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Use a graphing utility to graph the polar equation. Describe your viewing window. \(r=\frac{5 \pi}{8}\)

You and a friend live 4 miles apart (on the same "east-west" street) and are talking on the phone. You hear a clap of thunder from lightning in a storm, and 18 seconds later your friend hears the thunder. Find an equation that gives the possible places where the lightning could have occurred. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.)

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) for which the graph is traced only once. \(r=5+4 \cos \theta\)

Find the eccentricity of the ellipse. \(4 x^{2}+3 y^{2}-8 x+18 y+19=0\)

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

A projectile is launched at a height of \(h\) feet above the ground at an angle of \(\boldsymbol{\theta}\) with the horizontal. The initial velocity is \(v_{0}\) feet per second, and the path of the projectile is modeled by the parametric equations \(x=\left(v_{0} \cos \theta\right) t\) and \(y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}\) Use a graphing utility to graph the paths of a projectile launched from ground level at each value of \(\boldsymbol{\theta}\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and the range of the projectile. (a) \(\theta=15^{\circ}, \quad v_{0}=50\) feet per second (b) \(\theta=15^{\circ}, \quad v_{0}=120\) feet per second (c) \(\theta=10^{\circ}, \quad v_{0}=50\) feet per second (d) \(\theta=10^{\circ}, \quad v_{0}=120\) feet per second

A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet. (a) Draw a rectangular coordinate system on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identify the coordinates of the known points. (b) Find an equation of the semielliptical arch. (c) You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch?

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.

Halley's comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately 0.967 . The length of the major axis of the orbit is approximately 35.88 astronomical units. (An astronomical unit is about 93 million miles.) (a) Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the \(x\) -axis. (b) Use a graphing utility to graph the equation of the orbit. (c) Find the greatest (aphelion) and smallest (perihelion) distances from the sun's center to the comet's center.

The first artificial satellite to orbit Earth was Sputnik I (launched by the former Soviet Union in 1957 ). Its highest point above Earth's surface was 947 kilometers, and its lowest point was 228 kilometers (see figure). The center of Earth was at one focus of the elliptical orbit, and the radius of Earth is 6378 kilometers. Find the eccentricity of the orbit.