Foundations of Algebra and Geometry
Chapter 8 Answers
Using Ratios to Compare
As they move upward, mountain climbers encounter increases in the amount of snow and decreases in the boiling point of water.
Part B, Expressing Change
1. Using the Internet, answer the following question.
a. Find the average annual snowfall in Jackson Hole, Wyoming. Scroll down the page to find your answer.
[396 in.]b. Complete the table.
No. of Years (n ) 1 2 3 4 Total Snowfall (s ) Answer:
No. of Years (n ) 1 2 3 4 Total Snowfall (s ) 396 792 1188 1584 c. Write an equation relating n and s. Name the independent variable.
[s = 396n ; n]d. Suppose snow built up on Alaska's Malaspina Glacier at the same rate that it falls in Jackson Hole. Write an equation for the thickness, t, of the glacier, in inches, after y years. Use the minimum thickness given for the glacier. Hint: Be sure to convert the thickness of the glacier into inches.
[t = 12,000 + 396y]
Part D, Making Connections
2. Using the Internet, answer the following questions.
a. Explain how the boiling point of Water changes in relation to a 5,000 foot change in altitude. Round your answer to the nearest degree.
[Boiling point is 212° F at sea level and decreases 9° F for each 5000 ft of altitude.]b. What is the elevation of Butte, MT ? (Enter "Butte" in the first blank. Disregard "to ______".) What is the boiling point of water in Butte?
c. Find the boiling point of water in your town.
[Answers will vary.]d. The equation b = 212 9 (e/5000) expresses the relationship between the boiling point of water, b, and the elevation, e. Use the equation to complete the table.
Elevation (ft) 0 2500 5000 75000 10,000 Boiling point (°F) Answer:
Elevation (ft) 0 2500 5000 75000 10,000 Boiling point(°F) 212 207.5 203 198.5 194 e. Graph the equation from d.
Answer:
In the following investigations, you'll use Olympic sprinting records to explore linear functions.
Part A, Understanding Linear Functions
1. Using the Internet, answer the following questions.
a. Find the winning time in the men's 100-meter run in each Olympic Games from 1904 to 1936. (Note: Because of World War I, there were no games in 1916). Then complete the table.
Year 1904 1908 1912 1916 1920 1924 1928 1932 1936 Time (sec) Answer:
Year 1904 1908 1912 1916 1920 1924 1928 1932 1936 Time (sec) 11.0 10.8 10.8 10.8 10.6 10.8 10.3 10.3
b. Do the values in the table represent a linear function? Explain.
[No. Equal changes in years are not matched by equal changes in times.]c. Draw a scatter plot of the values in the table. Put "Year" on the horizontal axis. Put "Time" on the vertical axis and show only the region from 10 sec to 11 sec.
Answer:
d. Explain how the plot shows that the function is not linear.
[No line contains all the points.]e. Find and draw a sloping trend line that contains three points on the plot.
Answer:
A horizontal line can be drawn through the four points representing times of 10.8 sec. But the question asks for a sloping line, not a horizontal one. One such line can be drawn representing values for 1904, 1912, and 1932.
f. Your trend line represents an ideal model of 100-meter-run times. Your data illustrates the rule that human behavior does not usually match mathematical models. Look at the five points that are not on your trend line. For the winning performances to match the model, what times should the athletes have attained? Rewrite the table from 1a showing three actual times and five ideal times. Indicate the times you have changed.
Answer:
Changes are in italics.
Year 1904 1908 1912 1916 1920 1924 1928 1932 1936 Time (sec) 11.0 10.9 10.8 10.6 10.5 10.4 10.3 10.2 g. Do the values in the new table represent a linear function? Explain.
[Yes. Equal changes in years are matched by equal changes in times.]
Part C, Making Connections
2. Using the Internet, answer the following questions.
a. American runner Bob Hayes won the gold medal in the 100-meter run in the 1964 Olympics. Find his winning time.
[10.0 sec]b. Express Hayes's velocity in meters per second.
[10 m/sec]c. Write and graph an equation y = vx, where v is Hayes's velocity (m/sec), x is the length of time he has run (sec), and y is the distance (m) he has run.
Answer:
d. Give the slope and y-intercept of the line.
[slope = 10; y-intercept = 0]e. Suppose that you and Bob Hayes both started running 100-meters at the same time but that Hayes gave you a 20-yd head start. After 10 seconds, you're still 20 meters from the finish line. On your graph from 2c, draw the line representing your run.
Answer:
f. What is the equation of the line representing your run?
[y = 6x + 20]g. Where were you 3.5 sec after the race started?
[y = 6(3.5) + 20 = 41 -> You were at (3.5, 41), 41 m from the 0-m starting line.]h. How can you use the equation to find your velocity? What was your velocity?
[Your velocity is the slope of the line; 6 m/sec.]i. Where do the lines intersect? What happened during the race at that point?
[The lines intersect at (5 sec, 50 m). At that point, Hayes passed you.]
Functions with curved graphs are often encountered in astrophysics, the study of the behavior of rockets and heavenly bodies.
Part A, Quadratic Functions
1. Using the Internet, answer the following questions.
a. Robert H. Goddard was the founding father of American rocketry. Find the maximum speed and height of the rockets he built between 1930 and 1935 .
[550 mi/h, 1.5 mi]b. Convert the above velocity to ft/sec by multiplying 550 by 5280, the number of feet in a mile. Then divide the product by the number of seconds in an hour. The result is the maximum speed of Goddard's rockets in ft/sec. Round to the nearest 100 ft.
c. The function y = vx 16x2 gives the height, y, of a rocket x seconds after launch, where v is the initial velocity. Write the formula for Goddard's rocket.
[y = 800x 16x2]d. Graph the function by plotting points at 5 second intervals, beginning at x = 0. Continue until the rocket returns to earth.
Answer:
These points are plotted: (0, 0), (5, 3600), (10, 6400), (15, 8400), (20, 9600), (25, 10000), (30, 9600), (35, 8400), (40, 6400), (45, 3600), (50, 0)
e. What is the maximum height reached by the rocket?
[10,000 ft]f. Compare the maximum height you found with that given in the story about Goddard. Explain the discrepancy.
[The maximum height mentioned in the story is 1.5 mi = 7,920 ft. The graph models an ideal flight of a rocket in a vacuum. Goddard's rocket did not reach the ideal height because of air drag, design deficiencies, and other factors.]
Part C, Making Connections
2. About 400 years ago, the German astronomer Johannes Kepler discovered a relationship between a planet's distance from the sun and the length of time the planet takes to orbit the sun. The relationship can be expressed as a square root function.
a. Find Mars' average distance from the Sun and the length of time it takes to orbit the Sun.
[142,000,000 mi; 687 days]b. The earth's average distance from the Sun is called an astronomical unit (AU). At the above site, find the length of 1 AU. Then express Mars' distance in AUs. Round to the nearest hundredth.
[93,000,000 mi; 1.53 AU]c. Express Mars' orbital time in years (1 year = 365 days). Round to the nearest hundredth.
[1.88 years]d. Cube Mars' distance (AU). Take the square root of the result. Compare with the planet's orbital time. What do you find?
Answer:
e. Complete the statement of Kepler's discovery: A planet's orbital time in years equals ______________.
[the square root of the cube of its distance from the Sun, in AUs.]f. Test the relationship using figures for Saturn.
