Problem $1(15$ points$)$: Consider an idealized muscle.

$1\mathrm{.}1)[6$ points$]$ Express the force in the muscle when activated as a linear relationship between force $\backslash ($ F $\backslash )$ and active muscle length $\backslash ($ l $\backslash )-$ that is$,\backslash ($ F$=$m $\backslash$cdot l$+$b $\backslash ),$ where $\backslash ($ m $\backslash )$ is the slope and $\backslash ($ b $\backslash )$ is the y intercept. What are $\backslash ($ m $\backslash )$ and $\backslash ($ b $\backslash )?$

$1\mathrm{.}2)[3$ points$]$ Sketch the force$-$length relationship from part $(1\mathrm{.}2),$ marking $\backslash ($ l$\_\{$c$\}\backslash )$ on the x axis and $\backslash ($ c $\backslash )$ on the $\backslash ($ y $\backslash )$ axis.

$1\mathrm{.}3)[3$ points$]$ In reality, the force$-$length equation above is limited only for a small range of extension ratios. What are the practical limits to this linear relationship? $($That is$,$ why does the full force$-$length curve look like the figure below?$)$

$1\mathrm{.}4)[3$ points$]$ Estimate $\backslash ($ c $\backslash )$ from the graph below.