$($a$)$ The rose petals for $k=$; $($b$)$ The rose petals for $k=\sqrt[\phantom{2}]{2}.$

$x=$acos$(kt)cos(t)$

$y=$acos$(kt)sin(t)$

where $a=3,=2,k=$ for Fig. $6*17(a)$ and $k=\sqrt[\phantom{2}]{2}$ for Fig. $6*17$

Derive the curvature formula for the curve in Fig. $6\mathrm{.}17($a$).$

Use

${\displaystyle \underset{0}{\overset{30}{}}}|r^{\text{'}}(t)|dt$

calculate the length of the curve in Fig. $6\mathrm{.}17($a$).$ Use R command integrate$()$ to evaluate the final integral to obtain a numerical value.$(b),$ and $t$ goes from $0$ to $30.$