SOLUTION: A rectangular garden with dimensions 18yd by 24yd is to have a sidewalk of uniform width placed completely around the inside of its border in such a way that the area of the remain

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Question 445401: A rectangular garden with dimensions 18yd by 24yd is to have a sidewalk of uniform width placed completely around the inside of its border in such a way that the area of the remaining garden is 216 yd. how wide is the sidewalk?
Found 2 solutions by mananth, chriswen:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
Let the width of sidewalk be x
..
Length of ground =24
Width = 18

Area = 432 M^2
Area of sidewalk = 68M^2
(24-2x)(18-2x)=216
432-48x-36x+4X^2=216
4X^2-84x-216=0
/4
x^2-21x-54=0
(x-18)(x-3)=-
x=18 OR x=3
the with is obviously 3 yards

Answer by chriswen(106) About Me  (Show Source):
You can put this solution on YOUR website!
Let x be the width of the border.
...
SO, 18-2x is the width after the sidewalk.
and 24-2x is the length of the garden after the sidewalk.
(since the sidewalk on both sides make the width and length smaller).
...
(18-2x)(24-2x)=216
2*(9-x)*2*(12-x)=216
(9-x)(12-x)=216/4
(9-x)(12-x)=54
...
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • Graphical form: %289-x%29%2A%2812-x%29 simplifies to x%5E2-21%2Ax%2B108
  • Text form: (9-x)*(12-x) simplifies to x^2-21*x+108
  • Cartoon (animation) form: simplify_cartoon%28+%289-x%29%2A%2812-x%29+%29
    For tutors: simplify_cartoon( (9-x)*(12-x) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at %28highlight_red%28+9+%29-x%29%2A%2812-x%29.
Moved 9 to the right of expression
It becomes %28-x%2Bhighlight_green%28+9+%29%29%2A%2812-x%29.
Look at %28-x%2B9%29%2A%28highlight_red%28+12+%29-x%29.
Moved 12 to the right of expression
It becomes %28-x%2B9%29%2A%28-x%2Bhighlight_green%28+12+%29%29.
Look at highlight_red%28+%28-x%2B9%29%2A%28-x%2B12%29+%29.
Expanded term %28-x%2B12%29 by using associative property on %28-x%2B9%29
It becomes -highlight_green%28+%28-x%2B12%29%2Ax+%29%2Bhighlight_green%28+%28-x%2B12%29%2A9+%29.
Look at -highlight_red%28+%28-x%2B12%29%2Ax+%29%2B%28-x%2B12%29%2A9.
Expanded term -x by using associative property on %28-x%2B12%29
It becomes highlight_green%28+x%2Ax+%29-highlight_green%28+x%2A12+%29%2B%28-x%2B12%29%2A9.
Look at highlight_red%28+x+%29%2Ahighlight_red%28+x+%29-x%2A12%2B%28-x%2B12%29%2A9.
Reduce similar several occurrences of highlight_red%28+x+%29 to highlight_green%28+x%5E2+%29
It becomes highlight_green%28+x%5E2+%29-x%2A12%2B%28-x%2B12%29%2A9.
Look at x%5E2-x%2A12%2Bhighlight_red%28+%28-x%2B12%29%2A9+%29.
Expanded term 9 by using associative property on %28-x%2B12%29
It becomes x%5E2-x%2A12-highlight_green%28+9%2Ax+%29%2Bhighlight_green%28+9%2A12+%29.
Look at x%5E2-x%2A12-9%2Ax%2Bhighlight_red%28+9+%29%2Ahighlight_red%28+12+%29.
Multiplied numerator integers
It becomes x%5E2-x%2A12-9%2Ax%2Bhighlight_green%28+108+%29.
Look at x%5E2-highlight_red%28+x%2A12+%29-highlight_red%28+9%2Ax+%29%2B108.
Eliminated similar terms highlight_red%28+-x%2A12+%29,highlight_red%28+-9%2Ax+%29 replacing them with highlight_green%28+%28-12-9%29%2Ax+%29
It becomes x%5E2%2Bhighlight_green%28+%28-12-9%29%2Ax+%29%2B108.
Look at x%5E2%2B%28-highlight_red%28+12+%29-highlight_red%28+9+%29%29%2Ax%2B108.
Added fractions or integers together
It becomes x%5E2%2B%28highlight_green%28+-21+%29%29%2Ax%2B108.
Look at x%5E2%2B%28highlight_red%28+-21+%29%29%2Ax%2B108.
Removed extra sign in front of -21
It becomes x%5E2%2B%28-highlight_green%28+21+%29%29%2Ax%2B108.
Look at x%5E2%2Bhighlight_red%28+%28-highlight_red%28+21+%29%29%2Ax+%29%2B108.
Remove unneeded parentheses around factor highlight_red%28+21+%29
It becomes x%5E2-highlight_green%28+21+%29%2Ax%2B108.
Result: x%5E2-21%2Ax%2B108

Universal Simplifier and Solver


Done!

...
x^2-21*x+108=54
x^2-21*x+108-54=0
x^2-21*x+54=0
...
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-21x%2B54+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-21%29%5E2-4%2A1%2A54=225.

Discriminant d=225 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--21%2B-sqrt%28+225+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-21%29%2Bsqrt%28+225+%29%29%2F2%5C1+=+18
x%5B2%5D+=+%28-%28-21%29-sqrt%28+225+%29%29%2F2%5C1+=+3

Quadratic expression 1x%5E2%2B-21x%2B54 can be factored:
1x%5E2%2B-21x%2B54+=+1%28x-18%29%2A%28x-3%29
Again, the answer is: 18, 3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-21%2Ax%2B54+%29

...
So there are two answers 18 and 3 but, we can't have a negative number.
When we had it in factored form (9-x)(12-x), x now has to be less than 9 or else one of these will be a negative number and you can't have a negative width.
...
So therefore the width of the sidewalk is 3 feet.