Modelling and problem solving

I need help with the following problems:

1.A piece of wire 12 cm long is cut into 2 pieces. One piece is used to form a square shape and the other to form a rectangle shape of which the length is twice its width. Find the length of the side of the square if the combined area of the 2 shapes is 4.25cm^2.

2. Using a square sheet of paper:

choose a point P on the top edge

join P to the bottom right hand corner

create a 45 degree angle in the top left hand corner using P

what position of the original point P will make the area OF THE QUADRILATERAL A MAXIMUM ?

3. A shape that has been of interest to architects and artisits over the centuries is the 'golden rectangle'. Many have thought that it gave the perfect proportions for buildings. The rectangle is such is such that if a square is drawn on one of the longer sides then the new rectangle is similar to rectangle APQD. find the value of X. (this is known as the golden ratio).

The outer rectangle is labelled APQD. It has sides 1 (the longer side) and width of x. The square has sides x. The inner rectangle has sides 1 - x and x.

Looking forward to hear from someone. Thanks.

1 Attachment(s)

Re: Modelling and problem solving

Hi Elmidge. Let me help you with at least point 3.

http://s8.postimage.org/f4y86fbyd/goldenre.png

Let $\displaystyle \phi$ denote 'The golden ratio'.

$\displaystyle \frac{a+b}{a}=\frac{a}{b}={\phi}$

Attachment 27096

Let's solve this algebraically.

$\displaystyle \frac{a}{a}+\frac{b}{a}={a}{b}=\phi\\\\ 1+\frac{b}{a}=\frac{a}{b}=\phi\\\\ 1+\frac{b}{a}=1+\frac{1}{{a}/{b}}=\phi\\\\1+\frac{1}{\phi}=\phi$

Multiply both sides by $\displaystyle \phi$ to get:

$\displaystyle \phi^2=\phi+1\Rightarrow\phi^2-\phi-1=0$

plug into quadratic formula:

$\displaystyle \frac{1\pm\sqrt{1-\left(-4\right)}}{2}}\Rightarrow\frac{1\pm\sqrt{5}}{2}} \Rightarrow \phi=1,61803...$

Please let me know if any of the steps confuse you.