$Find the length of the arc of the curve from point $P$ to point $Q.$Find an equation for the conic that satisfies the given conditions.Find the slope of the tangent line to the given polar curve at the point specified by the value of $.$

$r=5+2cos(),=\frac{}{3}$Consider the parametric equations below.

$x=2+3t,y=5-t^2$

$($a$)$ Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $t$ increases.

$($b$)$ Eliminate the parameter to find a Cartesian equation of the curve.The given curve is rotated about the $y-$axis. Find the area of the resulting surface.

$y=\frac{1}{4}{x}^2-\frac{1}{2}ln(x),4x5$The masses $m_i$ are located at the points $P_i.$ Find the moments $M_x$ and $M_y$ and the center of mass of the system.

hyperbola, vertices $(-3,1),(9,1),$ foci $(-4,1),(10,1)$

$x^2=(y-8{)}^3,P(1,9),Q(27,17)$Find the exact length of the polar curve.

$r={}^2,0\frac{}{4}$Consider the equation below.

$r=\frac{11}{6+3cos()}$

$($a$)$ Find the eccentricity.

$e=$

$($b$)$ Identify the conic.

ellipse

parabola

hyperbola

none of the above

$($c$)$ Give an equation of the directrix $($in Cartesian coordinates$).$

$($d$)$ Sketch the conic.