The golden ratio, \(r\) , was considered by the Ancient Greeks to be the perfect ratio between the lengths of two adjacent sides of a rectangle. The exact value of \(r\) is \(\frac{{1 + \sqrt 5 }}{2}\).

Write down the value of \(r\)

i)     correct to \(5\) significant figures;

ii)    correct to \(2\) decimal places.

Phidias is designing rectangular windows with adjacent sides of length \(x\) metres and \(y\) metres. The area of each window is \(1\,{{\text{m}}^2}\).

Write down an equation to describe this information.

Phidias designs the windows so that the ratio between the longer side, \(y\) , and the shorter side, \(x\) , is the golden ratio, \(r\).

Write down an equation in \(y\) , \(x\) and \(r\) to describe this information.

Find the value of \(x\) .

i)     \(1.6180\)      (A1)

ii)    \(1.62\)          (A1)(ft)       (C2)

Note: Follow through from part (a)(i).

\(xy = 1\)         (A1)     (C1)

\(\frac{y}{x} = r\)  OR \(\frac{y}{x} = \frac{{1 + \sqrt 5 }}{2}\) OR equivalent         (A1)    (C1)

Note: Accept \(\frac{y}{x} = \)  their part (a)(i) or (a)(ii).

\({x^2}r = 1\) or eqivalent                (M1)

\(x = 0.786\,\,\,(0.78615...)\)          (A1)(ft)     (C2)

Note: Award (M1) for substituting their part (c) into their equation from part (b). Follow through from parts (a), (b) and (c). Use of \(r = 1.62\) gives \(0.785674...\)

Question 13: Golden ratio
This question was partially answered by all but the best candidates. Parts (a) and (b) yielded the most success. Only the best candidates were successful in part (d).

Question 13: Golden ratio
This question was partially answered by all but the best candidates. Parts (a) and (b) yielded the most success. Only the best candidates were successful in part (d).

Question 13: Golden ratio
This question was partially answered by all but the best candidates. Parts (a) and (b) yielded the most success. Only the best candidates were successful in part (d).

Question 13: Golden ratio
This question was partially answered by all but the best candidates. Parts (a) and (b) yielded the most success. Only the best candidates were successful in part (d).