SOLUTION: I am having some trouble with this word problem, can some one help out and also show me the steps, thanks a lot Hazel has a screen door whose height is 4 feet more than its widt

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Question 155442: I am having some trouble with this word problem, can some one help out and also show me the steps, thanks a lot
Hazel has a screen door whose height is 4 feet more than its width. She wishes to stabilize the door by attaching a steel cable diagonally. If the cable measures (Sqrt 194)/2 ft, what are the dimensions of the door?
A) 2 1/4 ft by 6 1/4 ft
B) 2 1/2 ft by 6 1/2 ft
C) 3 ft by 7 ft
D) 3 1/2 ft by 7 1/2 ft
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = width


Since the "height is 4 feet more than its width", this means that the height is x%2B4 feet.

a%5E2%2Bb%5E2=c%5E2 Start with Pythagoreans Theorem. Note: "a" and "b" are the legs of the triangle (in this case the width and height of the door) and "c" is the hypotenuse (which in this problem is the diagonal of the door)


x%5E2%2B%28x%2B4%29%5E2=%28sqrt%28194%29%2F2%29%5E2 Plug in a=x (this is the width), b=x%2B4 (this is the height), and c=sqrt%28194%29%2F2 (which is given as the diagonal)


x%5E2%2B%28x%2B4%29%5E2=194%2F4 Square sqrt%28194%29%2F2 to get 194%2F4


x%5E2%2B%28x%2B4%29%5E2=97%2F2 Reduce


x%5E2%2Bx%5E2%2B8x%2B16=97%2F2 FOIL


2x%5E2%2B8x%2B16=97%2F2 Combine like terms.


4x%5E2%2B16x%2B32=97 Multiply every term by the LCD 2 to clear the fraction


4x%5E2%2B16x-65=0 Subtract 97 from both sides.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc=0 where a=4, b=16, and c=-65


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-%2816%29+%2B-+sqrt%28+%2816%29%5E2-4%284%29%28-65%29+%29%29%2F%282%284%29%29 Plug in a=4, b=16, and c=-65


x+=+%28-16+%2B-+sqrt%28+256-4%284%29%28-65%29+%29%29%2F%282%284%29%29 Square 16 to get 256.


x+=+%28-16+%2B-+sqrt%28+256--1040+%29%29%2F%282%284%29%29 Multiply 4%284%29%28-65%29 to get -1040


x+=+%28-16+%2B-+sqrt%28+256%2B1040+%29%29%2F%282%284%29%29 Rewrite sqrt%28256--1040%29 as sqrt%28256%2B1040%29


x+=+%28-16+%2B-+sqrt%28+1296+%29%29%2F%282%284%29%29 Add 256 to 1040 to get 1296


x+=+%28-16+%2B-+sqrt%28+1296+%29%29%2F%288%29 Multiply 2 and 4 to get 8.


x+=+%28-16+%2B-+36%29%2F%288%29 Take the square root of 1296 to get 36.


x+=+%28-16+%2B+36%29%2F%288%29 or x+=+%28-16+-+36%29%2F%288%29 Break up the expression.


x+=+%2820%29%2F%288%29 or x+=++%28-52%29%2F%288%29 Combine like terms.


x+=+5%2F2 or x+=+-13%2F2 Simplify.


So the possible widths are x+=+5%2F2 or x+=+-13%2F2 (which in decimal form are x=2.5 or x=-6.5 respectively)


However, since a negative width doesn't make sense, this means that the only solution is x+=+5%2F2 (which is the mixed fraction x=2%261%2F2)


So the width is 2 and a half feet


x%2B4 Go back to the expression that represents the height


2%261%2F2%2B4 Plug in x=2%261%2F2)


6%261%2F2 Add


So the height of the door is 6 and a half feet.


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Answer:
So the dimensions of the door are 2%261%2F2 feet by 6%261%2F2 which means that the answer is B) 2 1/2 ft by 6 1/2 ft