91Ó°ÊÓ

A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveyor. (c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor.

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 y^{2}-2 x^{2}-4 y-8 x-15=0$$

Vertices: \((5,0),(5,12)\); endpoints of the minor axis: \((1,6),(9,6)\)

In Exercises 45-58, find any points of intersection of the graphs algebraically and then verify using a graphing utility. $$ \begin{array}{r} 4 x^{2}+9 y^{2}-36 y=0 \\ x^{2}+9 y-27=0 \end{array} $$

In Exercises 53 and 54, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency. Parabola Tangent Line $$ \begin{array}{cc} \text { Parabola } & \text { Tangent Line } \\ x^{2}+12 y=0 & x+y-3=0\end{array} $$

Convert the polar equation to rectangular form. $$r=10$$

Involute of circle: $$ \begin{aligned} &x=\frac{1}{2}(\cos \theta+\theta \sin \theta) \\ &y=\frac{1}{2}(\sin \theta-\theta \cos \theta) \end{aligned} $$

In Exercises 55-58, find an equation of the tangent line to the parabola at the given point, and find the \(x\)-intercept of the line. $$ x^{2}=2 y,(4,8) $$

Find an equation of the ellipse with vertices \((\pm 5,0)\) and eccentricity \(e=\frac{3}{5}\).

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$25 x^{2}-10 x-200 y-119=0$$