Chapter 10

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r=2(1-2 \sin \theta)$$

Find a set of parametric equations to represent the graph of the given rectangular equation using the parameters (a) \(t=x\) and (b) \(t=2-x.\) $$y=\ln (x+2)$$

Three listening stations located at (3300,0),(3300,1100) and (-3300,0) monitor an explosion. The last two stations detect the explosion 1 second and 4 seconds after the first, respectively. Determine the coordinates of the explosion. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: (0,3)

Convert the rectangular equation to polar form. Assume \(a<0\) $$3 x-y+2=0$$

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r=2 \cos \frac{3 \theta}{2}$$

Use a graphing utility to graph the curve represented by the parametric equations. Hypocycloid: \(x=3 \cos ^{3} \theta, y=3 \sin ^{3} \theta\)

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. $$\text { Focus: }\left(-\frac{1}{2}, 0\right)$$

Halley's comet has an elliptical orbit with the sun at one focus. The eccentricity of the orbit is approximately 0.97. The length of the major axis of the orbit is about 35.67 astronomical units. (An astronomical unit is about 93 million miles.) Find the standard form of the equation of the orbit. Place the center of the orbit at the origin and place the major axis on the \(x\) -axis.

Convert the rectangular equation to polar form. Assume \(a<0\) $$3 x+5 y-2=0$$

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r=3 \sin \frac{5 \theta}{2}$$

Use a graphing utility to graph the curve represented by the parametric equations. Curtate cycloid: \(x=8 \theta-4 \sin \theta, y=8-4 \cos \theta\)

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. $$\text { Focus: }\left(-\frac{3}{2}, 0\right)$$

The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length of the major axis is \(2 a\) (see figure). Show that the polar equation of the orbit of a planet is $$r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}$$ where \(e\) is the eccentricity.

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r=9 \sin \theta$$

Use a graphing utility to graph the curve represented by the parametric equations. Witch of Agnesi: \(x=2 \cot \theta, y=2 \sin ^{2} \theta\)

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(y=4\)

A lithotripter machine uses an elliptical reflector to break up kidney stones nonsurgically. A spark plug in the reflector generates energy waves at one focus of an ellipse. The reflector directs these waves toward the kidney stone positioned at the other focus of the ellipse with enough energy to break up the stone, as shown in the figure. The lengths of the major and minor axes of the ellipse are 280 millimeters and 160 millimeters, respectively. How far is the spark from the kidney stone?

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r^{2}=25 \cos 2 \theta$$

Use a graphing utility to graph the curve represented by the parametric equations. Folium of Descartes: \(x=\frac{3 t}{1+t^{3}}, y=\frac{3 t^{2}}{1+t^{3}}\)

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(y=-1\)

Convert the rectangular equation to polar form. Assume \(a<0\) $$\left(x^{2}+y^{2}\right)^{2}=9\left(x^{2}-y^{2}\right)$$

Use a graphing utility to graph the curve represented by the parametric equations. Cycloid: \(x=\theta+\sin \theta, y=1-\cos \theta\)

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$9 x^{2}+4 y^{2}-18 x+16 y-119=0$$

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(x=-2\)

Use a graphing utility to graph the curve represented by the parametric equations. Prolate cycloid: \(x=2 \theta-4 \sin \theta, y=2-4 \cos \theta\)

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(x^{2}+y^{2}-4 x-6 y-23=0\)

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(x=5\)

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-6 x=0$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(16 x^{2}-9 y^{2}+32 x+54 y-209=0\)

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Horizontal axis and passes through the point (3,3)

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-8 y=0$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(x^{2}+4 x-8 y+20=0\)

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Vertical axis and passes through the point (-8,-2)

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-2 a x=0$$

The sound pickup pattern of a microphone is modeled by the polar equation \(r=5+5 \cos \theta,\) where \(|r|\) measures how sensitive the microphone is to sounds coming from the angle \(\theta\). (a) Sketch the graph of the model and identify the type of polar graph. (b) At what angle is the microphone most sensitive to sound?

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y=\frac{1}{2} x^{2}$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(y^{2}+12 x+4 y+28=0\)

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-2 a y=0$$

The area of the lemniscate \(r^{2}=a^{2} \cos 2 \theta\) is \(a^{2} .\) Sketch the graph of \(r^{2}=16 \cos 2 \theta .\) Then find the area of one loop of the graph.

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y=-2 x^{2}$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(4 x^{2}+25 y^{2}+16 x+250 y+541=0\)

Determine whether the statement is true or false. Justify your answer. The graph of \(r=6 \sin 5 \theta\) is a rose curve with five petals.

Consider a projectile 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 $$x=\left(v_{0} \cos \theta\right) t \text { and } y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}.$$ The center field fence in a baseball stadium is 7 feet high and 408 feet from home plate. A baseball is hit at a point 3 feet above the ground. It leaves the bat at an angle of \(\theta\) degrees with the horizontal at a speed of 100 miles per hour (see figure). (a) Write a set of parametric equations that model the path of the baseball. (b) Use a graphing utility to graph the path of the baseball when \(\theta=15^{\circ} .\) Is the hit a home run? (c) Use the graphing utility to graph the path of the baseball when \(\theta=23^{\circ} .\) Is the hit a home run? (d) Find the minimum angle required for the hit to be a home run.

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y^{2}=-6 x$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(x^{2}+y^{2}+2 x-6 y=0\)

On November 27, \(1963,\) the United States launched a satellite named Explorer \(18 .\) Its low and high points above the surface of Earth were about 119 miles and 122,800 miles, respectively (see figure). The center of Earth is at one focus of the orbit. (a) Find the polar equation of the orbit (assume the radius of Earth is 4000 miles). (b) Find the distance between the surface of Earth and the satellite when \(\theta=60^{\circ}\). (c) Find the distance between the surface of Earth and the satellite when \(\theta=30^{\circ}\).

Determine whether the statement is true or false. Justify your answer. The graph of \(r=4+2 \cos \theta\) is a dimpled limaçon.

Consider a projectile 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 $$x=\left(v_{0} \cos \theta\right) t \text { and } y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}.$$ An archer releases an arrow from a bow at a point 5 feet above the ground. The arrow leaves the bow at an angle of \(15^{\circ}\) with the horizontal and at an initial speed of 225 feet per second. (a) Write a set of parametric equations that model the path of the arrow. (b) Assuming the ground is level, find the distance the arrow travels before it hits the ground. (Ignore air resistance.) (c) Use a graphing utility to graph the path of the arrow and approximate its maximum height. (d) Find the total time the arrow is in the air.

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y^{2}=3 x$$