Question 155442
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 <b>every</b> 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