91Ó°ÊÓ

Using a Graphing Utility to Find Polar Coordinates In Exercises \(61-70\) , use a graphing utility to find one set of polar coordinates of the point given in rectangular coordinates. $$\left(-\frac{7}{9},-\frac{3}{4}\right)$$

Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towen that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table by finding the height y of the suspension cables over the roadway at a distance of \(x\) meters from the center of the bridge.

Projectile Motion In Exercises 75 and \(76,\) consider the path of a projectile projected horizontally with a velocity of \(v\) feet per second at a height of s feet, where the model for the path is $$x^{2}=-\frac{v^{2}}{16}(y-s)$$ In this model (in which air resistance is disregarded), \(y\) is the height (in feet) of the projectile and \(x\) is the horizontal distance (in feet) the projectile travels. A ball is thrown from the top of a 100 -foot tower with a velocity of 28 feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontally before striking the ground?

The points represent the vertices of a triangle. (a) Draw triangle \(A B C\) in the coordinate plane, (b) find the altitude from vertex \(B\) of the triangle to side \(A C,\) and \((c)\) find the area of the triangle. \(A(-3,0), B(0,-2), C(2,3)\)

Points of lntersection A circle and a parabola can have \(0,1,2,3,\) or 4 points of intersection. Sketch the circle \(x^{2}+y^{2}=4 .\) Discuss how this circle could intersect a parabola with an equation of the form \(y=x^{2}+C .\) Then find the values of \(C\) for each of the five cases described below. Use a graphing utility to verify your results. (a) No points of intersection (b) One point of intersection (c) Two points of intersection (d) Three points of intersection (e) Four points of intersection

Write an equation for the rose curve \(r=2 \sin 2 \theta\) after it has been rotated through the given angle. $$\begin{array}{ll}{\text { (a) } \frac{\pi}{6}} & {\text { (b) } \frac{\pi}{2} \quad \text { (c) } \frac{2 \pi}{3} \quad \text { (d) } \pi}\end{array}$$

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.

Projectile Motion A projectile is launched at a height of \(h\) feet above the ground at an angle of \(\theta\) with the horizontal. The initial velocity is \(v_{0}\) feet per second, and the path of the projectile is modeled by the parametric equations $$ \begin{array}{l}{x=\left(v_{0} \cos \theta\right) t} \\ {\text { and }} \\\ {y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}}\end{array} $$ In Exercises 93 and \(94,\) use a graphing utility to graph the paths of a projectile launched from ground level at each value of \(\theta\) and \(v_{0 .}\) . For each case, use the graph to approximate the maximum height and the range of the projectile. \(\begin{array}{ll}{\text { (a) } \theta=60^{\circ},} & {v_{0}=88 \text { feet per second }} \\ {\text { (b) } \theta=60^{\circ},} & {v_{0}=132 \text { feet per second }} \\ {\text { (c) } \theta=45^{\circ},} & {v_{0}=88 \text { feet per second }} \\ {\text { (d) } \theta=45^{\circ},} & {v_{0}=132 \text { feet per second }}\end{array}\)

Projectile Motion Eliminate the parameter \(t\) from the parametric equations $$ \begin{array}{l}{x=\left(v_{0} \cos \theta\right) t} \\ {\text { and }} \\\ {y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}}\end{array} $$ for the motion of a projectile to show that the rectangular equation is $$ y=-\frac{16 \sec ^{2} \theta}{v_{0}^{2}} x^{2}+(\tan \theta) x+h $$

Converting a Polar Equation to Rectangular Form In Exercises \(91-116,\) convert the polar equation to rectangular form. $$r^{2}=\cos \theta$$