 10 since we want the border to be 10 square meters

So far we have
C = -4x^2+30x
C = 10

Equate the two right hand sides. Solve for x
-4x^2+30x = 10
-4x^2+30x-10 = 0
-2(2x^2-15x+5) = 0
From here use the quadratic formula. Plug in a = 2, b = -15, c = 5

x+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29 or x+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-15%29%2Bsqrt%28%28-15%29%5E2-4%282%29%285%29%29%29%2F%282%282%29%29 or x+=+%28-%28-15%29-sqrt%28%28-15%29%5E2-4%282%29%285%29%29%29%2F%282%282%29%29

x+=+%2815%2Bsqrt%28185%29%29%2F%284%29 or x+=+%2815-sqrt%28185%29%29%2F%284%29

x+=+7.15036762718386 or x+=+0.349632372816139

x+=+7.15 or x+=+0.35

Both x solutions are positive; however, we need to make sure that 5-2x and 10-2x are positive as well.

If x = 7.15, then
5-2x = 5-2*7.15 = -9.3
We get a negative result, so we ignore x = 7.15

If x = 0.35, then
5-2x = 5-2*0.35 = 4.3
10-2x = 10-2*0.35 = 9.3
Both results are positive, so x = 0.35 is a valid border width.

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Answer: The border is approximately 0.35 meters wide.