Exam 10: Parametric Equations and Polar Coordinates

Find an equation of the parabola with focus and directrix

In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located at A and B transmit simultaneous signals to a ship or an aircraft located at P.The onboard computer converts the time difference in receiving these signals into a distance difference and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola (see the figure).Suppose that station B is located mi due east of station A on a coastline.A ship received the signal from B microseconds ( before it received the signal from A.Assuming that radio signals travel at a speed of and if the ship is due north of B, how far off the coastline is the ship? Round yourAnswer to the nearest mile.Select the correct Answer

Eliminate the parameter to find a Cartesian equation of the curve.

Write a polar equation in r and of a hyperbola with the focus at the origin, with the eccentricity and directrix

Select the correct Answer: for each question -Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Select the correct Answer: for each question -Suppose a planet is discovered that revolves around its sun in an elliptical orbit with the sun at one focus.Its perihelion distance (minimum distance from the planet to the sun) is approximately km, and its aphelion distance (maximum distance from the planet to the sun) is approximately km.Approximate the eccentricity of the planet's orbit.Round to three decimal places.

Select the correct Answer: for each question. -Write a polar equation in and of an ellipse with the focus at the origin, with the eccentricity and directrix

Select the correct Answer: for each question -Find the area of the region enclosed by one loop of the curve.

Select the correct Answer: for each question. -The planet Mercury travels in an elliptical orbit with eccentricity Its minimum distance from the Sun is km.If the perihelion distance from a planet to the Sun is and the aphelion distance is , find the maximum distance (in km) from Mercury to the Sun.

Select the correct Answer: for each question. -The graph of the following curve is given.Find the area that it encloses.

Find parametric equations to represent the line segment from Select the correct Answer

Find the area of the region that lies inside the first curve and outside the second curve.

Select the correct Answer: for each question. -Use a graph to estimate the values of for which the curves and intersect.Round yourAnswer to two decimal places.

The orbit of Hale-Bopp comet, discovered in 1995, is an ellipse with eccentricity and one focus at the Sun.The length of its major axis is AU.[An astronomical unit (AU) is the mean distance between Earth and the Sun, about 93 million miles.] Find the maximum distance from the comet to the Sun.(The perihelion distance from a planet to the Sun is and the aphelion distance is Find theAnswer in AU and round to the nearest hundredth.

Select the correct Answer: for each question. -A cross-section of a parabolic reflector is shown in the figure.The bulb is located at the focus and the opening at the focus is 18 cm.Find an equation of the parabola.Let be the origin.Find the diameter of the opening cm from the vertex.

Find the surface area generated by rotating the lemniscate about the line

Describe the motion of a particle with position as t varies in the given interval

Select the correct Answer: for each question -Find the surface area generated by rotating the lemniscate about the line

Select the correct Answer: for each question. -Find the exact area of the surface obtained by rotating the given curve about the x-axis.

Match the equation with the correct graph.