golden mean, golden section, or extreme, mean ratio equal to

[afritts1987]

I found the following definition in The Harper Collins Dictionary of Mathematics : " golden mean , golden section , or extreme and mean ratio , n. the proportion of the division of a line so that the smaller is to the larger as the larger is to the whole, or of the sides of a rectangle so that the ratio of their difference to the smaller equals that of the smaller to the larger, supposed in classical esthetic theory to be uniquely pleasing to the eye."
Those Greek were onto something! Apparently that "extreme and mean ratio" is a specific number. In fact it is _________________.

I have NO idea how to do this problem. I know that this message bored requires that we show work but I don't even know how to START this problem. The only hint my teacher gave me is "set up and solve a quadratic equation" but I dont even see any numbers involved

 

[Deleted member 4993]

Re: Strange Word Problem

I found the following definition in The Harper Collins Dictionary of Mathematics :
" golden mean , golden section , or extreme and mean ratio , n. the proportion of the division of a line so that the smaller is to the larger as the larger is to the whole, or of the sides of a rectangle so that the ratio of their difference to the smaller equals that of the smaller to the larger, supposed in classical esthetic theory to be uniquely pleasing to the eye."
Those Greek were onto something! Apparently that "extreme and mean ratio" is a specific number. In fact it is _________________.


I have NO idea how to do this problem. I know that this message bored requires that we show work but I don't even know how to START this problem. The only hint my teacher gave me is "set up and solve a quadratic equation" but I dont even see any numbers (algebra may not involve any number - may be all letters or symbols) involved

Like most algebra word problem - start with naming variables.

let
length of the larger section of the line be = L
length of the smaller section of the line be = S

length of the line be = L + W

"...the proportion of the division of a line so that the smaller is to the larger as the larger is to the whole...."

Now continue....

 

[soroban]

Re: Strange Word Problem

Hello, afritts1987!

Golden mean: the proportion of the division of a line so that the smaller is to the larger
as the larger is to the whole.

Make a diagram . . .

      : - - - - - 1 + x - - - - - :
    A *---------*---------------- * B
      :  - 1 -  :  - - - x - - -  :

Draw a line \(\displaystyle AB\). .Divide it into two parts.
Let the smaller part be \(\displaystyle 1\) and the larger part be \(\displaystyle x.\)
Then the entire line is \(\displaystyle 1 + x\)

\(\displaystyle \text{It says: }\;\underbrace{\text{smaller}}_1\;\underbrace{\text{is to}}_{\div}\;\underbrace{\text{larger}}_x\;\underbrace{\text{ as }}_=\;\underbrace{\text{larger}}_x\;\underbrace{\text{is to}}_{\div}\;\underbrace{\text{whole}}_{1+x}\)

\(\displaystyle \text{So we have: }\;\frac{1}{x} \:=\:\frac{x}{1+x} \quad\Rightarrow\quad 1 + x \:=\:x^2 \quad\Rightarrow\quad x^2 - x - 1 \:=\:0\)

\(\displaystyle \text{The Quadratic Formula gives us: }\;x \;=\;\frac{1 \pm\sqrt{5}}{2}\)

\(\displaystyle \text{Since }x\text{ is positive: }\;x \;=\;\frac{1 + \sqrt{5}}{2} \;=\;1.618033989...\)

\(\displaystyle \text{This quantity, the Golden Mean, is denoted: }\:\phi\text{ (phi)}\)

 

[Denis]

Re: Strange Word Problem

fritts, you could have googled "golden ratio" and easily get this on your own... :shock: