An Illustrative Example

Figure 1: An illustrating example: The ladder in the mine.

What is the longest ladder ($L_1+L_2$)? (see the Fig. 1)

$$L_1=\frac{w_1}{Sin b},L_2=\frac{w_2}{Sin c}, b=\pi-a-c,L=L_1+L_2=\frac{w_1}{sin(\pi-a-c)}+\frac{w_2}{sin c}$$

The maximum length of the ladder$\Rightarrow$ ${\frac{dL}{dc}}\rfloor_{c=C}=0$ $\Rightarrow$ calculus way
MATLAB way is as the following: (see the Fig. 2)

a=123*2*pi*/360
L=inline('9/sin(pi-2.1468-c)+7/sin(c)')
fplot(L,[0.4,0.5]); grid on
fminbnd(L,0.4,0.5)
L(0.4677)
fminbnd(L,0.4,0.5,optimset('Display','iter'))

Figure 2: An illustrating example: The ladder in the mine. Solution with MATLAB