you are to make a golden rectangle with 6 inches of string. What's the width and length?

Question

you are to make a golden rectangle with 6 inches of string. What's the width and length? If this helps, we're talking about the formula (1+ square root of five)/2

Answer 1

This problem has been solved at the following link:

 

https://www.wyzant.com/resources/answers/400051/you_are_to_make_a_golden_rectangle_with_6_inches_of_string_what_39_s_the_width_and_length

 

 

The answers given produce a perimeter of 12.

Perimeter = 2 * Length + 2 * width

   6 = 2 * ( length + width)

  3 = length + width

 

 So if x = width then length is 3-x

 

 (3-x)/x = (1 + sqrt(5))/2  <--- ratio of length to width is the golden ratio

 

2(3-x) = (1 + sqrt(5))x <-- cross multiplies

 

6 - 2x = (1 + sqrt(5))x

 

6 = (3 + sqrt(5))x  <--- adds 2x to both sides and combines like terms

 

x = 6/(3 + sqrt(5))  <--- solves for x

 

x  = 6(3 - sqrt(5))/(3 + sqrt(5))(3 - sqrt(5))  <--- rationalizes the denominator

 

x = 6(3 - sqrt(5))/ (9 - 5)  <--- FOILs the denominator; cross terms cancel as expected

 

x = 6/4(3 - sqrt(5))  <--- 6/4 = 3/2

 

x = 3/2(3 - sqrt(5))  <--- THIS IS THE WIDTH!!!!

 

The length is then  3 - 3/2(3 - sqrt(5))

 

                           3 - 9/2 + 3/2*sqrt(5)  <--- distributes -3/2

 

                      6/2 - 9/2 + 3/2 * sqrt(5) <-- common denominator

 

                      - 3/2 + 3/2*Sqrt(5)  <-- adds the numerators : 6/2 - 9/2 = -3/2

 

                         3/2*Sqrt(5) - 3/2  <--- definition of subtraction

 

                        = 3/2( sqrt(5) - 1)  <--- factors out 3/2; THIS IS THE LEGNTH!!!!

 

 

Now the check step:

 

Doubling the length: 3*sqrt(5) - 3  <--  2 x 3/2 = 3, then distributes the 3

 

Doubling the width: 3(3 - sqrt(5)) = 9 - 3*sqrt(5)  <--- 2 x 3/2 = 3, then distributes the 3

 

Adding them together, 3*sqrt(5) cancels, and 9 - 3 = 6, which is

the perimeter ( double the length + double the width)

 

 

Answer 2

  

The golden ratio, as you have kindly stated, is phi =  (1 + sqrt(5))/2

and is represented by the greek letter phi (pronounced either "FEE" or FI as in W-fi or fire "FIghter")

 

The golden rectangle is the rectangle whose adjacent side lengths are in 

the ratio of the golden ratio

 

Perimeter = 2 x length + 2 x width

6 = 2 x length + 2 x width

6 = 2 x ( length + width)

3 = length + width  <--- call this equation ALPHA

 

In order for this rectangle to be "GOLDEN", length/width = PHI

 

         Specifically,  length / width = 1 + sqrt(5) /2

 

       Cross multiplying, it is 2 * length = width * (1 + sqrt(5)) <--- call this equation BETA

 

             Solving equation ALPHA for length: length = 3 - width

 

                         2 * ( 3  - width ) = width * (1 + sqrt(5))

                      

                    6 - 2 * width = (1 + sqrt(5)) * width

 

                     6 = (1 + sqrt(5)) * width + 2 * width

 

                   6 = (3 + sqrt(5)) * width

 

             6 / (3 + sqrt(5)) = width

 

           6 * (3 - sqrt(5)) / (3 + sqrt(5))(3 - sqrt(5)) = width

 

       6 * (3 - sqrt(5)) / ( 9 - 5) = width

 

    6/4 * (3 - sqrt(5)) = width

 

    3/2 * (3 - sqrt(5)) = width

 

The length is then 3 - width = 3 - 3/2(3 - sqrt(5)) =

                                  

                                           3 - 9/2 + 3/2*sqrt(5) =

 

                                          6/2 - 9/2 + 3/2*Sqrt(5) =

 

                                           3/2*sqrt(5) - 3/2 = 3/2( sqrt(5) - 1)

 

So the width is 3/2 * ( 3 - sqrt(5)) and the length is 3/2*(sqrt(5)-1)

 

As a check, The perimeter is 2 * length + 2 * width =

 

                   2*(3/2)(sqrt(5) - 1) + 2*(3/2)*(3 - sqrt(5)) =

                  3 * (sqrt(5) - 1) + 3*(3 - sqrt(5)) =

                  3*Sqrt(5) - 3 + 9 - 3*sqrt(5) = 6

 

 

The dimesions of this golden rectangle is given in bold above

and are checked, verified, and proven to be correct.

Answer 3

Semiperimeter is 3" and that should be divided into width and length in the ratio 1:φ where phi is the Golden ratio which you cited.  (It's approx. 1.618033).

Answer 4

Let x = width   Then, the length = 3-x

 

(3-x)/x = (1+√5)/2

 

6-2x = x + √5x

 

3x+√5x = 6

 

x(3+√5) = 6

 

x = 6/(3+√5) ≈ 1.146" = width

 

Length = 3-x = 1.854"

 

Answer 5

You have 3 inches for a length and a width.

w + (1 + √5)w/2 = 3

2w + (1 + √5)w = 6

w(2 + 1 + √5) = 6

w(3 + √5) = 6

 

Can you complete?