Anthonete is weaving a towel.the length of thatowel 2 foot greater tha twice its width. the area covered by the towel is 25 square feet, how long is the towel?

Question

Please complete the equation in solving the word problem 

Answer 1

 

Area = length x width

 

25 = length x width

 

25 = l  x  w  where l is the length and w is the width

 

25 = (2w + 2) * w    <--- length is 2 feet greater than twice the width

 

25 = 2w^2 + 2w  <-- distributes the 2

 

0 = 2w^2 + 2w - 25 <--- subtracts 25 from both sides

 

Quadratic formula with A=2, B=2, and C = -25

 

w = [ -2 +or- sqrt( 2^2 - 4*2*(-25))]/(2*2)

 

   = [ -2 +or- sqrt( 4 + 200)]/4 <--- subtracting negative is the same as adding the positive; 2^2 = 4

 

  = [-2 +or- sqrt(204)]/4

 

  Prime factoring 204  ---> 204 = 2 * 102 = 2 * 2 * 51 = 2 * 2 * 3 * 17

  sqrt(204) = 2*sqrt(51)

 

 So the two candidates for the width are [ -2 +or- 2 sqrt(51)]/4

 

The 2 factors out of the numerator and cancels the 4 in the denominator

 

The two candidates are now: [-1 +or- sqrt(51) ]/2

 

The width of the rectangle MUST be positive.

So the negative sign is disqualified and rejected

because it will make the width negative.

 

So the width MUST be (-1 + sqrt(51))/2

     or (sqrt(51)-1)/2

 

The length L = 2*w+2 = 2(sqrt(51)-1)/2 + 2 = sqrt(51)- 1+2 = sqrt(51) +1

 

The area is ( sqrt(51) + 1) (sqrt(51 - 1)/2

                   

                ( sqrt(51) + 1) (sqrt(51 - 1) * 1/2

 

                (  51 - sqrt(51) + sqrt(51) - 1 ) * 1/2  <-- FOIL

 

                 (51 - 1)/2  <--- cross terms cancel per difference of squares pattern

 

                50/2=

 

             25

 

The values for width and length are highlighted in bold above and PROVEN to be correct

Answer 2

length of the towel is 2 foot greater tha twice its width

 L = 2w + 2

 

area covered by the towel is 25 ft²

Lw = 25

 

(2w + 2)w = 25

2w² + 2w - 25 = 0

 

from quadratic equation with a=2, b=2, c = -25

w = [-2±√(2²-4(2)(-25))]/[2(2)]

 

w = (-2 ±√204)/4

w = (-2 ±2√51)/4 = (-1 ±√51)/2

 

Since the width will be positive:  w = (-1 + √51)/2

 

L = 2w + 2 = -1 + √51 + 2 = 1 + √51 ≅ 8.14 ft