Solved: Architecture A semielliptical archway is to be formed over the entrance to an

In Exercises 14, find the inclination (in radians and degrees) of the line with the given characteristics. Passes through the points and

In Exercises 14, find the inclination (in radians and degrees) of the line with the given characteristics. Passes through the points and

In Exercises 14, find the inclination (in radians and degrees) of the line with the given characteristics. Equation:

In Exercises 14, find the inclination (in radians and degrees) of the line with the given characteristics. Equation:

In Exercises 58, find the angle (in radians and degrees) between the lines.

In Exercises 58, find the angle (in radians and degrees) between the lines.

In Exercises 58, find the angle (in radians and degrees) between the lines.

In Exercises 58, find the angle (in radians and degrees) between the lines. Point Line

In Exercises 9 and 10, find the distance between the point and the line.

In Exercises 9 and 10, find the distance between the point and the line.

In Exercises 11 and 12, state what type of conic is formed by the intersection of the plane and the double-napped cone.

In Exercises 11 and 12, state what type of conic is formed by the intersection of the plane and the double-napped cone.

In Exercises 1316, find the standard form of the equation of the parabola with the given characteristics. Then graph the parabola. Vertex:

In Exercises 1316, find the standard form of the equation of the parabola with the given characteristics. Then graph the parabola. Vertex: Focus: Focus:

In Exercises 1316, find the standard form of the equation of the parabola with the given characteristics. Then graph the parabola. Vertex:

In Exercises 1316, find the standard form of the equation of the parabola with the given characteristics. Then graph the parabola. Vertex: Directrix: Directrix:

In Exercises 17 and 18, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line.

In Exercises 17 and 18, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line.

Architecture A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters (see figure). How wide is the archway at ground level? FIGURE FOR 19 FIGURE FOR 20

Flashlight The light bulb in a flashlight is at the focus of its parabolic reflector, 1.5 centimeters from the vertex of the reflector (see figure). Write an equation of a cross section of the flashlights reflector with its focus on the positive -axis and its vertex at the origin.

In Exercises 2124, find the standard form of the equation of the ellipse with the given characteristics. Then graph the ellipse. Vertices: foci:

In Exercises 2124, find the standard form of the equation of the ellipse with the given characteristics. Then graph the ellipse. Vertices: foci:

In Exercises 2124, find the standard form of the equation of the ellipse with the given characteristics. Then graph the ellipse. Vertices: endpoints of the minor axis:

In Exercises 2124, find the standard form of the equation of the ellipse with the given characteristics. Then graph the ellipse. Vertices: endpoints of the minor axis:

Architecture A semielliptical archway is to be formed over the entrance to an estate. The arch is to be set on pillars that are 10 feet apart and is to have a height (atop the pillars) of 4 feet. Where should the foci be placed in order to sketch the arch?

Wading Pool You are building a wading pool that is in the shape of an ellipse. Your plans give an equation for the elliptical shape of the pool measured in feet as Find the longest distance across the pool, the shortest distance, and the distance between the foci.

In Exercises 2730, find the center, vertices, foci, and eccentricity of the ellipse.

In Exercises 2730, find the center, vertices, foci, and eccentricity of the ellipse.

In Exercises 2730, find the center, vertices, foci, and eccentricity of the ellipse.

In Exercises 2730, find the center, vertices, foci, and eccentricity of the ellipse.

In Exercises 3134, find the standard form of the equation of the hyperbola with the given characteristics. Vertices: foci:

In Exercises 3134, find the standard form of the equation of the hyperbola with the given characteristics. Vertices: foci:

In Exercises 3134, find the standard form of the equation of the hyperbola with the given characteristics. Foci: asymptotes:

In Exercises 3134, find the standard form of the equation of the hyperbola with the given characteristics. Foci: asymptotes:

In Exercises 3538, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid.

In Exercises 3538, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid.

In Exercises 3538, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid.

In Exercises 3538, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid.

LORAN Radio transmitting station A is located 200 miles east of transmitting station B. A ship is in an area to the north and 40 miles west of station A. Synchronized radio pulses transmitted at 186,000 miles per second by the two stations are received 0.0005 second sooner from station A than from station B. How far north is the ship?

Locating an Explosion Two of your friends live 4 miles apart and on the same east-west street, and you live halfway between them. You are having a three-way phone conversation when you hear an explosion. Six seconds later, your friend to the east hears the explosion, and your friend to the west hears it 8 seconds after you do. Find equations of two hyperbolas that would locate the explosion. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.)

In Exercises 4144, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

In Exercises 4144, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

In Exercises 4144, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

In Exercises 4144, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

In Exercises 4548, rotate the axes to eliminate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes.

In Exercises 4548, rotate the axes to eliminate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes.

In Exercises 4548, rotate the axes to eliminate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes.

In Exercises 4548, rotate the axes to eliminate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes.

In Exercises 4952, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for and (c) use a graphing utility to graph the equation.

In Exercises 4952, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for and (c) use a graphing utility to graph the equation.

In Exercises 4952, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for and (c) use a graphing utility to graph the equation.

In Exercises 4952, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for and (c) use a graphing utility to graph the equation.

In Exercises 53 and 54, complete the table for each set of parametric equations. Plot the points and sketch a graph of the parametric equations. and

In Exercises 53 and 54, complete the table for each set of parametric equations. Plot the points and sketch a graph of the parametric equations. and

In Exercises 5560, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. (c) Verify your result with a graphing utility.

In Exercises 5560, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. (c) Verify your result with a graphing utility.

In Exercises 5560, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. (c) Verify your result with a graphing utility.

In Exercises 5560, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. (c) Verify your result with a graphing utility.

In Exercises 5560, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. (c) Verify your result with a graphing utility.

In Exercises 5560, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. (c) Verify your result with a graphing utility. y _ 2 _ 5 sin _

Find a parametric representation of the circle with center and radius 6.

Find a parametric representation of the ellipse with center major axis horizontal and eight units in length, and minor axis six units in length.

Find a parametric representation of the hyperbola with vertices and foci

Involute of a Circle The involute of a circle is described by the endpoint of a string that is held taut as it is unwound from a spool (see figure). The spool does not rotate. Show that a parametric representation of the involute of a circle is

In Exercises 6568, plot the point given in polar coordinates and find two additional polar representations of the point, using

In Exercises 6568, plot the point given in polar coordinates and find two additional polar representations of the point, using

In Exercises 6568, plot the point given in polar coordinates and find two additional polar representations of the point, using

In Exercises 6568, plot the point given in polar coordinates and find two additional polar representations of the point, using

In Exercises 6972, a point in polar coordinates is given. Convert the point to rectangular coordinates.

In Exercises 6972, a point in polar coordinates is given. Convert the point to rectangular coordinates.

In Exercises 6972, a point in polar coordinates is given. Convert the point to rectangular coordinates.

In Exercises 6972, a point in polar coordinates is given. Convert the point to rectangular coordinates.

In Exercises 7376, a point in rectangular coordinates is given. Convert the point to polar coordinates.

In Exercises 7376, a point in rectangular coordinates is given. Convert the point to polar coordinates.

In Exercises 7376, a point in rectangular coordinates is given. Convert the point to polar coordinates.

In Exercises 7376, a point in rectangular coordinates is given. Convert the point to polar coordinates.

In Exercises 7782, convert the rectangular equation to polar form.

In Exercises 7782, convert the rectangular equation to polar form.

In Exercises 7782, convert the rectangular equation to polar form.

In Exercises 7782, convert the rectangular equation to polar form.

In Exercises 7782, convert the rectangular equation to polar form.

In Exercises 7782, convert the rectangular equation to polar form.

In Exercises 8388, convert the polar equation to rectangular form.

In Exercises 8388, convert the polar equation to rectangular form.

In Exercises 8388, convert the polar equation to rectangular form.

In Exercises 8388, convert the polar equation to rectangular form.

In Exercises 8388, convert the polar equation to rectangular form.

In Exercises 8388, convert the polar equation to rectangular form.

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 8998, determine the symmetry of the maximum value of , and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary).

In Exercises 99102, identify the type of polar graph and use a graphing utility to graph the equation.

In Exercises 99102, identify the type of polar graph and use a graphing utility to graph the equation.

In Exercises 99102, identify the type of polar graph and use a graphing utility to graph the equation.

In Exercises 99102, identify the type of polar graph and use a graphing utility to graph the equation.

In Exercises 103106, identify the conic and sketch its graph.

In Exercises 103106, identify the conic and sketch its graph.

In Exercises 103106, identify the conic and sketch its graph.

In Exercises 103106, identify the conic and sketch its graph.

In Exercises 107110, find a polar equation of the conic with its focus at the pole. Parabola Vertex:

In Exercises 107110, find a polar equation of the conic with its focus at the pole. Parabola Vertex:

In Exercises 107110, find a polar equation of the conic with its focus at the pole. Ellipse Vertices:

In Exercises 107110, find a polar equation of the conic with its focus at the pole. Hyperbola Vertices: _1, 0_, _7, 0_

Explorer 18 On November 26, 1963, the United States launched Explorer 18. Its low and high points above the surface of Earth were 119 miles and 122,800 miles, respectively (see figure). The center of Earth was at one focus of the orbit. Find the polar equation of the orbit and find the distance between the surface of Earth (assume Earth has a radius of 4000 miles) and the satellite when

Asteroid An asteroid takes a parabolic path with Earth as its focus. It is about 6,000,000 miles from Earth at its closest approach. Write the polar equation of the path of the asteroid with its vertex at Find the distance between the asteroid and Earth when True or False?

True or False? In Exercises 113116, determine whether the statement is true or false. Justify your answer. When in an equation of the form the graph of the equation can be a parabola only if also.

True or False? In Exercises 113116, determine whether the statement is true or false. Justify your answer. The graph of is a hyperbola.

True or False? In Exercises 113116, determine whether the statement is true or false. Justify your answer. Only one set of parametric equations can represent the line

True or False? In Exercises 113116, determine whether the statement is true or false. Justify your answer. There is a unique polar coordinate representation of each point in the plane.

Consider an ellipse with the major axis horizontal and 10 units in length. The number in the standard form of the equation of the ellipse must be less than what real number? Explain the change in the shape of the ellipse as approaches this number.

The graph of the parametric equations and is shown in the figure. How would the graph change for the equations and FIGURE FOR 118

A moving object is modeled by the parametric equations and where is time (see figure). How would the path change for the following? (a) (b)

Identify the type of symmetry each of the following polar points has with the point in the figure. (a) (b) (c)

What is the relationship between the graphs of the rectangular and polar equations? (a) (b)

Geometry The area of the ellipse in the figure is twice the area of the circle. What is the length of the major axis? (Hint: The area of an ellipse is )