Exam 10: Parametric Equations and Polar Coordinates

Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places.

Find the point(s) on the curve where the tangent is horizontal.

If a projectile is fired with an initial velocity of meters per second at an angle above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations , , where g is the acceleration of gravity . If a gun is fired with and when will the bullet hit the ground?

Find an equation of the tangent line to the curve at the point corresponding to the value of the parameter. , ;

Find an equation of the tangent to the curve at the point by first eliminating the parameter. , ;

Sketch the polar curve with the given equation.

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

Find the area enclosed by the curve .

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 L = mi due east of station A on a coastline. A ship received the signal from B microseconds (µs) before it received the signal from A. Assuming that radio signals travel at a speed of ft /µs and if the ship is due north of B, how far off the coastline is the ship? Round your answer to the nearest mile.

Find parametric equations to represent the line segment from .

Find a polar equation for the curve represented by the given Cartesian equation.

Find an equation for the conic that satisfies the given conditions. hyperbola, foci (0, ± ) , vertices (0, ± )