390 CHAPTER 6 Rational Expressions

390 CHAPTER 6Rational Expressions66. A doctor recorded a body-mass index of 47 on a patient’schart. Later, a nurse notices that the doctor recorded thepatient’s weight as 240 pounds but neglected to record thepatient’s height. Explain how the nurse can use the information from the chart to find the patient’s height. Then find theheight.In physics, when the source of a sound is traveling toward an observer, the relationship between the actual pitch a of the sound andthe pitch h that the observer hears due to the Doppler effect is deascribed by the formula h , where s is the speed of thes1 770sound source in miles per hour. Use this formula to answer Exercise 67 and 68.67. An emergency vehicle has a single-tone siren with the pitchof the musical note E. As it approaches an observer standingby the road, the vehicle is traveling 50 mph. Is the pitch thatthe observer hears due to the Doppler effect lower or higherthan the actual pitch? To which musical note is the pitch thatthe observer hears closest?Pitch of an Octave of MusicalNotes in Hertz (Hz)NotePitchMiddle 8. Suppose an emergency van has a single-tone siren with thepitch of the musical note G. If the van is traveling at 80 mphapproaching a standing observer, name the pitch the observerhears (rounded to the nearest tenth) and the musical noteclosest to that pitch.In electronics, the relationship among the resistances R 1 and R 2 oftwo resistors wired in a parallel circuit and their combined resis111tance R is described by the formula . Use thisRR1R2formula to solve Exercises 69 through 71.69. If the combined resistance is 2 ohms and one of the tworesistances is 3 ohms, find the other resistance.70. Find the combined resistance of two resistors of 12 ohmseach when they are wired in a parallel circuit.71. The relationship among resistance of two resistors wired ina parallel circuit and their combined resistance may be extended to three resistors of resistances R 1, R 2, and R 3. Writean equation you believe may describe the relationship anduse it to find the combined resistance if R 1 is 5, R 2 is 6, andR 3 is 2.72. For the formula1111- , find x if y 2, z 7, yzxwand w 6.Note: Greater numbers indicate higher pitches (acoustically).(Source: American Standards Association)6.7Variation and Problem SolvingOBJECTIVEOBJECTIVES1 Solve Problems InvolvingDirect Variation.2 Solve Problems InvolvingInverse Variation.1Solving Problems Involving Direct VariationA very familiar example of direct variation is the relationship of the circumferenceC of a circle to its radius r. The formula C 2pr expresses that the circumference isalways 2p times the radius. In other words, C is always a constant multiple 12p2 of r.Because it is, we say that C varies directly as r, that C varies directly with r, or that C isdirectly proportional to r.3 Solve Problems Involving JointVariation.4 Solve Problems InvolvingCombined Variation.C 2prconstant

Section 6.7Variation and Problem Solving 391Direct Variationy varies directly as x, or y is directly proportional to x, if there is a nonzero constantk such thaty kxThe number k is called the constant of variation or the constant of proportionality.In the above definition, the relationship described between x and y is a linear one.In other words, the graph of y kx is a line. The slope of the line is k, and the linepasses through the origin.For example, the graph of the direct variation equation C 2pr is shown. Thehorizontal axis represents the radius r, and the vertical axis is the circumference C.From the graph, we can read that when the radius is 6 units, the circumference isapproximately 38 units. Also, when the circumference is 45 units, the radius is between7 and 8 units. Notice that as the radius increases, the circumference increases.C50454035C30increases252015105C 2pr1 2 3 4 5 6 7 8 9 10ras r increasesE X A ircumference and area of each circle. See the insidecover for a list of geometric formulas.55.56.4 in.6 cm

Chapter 6 Vocabulary Check 39971. The horsepower to drive a boat varies directly as the cubeof the speed of the boat. If the speed of the boat is to double,determine the corresponding increase in horsepowerrequired.58.57.7m9 cm72. The volume of a cylinder varies jointly as the height and thesquare of the radius. If the height is halved and the radius isdoubled, determine what happens to the volume.73. Suppose that y varies directly as x. If x is doubled, what isthe effect on y?Find each square root. See Section 1.3.59. 28160. 23661. 21163.A4465.A962. 24164.A 252566.A 12174. Suppose that y varies directly as x 2 . If x is doubled, what isthe effect on y?kComplete the following table for the inverse variation y x overeach given value of k. Plot the points on a rectangular coordinatesystem.CONCEPT EXTENSIONSy Solve. See the Concept Check in this section. Choose the typeof variation that each equation represents. a. Direct variationb. Inverse variation c. Joint variation267. y x375. k 312124kx76. k 177. k 0.668. y x270. xy 1169. y 9abChapter 614x1278. k 5Vocabulary CheckFill in each blank with one of the words or phrases listed below.rational expressionequationcomplex fractionoppositessynthetic divisionleast common denominatorexpressionlong divisionjointlydirectlyinversely1. A rational expression whose numerator, denominator, or both contain one or more rational expressions is called a(n).2. To divide a polynomial by a polynomial other than a monomial, we use3. In the equation y kx, y varies4. In the equation y as x.k, y variesx5. The.as x.of a list of rational expressions is a polynomial of least degree whose factors include thedenominator factors in the list.6. When a polynomial is to be divided by a binomial of the form x - c, a shortcut process calledmaybe used.7. In the equation y kxz, y variesas x and z.8. The expressions 1x - 52 and 15 - x2 are called9. A(n).is an expression that can be written as the quotientPof two polynomials P and Q as long asQQ is not 0.10. Which is an expression and which is an equation? An example of anexample of anis25 2.xxis22 2 7, and anxx

394 CHAPTER 6 Rational Expressions Joint Variation If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional to, the other variables. If y kxz then the number k is the constant of variation or the constant of proportionality. 3. SOLVE. P 1344 2.5 Let V 2.5. 537.6 4. INTERPRET. Check: Check the proposed .