Advanced Algebra
Chapter 5 Answers
Quadratic Functions and Relations
To see the light emitted by astronomical objects, astronomers use telescopes in the familiar telescope shape. To "see" the radio waves emitted by such objects, astronomers use radio telescopes in the shape of a parabola.
Part B, Sketching Parabolas: f (x) = ax2
1. During the 1960s, NASA built radio telescopes around the world to study deep space and to track Apollo moon missions.
a. Look at this Photo of a telescope found at the Second Deep Space Station in Spain. Sketch the parabolic radio dish on a coordinate axes with the parabola opening upward and the vertex at the origin OR print the photo and draw a coordinate axis on the photo with the parabola opening upward and the vertex at the origin.
b. The size given for the telescope is the diameter of the dish. Use this value to estimate the depth of the dish. Identify two points on the graph of the parabola, in addition to the origin.
[Answer for 1a. and 1b.
The y-value of 6 for the two points with x-values of 13 and 13 is an estimate and may vary.]
c. Estimate the x-value (s) of the point (s) on the graph with a y- value of 3.
[Estimates may vary; one possible answer is 10 and 10]d. Find the equation of the parabola.
Part C, Translations: f(x) = a(x h)2 + k
2. Study the photos of New Mexico's Very Large Array (VLA) of radio telescopes.
a.Estimate the depth of each dish.
[Estimates will vary; one possible answer is 8 m.]b. In the photo of the dishes in "Y-formation" estimate the distance between two dishes at the far right.
[Estimates will vary; one possible answer is 50 m.]c. Sketch a VLA dish opening upward with its vertex at the origin. To its right, at the distance you estimated in b, sketch a second dish with its origin on the x-axis.
[Answer assumes a 50-m distance between dishes.]
d. Suppose the equation of the first dish is y = ax2. What is the equation of the second dish?
[Answers will vary. For a 50-m separation distance, y = a ( x - 75)2.]
Part D, Completing the Square
3. Find and study the photo of the 64-Meter Antenna at Parkes, Australia.
a. Sketch the radio telescope, including the tower that holds it, on an xy -coordinate axes. Show the dish in vertical position (the parabola opening upward). Place the base of the tower on thex-axis, centered at the origin.
b. Find the equation of the dish. Write the equation in both standard quadratic form and completed-square form.
[Equations will depend on estimates of depth of dish and height of building, h. For a 10-m depth, y = 0.01x2 +h.]
Part E, Making Connections
4. Study the picture of the radio telescope at the following Web site.
a. Sketch the Effelsberg radio telescope. Include in your sketch the tall structure that stands in the middle of the dish.
b. On your sketch, show the paths of at least four incoming radio waves.
[Answer to 4a. and 4b.]
The English scientist Isaac Newton (1642-1727) discovered that objects attract one another with a gravitational force that depends on the masses of the objects and the distances between their centers. Because of gravity, a dropped object falls toward Earth's center.
Part A, Solving Quadratic Equations Graphically
1. Use the NAR Web site for the following questions.
a. Find the current National Association of Rocketry altitude record in the "F Altitude" event under Rule 19 in the Ages 14-18 division. Convert the record altitude to feet. Use 1 m = 3.28 ft.
[4690.4 ft as of 7/8/98]
b. The rocket was motionless as it began its free-fall descent from the record altitude. Use the equation h ( t ) = -16 t 2 + v 0t + h 0 to find how long after reaching its high point the rocket struck the ground (h (t ) = 0).
[For the record as of 1/1/98, t = 17.1 sec.]c. Sketch height h as a function of time t for the entire flight.
Part B, Solving Quadratic Equations by Factoring
2. Saturn's moon Titan is the second largest moon in the solar system.
a. Find the mass and radius of Earth and Titan.
[Earth: mass, 5.976 ¥ 1024 kg; radius, 6378 km; Titan: mass, 1.35 ¥ 1023 kg, radius 2575 km]b. In the metric system, the acceleration of gravity at a body's surface is a = 4.9 ¥ mass BODY (radius EARTH) 2 / massEARTH ¥ (radius BODY) 2. Find a for Titan and round it to the nearest whole number. Then write Titan's equation of free fall, h (t ) = at 2+ v 0 t + h 0.
[a = 1; h (t ) = t 2 + v 0 t + h 0]c. A rock is ejected from a volcano on Titan at an initial height of 200 m and an initial velocity of 10 m/sec. Write the equation of free fall you can use to find how long it will take the rock to reach height 0. Write the coefficient of t 2 as a positive number.
[0 = -t2 + 10t + 200]d. Solve the equation by factoring.
[t = 20 seconds.]
Part C, Using the Quadratic Formula
3. Michael Collins was the Command Module Pilot on Apollo 11, the first successful manned lunar landing.
a. Read Collins's first comment under "Launch" to find his weight at takeoff.
[165 lb]b. Weight is a measure of Earth's gravitational pull on an object. An object that weighs w E pounds at Earth's surface will weigh w S pounds r miles above Earth's surface, wherer 2 + 7920r- - 15,681,600 /w E -wS, wS = 0
Use the quadratic formula to find how far above Earth's surface Astronaut Collins was when his weight measured 1 pound. [46,907 mi]
Part D, Classifying Solutions
Part E, Making Connections
4. Edwards Air Force Base in California is the touchdown site for the Space Shuttle.
a. Find the length of the runway at Edwards.
[14,000 ft]b. The distance, d, that the shuttle travels after touching down at Edwards is given by d = vt - v 2/2r where v is the shuttle's velocity, t is the number of seconds that elapse before the pilot applies the brakes, and r is the shuttle's deceleration rate. If the pilot's reaction time is 0.5 sec and the shuttle's deceleration rate is -12 ft/sec2, what is the maximum velocity at which the shuttle can land at one end of the runway without overshooting the other end?
[573.6 ft/sec. This is considerably faster than the normal shuttle landing speed of about 325 ft/sec (220mi/h).]
Mathematics is central to the study of health and medicine. In their investigations, medical researchers often encounter conic sections, sometimes in unexpected places.
Part A, Circles
1. Look at an image of a cauliflower Mosaic virus.
a. The image you see is a portion of an electron microscope scan of a photograph measuring approximately 24 cm by 19 cm. By scanning left-right and up-down, estimate the actual dimensions of a photo section that appears in your viewing screen. [Answers will vary; 13 ¥ 6 cm]
b. Measure the radius of a typical virus on your screen. Then estimate the radius of a virus as it appears in the photo.
[Answers will vary; about 0.6 cm]
c. Write the radius of the virus, as it appears in the photo, in nanometers (1 cm = 107 nm).
[Answers will vary; 6,000,000 nm]d. Use the magnification of the electron microscope image to approximate the actual radius of the virus in nanometers.
[Answers will vary; about 20 nm]e. If you graphed the virus on an xy-coordinate axis with the virus's center at the origin, what would its equation be? (1 unit = 1 nm)
[Answers will vary; x2 + y2 = 400]
Part B, Ellipses
2. Look at the image of the lithotripter beside the section "What Are the Advantages of Lithotripsy?"
a. Sketch the lithotripter, the kidney stone, and the shock waves as shown in the photo.
b. On your sketch, draw the major axis of the ellipse off which the shock waves bounce. Identify the vertices, foci, and center.
c. The distance from the vertex to the focus is 2 cm. The distance from the focus to the center is 8 cm. Write the equation of the ellipse.
[x2/100 + y2/36 = 1]
Part D, Hyperbolas/Making Connections
3. Estimate the speed and distance traveled of a deadly virus.
a. Find how long it took the 1918-1919 Influenza virus to spread from the coast of Europe, where it arrived on American troop ships, to Poland, a distance of approximately 1200 km. [about 80 days]
b. Approximate the speed at which the virus traveled, in miles per day.
[about 15 km/day]c. Public health officials in Allain detected the virus on May 5. Officials in Belnais, 200 km away, detected it on May 13.
How much farther from the point where the virus first struck Europe was Belnais than was Allain? (Call this distance d.) [about 120 km]d. Draw the line segment shown above. Draw a point on the segment whose distance from Belnais exceeds its distance from Allain by d km. How far is the point from Allain?
[about 40 km]e. Estimate the position and draw at least 6 other points whose distance from Belnais exceeds its distance from Allain by d km.
f. What conic section have you drawn a portion of? How do you know?
[a hyperbola; the difference of the distances of the plotted points from two fixed points (Allain and Belnais) is a constant (40 km)]g. How could health officials find where the virus struck Europe?
[Use data from a third town to draw a portion of another hyperbola. The point where the virus struck Europe is the intersection of the two curves.]
When two phenomena can be modeled by quadratic equations, you can analyze how they interact by solving a system of quadratic equations.
Part A, Quadratic-Linear Systems
1. Find out "How Far is it?" in the following questions.
a. Adair, Iowa, is due west of the Des Moines, Iowa, airport. Adair is at the extreme limit of the airport's radar. To determine the radius of the airport's radar circle, find the distance from Des Moines to Adair at this Web site.
[53 mi]
b. A plane over Adair enters the radar circle flying on a course with a slope of 1.8. Find the equation of the line representing the plane's course. Units are in miles and the airport is at (0, 0).
[y = 1.8x + 95.4]c. Write the equation of the radar circle.
[x2 + y2 = 532]d. Find the coordinates of the point where the plane will pass out of the radar circle.
[(28, 45)]
Part C, Quadratic-Quadratic Systems/Making Connections
2. Study comets, orbits, and quadratic systems in the following questions.
a. Some comets revolve around the sun in elliptical orbits. Check the Ephemerides (data) on two comets. Be sure to choose comets for which these values are given:
(1) a (half the length of the major axis) or z (reciprocal of a);
(2) e (the eccentricity).
(a and z are given in astronomical units (AU), where 1 AU equals Earth's average distance from the sun, about 93 million miles.) Round values to two decimal places.
[Answers will vary, depending on the comets chosen. Answers below are for comets 21P/Giacobini-Zimmer (a = 3.52, e = 0.71) and 88P/Howell (a = 3.14, e = 0.55), both visible during 1998.]b. Calculate b (half the length of the minor axis) for both comets using
. If you know z but not a , calculate a using a = 1/z. (Like a and z , b is given in AUs.)
[Answers will vary. GZ: b = 2.49; Howell: b = 2.62]c. Write the equations of both orbits, using AU's for units.
[Answers will vary. GZ: x2/3.522 + y2/2.492 = 1 ; Howell: x2/3.142 + y2/2.622 = 1]d. Sketch both orbits on graph paper, with the centers of the orbits at the origin. Choose an appropriate scale in AUs. Assume that both orbits lie in the same plane.
[Graphs will vary.]
e. Estimate whether the orbits intersect. Then confirm your estimate by solving the quadratic-quadratic system of orbital equations.
[Answers will vary. The orbits of comets Giacobini-Zimmer and Howell appear to intersect at four points (marked on graph).
Solution: (2.15, 1.91), (2.15, 1.91), (2.15, 1.91), (2.15, 1.91)
