Questions on Algebra: Quadratic Equation answered by real tutors!

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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: 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(9162) 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+4 feet.

a^2+b^2=c^2 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^2+(x+4)^2=(sqrt(194)/2)^2 Plug in a=x (this is the width), b=x+4 (this is the height), and c=sqrt(194)/2 (which is given as the diagonal)


x^2+(x+4)^2=194/4 Square sqrt(194)/2 to get 194/4


x^2+(x+4)^2=97/2 Reduce


x^2+x^2+8x+16=97/2 FOIL


2x^2+8x+16=97/2 Combine like terms.


4x^2+16x+32=97 Multiply every term by the LCD 2 to clear the fraction


4x^2+16x-65=0 Subtract 97 from both sides.


Notice we have a quadratic equation in the form of ax^2+bx+c=0 where a=4, b=16, and c=-65


Let's use the quadratic formula to solve for x


x = (-b +- sqrt( b^2-4ac ))/(2a) Start with the quadratic formula


x = (-(16) +- sqrt( (16)^2-4(4)(-65) ))/(2(4)) Plug in a=4, b=16, and c=-65


x = (-16 +- sqrt( 256-4(4)(-65) ))/(2(4)) Square 16 to get 256.


x = (-16 +- sqrt( 256--1040 ))/(2(4)) Multiply 4(4)(-65) to get -1040


x = (-16 +- sqrt( 256+1040 ))/(2(4)) Rewrite sqrt(256--1040) as sqrt(256+1040)


x = (-16 +- sqrt( 1296 ))/(2(4)) Add 256 to 1040 to get 1296


x = (-16 +- sqrt( 1296 ))/(8) Multiply 2 and 4 to get 8.


x = (-16 +- 36)/(8) Take the square root of 1296 to get 36.


x = (-16 + 36)/(8) or x = (-16 - 36)/(8) Break up the expression.


x = (20)/(8) or x = (-52)/(8) Combine like terms.


x = 5/2 or x = -13/2 Simplify.


So the possible widths are x = 5/2 or x = -13/2 (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 = 5/2 (which is the mixed fraction x=2&1/2)


So the width is 2 and a half feet


x+4 Go back to the expression that represents the height


2&1/2+4 Plug in x=2&1/2)


6&1/2 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&1/2 feet by 6&1/2 which means that the answer is B) 2 1/2 ft by 6 1/2 ft