Are you sure you want the length to be smaller than the width? It's not exactly the way they're defined. I'll solve it that way, but it's an odd question.
Equations
P = 2L + 2W
Area = L*W
Substitution and Solve
2L + 2W = 36
L*W = 56
Divide the first equation by 2
L + W = 36/2
L + W = 18
Solve for L
L = 18 - W
Put this result in the second equation
W(18 - W) = 56 Remove the brackets
18W - W^2 = 56 Transfer the Left Side to the Right.
0 = W^2 - 18W + 56 Factor
0 = (W - 14)(W - 4)
W - 14 = 0
W = 14 <<<<<Answer
W - 4 = 0
W = 4
Comment
Both 14 and 4 are in cm
W = 14 cm <<<< answer
L = 4 <<<<< answer
The full range is
![-\pi](https://tex.z-dn.net/?f=-%5Cpi%3Cx%3C%5Cpi)
(length
![2L=2\pi](https://tex.z-dn.net/?f=2L%3D2%5Cpi)
), so the half range is
![L=\pi](https://tex.z-dn.net/?f=L%3D%5Cpi)
. The half range sine series would then be given by
![f(x)=\displaystyle\sum_{n\ge1}b_n\sin\dfrac{n\pi x}L=\sum_{n\ge1}b_n\sin nx](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdisplaystyle%5Csum_%7Bn%5Cge1%7Db_n%5Csin%5Cdfrac%7Bn%5Cpi%20x%7DL%3D%5Csum_%7Bn%5Cge1%7Db_n%5Csin%20nx)
where
![b_n=\displaystyle\frac2L\int_0^Lf(x)\sin\dfrac{n\pi x}L\,\mathrm dx=\frac2\pi\int_0^\pi(\pi-x)\sin nx\,\mathrm dx](https://tex.z-dn.net/?f=b_n%3D%5Cdisplaystyle%5Cfrac2L%5Cint_0%5ELf%28x%29%5Csin%5Cdfrac%7Bn%5Cpi%20x%7DL%5C%2C%5Cmathrm%20dx%3D%5Cfrac2%5Cpi%5Cint_0%5E%5Cpi%28%5Cpi-x%29%5Csin%20nx%5C%2C%5Cmathrm%20dx)
Essentially, this is the same as finding the Fourier series for the function
![\begin{cases}g(x)=\begin{cases}\pi-x&\text{for }0](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dg%28x%29%3D%5Cbegin%7Bcases%7D%5Cpi-x%26%5Ctext%7Bfor%20%7D0%3Cx%3C%5Cpi%5C%5C-%5Cpi-x%26%5Ctext%7Bfor%20%7D-%5Cpi%3Cx%3C0%5Cend%7Bcases%7D%5C%5Cg%28x%2B2%5Cpi%29%3Dg%28x%29%5Cend%7Bcases%7D)
Integrating by parts yields
![b_n=\dfrac2\pi\left(\dfrac\pi n-\dfrac{\sin n\pi}{n^2}\right)=\dfrac2n](https://tex.z-dn.net/?f=b_n%3D%5Cdfrac2%5Cpi%5Cleft%28%5Cdfrac%5Cpi%20n-%5Cdfrac%7B%5Csin%20n%5Cpi%7D%7Bn%5E2%7D%5Cright%29%3D%5Cdfrac2n)
So the half range sine series for this function is simply
The Fibonacci sequence follows the rule that each successive term is the sum of the two previous terms
The sum of all exterior angles for every shape is 360 so just divide 360 by the measurement of one exterior angle (18) to get 20
The simplified rational expression for that is 2