Question

a patio is shaped like a golden rectangle. it's length ( the longer side) is 16ft. what is the patios width? write your answer in simplified radical form.

Answer 1

`a\ golden\ rectangle:\ \ \ (a+b)/(a)= (a)/(b) \\----------------\\a+b=16\ [ft]\ \ \ \ \Rightarrow \ \ \ b=16-a\\\\ (16)/(a) = (a)/(16-a) \ \ \ \ \Leftrightarrow\ \ \ 16(16-a)=a^2\\\\a^2+16a-256=0\\\\\ \ \ \Rightarrow\ \ \ \Delta=16^2-4\cdot(-256)=256\cdot5\ \ \ \Rightarrow\ \ \ √(\Delta) =16 √(5) \\\\a_1= (-16-16 √(5) )/(2) =-8-8 √(5) <0\\\\a_2= (-16+16 √(5) )/(2) =-8+8 √(5)=8( √(5) -1)>0\\\\`


`a=8( √(5) -1)\ \ \Rightarrow\ \ b=16-8( √(5) -1)=16-8√(5) +8=8(3- √(5) )\\\\Ans.\ This\ golden\ ractangle\ has\ sides:\\.\ \ \ \ \ \ \ 8( √(5) -1)\ [ft]\ \ and\ \ 8(3- √(5) )\ [ft]`