Advanced Algebra
Chapter 8 Answers
Polynomials and Polynomial Functions
The graph of a polynomial is a curve. Designers may be able to model the shapes of their designs with polynomials, then use the models to analyze and perhaps improve what they have made.
Part A, Graphing Polynomial Functions;
Part B, Maximums and Minimums
1. Look at the photo of the roller coaster known as Python.
a. Explain why the shape of the portion of Python shown in the photo cannot be modeled by a polynomial function. [The coaster intersects itself. Therefore, for certain values of (horizontal) x, there is more than one value of (vertical) y.]
b. Any function that models the curve of a roller coaster track must be continuous. Why? [A discontinuity would be a gap in the track. At such a point, the roller coaster car would fly off the track.]
c. The graph below models the start of the roller coaster Desperado. Find Desperado's height and longest drop. Then identify the relative maximum and the relative minimum in the interval shown on the graph.
[Relative maximum = 209; relative minimum = 16]d. Graph the "roller coaster" curve y = 0.75 x4 + 6.33 x3 17.25 x2 + 15.67 x. TRACE to find the absolute maximum, the relative maximum, and the relative minimum.
[Absolute maximum: ~ (0.7, 4.5); Relative maximum: ~ (3.4, 2.4); Relative minimum: ~ (2.2, 0.8)]
Part C, Zeros of a Function;
Part D, Making Connections
2. The Indianapolis 500 auto race track consists of four straightaways separated by quarter-circles at the corners.
a. Find the lengths of the straightaways, the semicircles, and the entire track. Draw a picture that shows the track and its dimensions.
[short straightways 1/8 mi; long straightways 5/8 mi; semicircles 1/4 mi; total 1/2 mi]b. Find the measurements and area of the rectangular "infield" (shaded area) of the track.
[The rectangle measures 0.625 mi by 0.443 mi. Its area is about 0.28 mi2.]c. A track designer has been hired to see if the area of the infield can be increased to make room for more spectators. The idea is to change the lengths of the long straightaways and to convert the end sections to semicircles. The length of the track must stay the same. Express the length of the track in terms of x and r. (straightaway = x; radius = r)
[2x + 2πr]d. Equate the above expression to the actual length of the track and solve for r.
e. Express the area of the infield A as a polynomial in x.
f. Graph the polynomial.
g. Find the measurements and area of the redesigned infield with maximum area. [The rectangle measures x by 2r, or approximately 0.625 mi by 0.398 mi. Its area is 2rx = 0.249 mi2.]
h. Should the new track design be adopted?
[The design does not increase the area of the infield. It should not be adopted.]
Polynomials can be used to model some natural phenomena. Associated polynomial equations can be solved to find optimal solutions to problems relating to the phenonomena.
Part A, Solving Polynomial Equations by Graphing;
Part B, Factors and Roots of Polynomials
1. Grunion are fish that come ashore briefly to lay their eggs in the sand.
a. When do grunion lay their eggs? [at high tide]
b. The polynomial function h = 0.0002 t4 0.003 t 3 0.09 t2 + t + 9 approximates the wave height h in feet at one coastal location at time t on August 19, 1998 for the 24-hour interval from t = 0, to t = 24. Graph the function in the indicated interval.
c. Suppose a decision is made to rope off the beach when the grunion are expected to come ashore, in order to protect the fish laying eggs. At what time will the grunion be arriving? About how high will the waves be? [just before 5 A.M.; about 11.5 ft]
Part C, The Fundamental Theorem;
Part D, Making Connections
2. A builder wondered if a 25-ft white pine log will produce enough lumber to build an entire house.
a. Large quantities of lumber are measured in board-feet. One board foot bf is the amount of lumber in a 12-in. square board that is 1-in. thick. About how many board feet of lumber would you need to build a 1500 ft2 house? [about 10,000 bf]
b. Find the normal range of heights and diameters of white pines. [height: 75-100 ft, diam: 2-4 ft]
c. The Doyle Log Rule gives the approximate lumber yield, y of a log bf given the radius r (in.) and length L (ft) of the log:
The builder's white pine log was 25-ft long and had a uniform diameter over its entire length. Use the length of the log and the number of board feet in a 1500 ft2 house to write a quadratic equation you can solve to find the radius that the tree must have if it is to supply all the lumber or the house.
d. Solve the equation for r by factoring. [0 = (r 42)(r + 38), or, r = 42 in.]
e. Will a 25-ft white pine log of normal radius supply enough lumber for the house? [No; maximum normal white pine radius is 2 ft, or 24 in, which falls short of the needed 42 in. radius.]
Several important formulas relating to ocean exploration involve division of one or more variables by others. To solve a problem involving such variables, you may need to graph a rational function or solve a rational equation.
Part A, Rational Expressions;
Part B, Adding and Subtracting Rational Expressions
1. A SCUBA diver descended at a rate of 10 ft/min to the wreck of the Kenosha off Long Island, New York.
a. How long did it take the diver to reach the ship? Use the greater depth in your calculation. [depth = 110 ft, so time = 11 min]
b. The diver spent 40 minutes exploring the Kenosha, moving at an average rate of 8 ft/min. How far did the diver travel? [320 ft]
c. In order to avoid a potentially fatal condition called the bends, divers must pause at specific depths for extended periods of time as they ascend to the surface. The U.S. Navy recommends that divers take about 33 minutes to ascend after a 40-min stay at the depth of the Kenosha. What is the recommended rate of ascent in ft/min? [110 ft / 33 min = 3 1/3 ft/min]
d. Find the diver's total distance traveled, total time, and average rate of speed. [540 ft; 84 min; 6 3/7 ft/min]
e. A diver descends at a rate of 5 ft/min to a shipwreck at a depth of d ft. Over the next 30 minutes the diver explores the ship, traveling a distance equal to 4 times the ship's depth. Finally, the diver ascends to the surface in 60 minutes. Find and simplify the diver's average rate of speed. [distance = 6d; time = d/5 + 90; rate = 30d / d + 450]
f. Is the average speed a rational expression? Explain. [Yes. It is a ratio of two polynomials.]
Part E, Making Connections
2. Charles's Law relates the pressure P, volume V, and temperature T of a gas under two sets of conditions:
A killer whale with a volume (V1) of 200 ft3 of oxygen in its lungs rests on the ocean surface. Follow these steps to discover how the oxygen volume changes when the whale dives.
a. Find P1, the air pressure at sea level, in pounds per square inch. [P1 =14.7 lb/in2]
b. At the ocean's surface, where the whale is basking, the temperature is a balmy 27° C. To use Charles's Law, temperature must be expressed on the Kelvin scale. Find the relationship between Kelvin temperature and Celsius temperature. Then find T1, the Kelvin temperature at sea level. [K = C + 273°; T1 = 300° K]
c. The total pressure at any point in the ocean is the sum of the air pressure at sea level plus the water pressure. Water pressure increases by 0.44 lb/in2 for every foot descended. Write an expression for P2, the total pressure at a depth of d feet. [P2 = 14.7 + 0.44d]
d. Write an expression for T2, the Kelvin temperature at depth d. Assume that temperature drops 0.01 degree for each foot of descent. [T2 = 300 0.01d]
e. Use Charles's Law to find a rational expression for V2, the oxygen volume at depth d.
f. Graph volume as a function of depth.
g. What happens to the volume of oxygen as the whale descends? [It decreases, rapidly at first, then more and more slowly, approaching but never reaching zero.]
h. At what depth is the volume of oxygen half of the original volume? [about 33 ft]
i. What is the volume of oxygen at a depth of 1000 ft? [about 6.25 ft3]