SOLUTION: Hi, Here is my problem Nancy walks 15 m diagonally across a rectangular field. She then returns to her starting position along the outside of the field. The total distance she w

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Question 169672: Hi, Here is my problem
Nancy walks 15 m diagonally across a rectangular field. She then returns to her starting position along the outside of the field. The total distance she walks is 36m. What are the dimensions of the field.
I have a good understanding of how to solve quadratic equations, but the word problems really confuse me. I made a diagram and used the pythagorean theorem to identify an equation, but it didn't factor in the end...Please help, if you could walk me through the steps that would be great
thanks
Found 2 solutions by Mathtut, jim_thompson5910:
Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
ok ........so when she walks diagonally across the field that ends up being the hypothenuse of the right triange which is 15 meters. what we dont know is the other 2 legs...but we have a hint since the perimeter of the triangel is 36m. So now we know that both legs add up to 21----->(36-15)....so we will call one leg a and the other 21-a.
:
a%5E2%2B%2821-a%29%5E2=15%5E2
:
a%5E2%2B441-42a%2Ba%5E2=225
:
2a%5E2-42a%2B216=0divide by 2
:
a%5E2-21a%2B108=0
:
using the quadratic we find that highlight%28a=9%29or highlight%28a=12%29
each leg can be either value. The dimensions of the field is 9m by 12m
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aa%5E2%2Bba%2Bc=0 (in our case 1a%5E2%2B-21a%2B108+=+0) has the following solutons:

a%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%2A108=9.

Discriminant d=9 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--21%2B-sqrt%28+9+%29%29%2F2%5Ca.

a%5B1%5D+=+%28-%28-21%29%2Bsqrt%28+9+%29%29%2F2%5C1+=+12
a%5B2%5D+=+%28-%28-21%29-sqrt%28+9+%29%29%2F2%5C1+=+9

Quadratic expression 1a%5E2%2B-21a%2B108 can be factored:
1a%5E2%2B-21a%2B108+=+1%28a-12%29%2A%28a-9%29
Again, the answer is: 12, 9. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-21%2Ax%2B108+%29

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=length of field and y=width of field


Remember, the perimeter of any triangle is simply the sum of its sides. In this case, two of the sides are the length and width (x and y) and the third side is the diagonal of the field


So the perimeter is P=x%2By%2B15

Now plug in P=36 to get 36=x%2By%2B15. After solving for y, we get y=21-x


Now since the diagonal splits the field into two triangles, where the length and width form the legs and the diagonal forms the hypotenuse, this means that we can use Pythagorean's Theorem

a%5E2%2Bb%5E2=c%5E2 Start with Pythagorean's Theorem


x%5E2%2By%5E2=15%5E2 Plug in a=x, b=y (the length and width) and c=15 (the diagonal)


x%5E2%2By%5E2=225 Square 15 to get 225


x%5E2%2B%2821-x%29%5E2=225 Plug in y=21-x (the previous isolated equation)


x%5E2%2B441-42x%2Bx%5E2=225 FOIL


x%5E2%2B441-42x%2Bx%5E2-225=0 Subtract 225 from both sides


2x%5E2-42x%2B216=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=2, b=-42, and c=216


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-42%29+%2B-+sqrt%28+%28-42%29%5E2-4%282%29%28216%29+%29%29%2F%282%282%29%29 Plug in a=2, b=-42, and c=216


x+=+%2842+%2B-+sqrt%28+%28-42%29%5E2-4%282%29%28216%29+%29%29%2F%282%282%29%29 Negate -42 to get 42.


x+=+%2842+%2B-+sqrt%28+1764-4%282%29%28216%29+%29%29%2F%282%282%29%29 Square -42 to get 1764.


x+=+%2842+%2B-+sqrt%28+1764-1728+%29%29%2F%282%282%29%29 Multiply 4%282%29%28216%29 to get 1728


x+=+%2842+%2B-+sqrt%28+36+%29%29%2F%282%282%29%29 Subtract 1728 from 1764 to get 36


x+=+%2842+%2B-+sqrt%28+36+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%2842+%2B-+6%29%2F%284%29 Take the square root of 36 to get 6.


x+=+%2842+%2B+6%29%2F%284%29 or x+=+%2842+-+6%29%2F%284%29 Break up the expression.


x+=+%2848%29%2F%284%29 or x+=++%2836%29%2F%284%29 Combine like terms.


x+=+12 or x+=+9 Simplify.


So the answers are x+=+12 or x+=+9



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Answer:


So the dimensions of the field are 12 m by 9 m