Revision Notes for Trigonometry, 11th Edition

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CHAPTERPROJECTS(DOWNLOADONLY)TRIGONOMETRY:AUNITCIRCLEAPPROACHELEVENTHEDITIONMichael SullivanChicago State University

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Platinum Plan:$100/month for up to 3000 K-bytesof service plus $0.04 for each addi-tional K-byte of serviceYou have been requested to write a report thatanswers the following questions in order to aid em-ployees in choosing the appropriate pricing plan.(a) IfCis the monthly charge forxK-bytes of serv-ice, expressCas a function ofxfor each of thethree plans.(b) Graph each of the three functions found inpart (a).(c) For how many K-bytes of service is the SilverPlan the best pricing option? When is the GoldPlan best? When is the Platinum Plan best?Explain your reasoning.(d) Write a report that summarizes your findings.During the past decade the availability and usageof wireless Internet services have increased. The in-dustry has developed a number of pricing proposalsfor such services. Marketing data have indicated thatsubscribers of wireless Internet services have tendedto desire flat fee rate structures as compared withrates based totally on usage.The Computer ResourceDepartment of Indigo Media (hypothetical) has en-tered into a contractual agreement for wirelessInternet services. As a part of the contractual agree-ment, employees are able to sign up for their ownwireless services. Three pricing options are available:Silver Plan:$20/month for up to 200 K-bytes ofservice plus $0.16 for each addi-tional K-byte of serviceGold Plan:$50/month for up to 1000 K-bytesof service plus $0.08 for each addi-tional K-byte of serviceP r o j e c ta tM o t o r o l a

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3.Cost of CableYou work for the Silver Satellite &Cable TV Company in the Research & DevelopmentDepartment. You have been asked to come up with aformula to determine the cost of running cable from aconnection box to a new cable household. The first ex-ample that you are working with involves the Stevenfamily, who own a rural home with a driveway 2 mileslong extending to the house from a nearby highway. Thenearest connection box is along the highway but 5 milesfrom the driveway.It costs the company $100 per mile to install cablealong the highway and $140 per mile to install cable offthe highway. Because the Steven’s house is surrounded byfarmland that they own, it would be possible to run thecable overland to the house directly from the connectionbox or from any point between the connection box to thedriveway.

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(a) Draw a sketch of this problem situation, assuming thatthe highway is a straight road and the driveway is alsoa straight road perpendicular to the highway. Includetwo or more possible routes for the cable.(b) Letxrepresent the distance in miles that the cableruns along the highway from the connection box be-fore turning off toward the house. Express the totalcost of installation as a function ofx. (You maychoose to answer part (c) before part (b) if you wouldlike to examine concrete instances before creatingthe equation.)(c) Make a table of the possible integral values ofxandthe corresponding cost in each instance. Does onechoice appear to cost the least?(d) If you charge the Stevens $800 for installation, wouldyou be willing to let them choose which way the cablewould go? Explain.(e) Using a graphing calculator, graph the function frompart (b) and determine the value ofxthat would makethe installation cost minimum.(f) Before proceeding further with the installation, youcheck the local regulations for cable companies andfind that there is pending state legislation that saysthat the cable cannot turn off the highway more than0.5 mile from the Steven’s driveway. If this legislationpasses, what will be the ultimate cost of installing theSteven’s cable?(g) If the cable company wishes to install cable in 5000homes in this area, and assuming that the figures forthe Steven’s installation are typical, how much willthe new legislation cost the company overall if theycannot use the cheapest installation cost, but insteadhave to follow the new state regulations?

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4.Oil SpillAn oil tanker strikes a sand bar that rips a holein the hull of the ship. Oil begins leaking out of the tanker

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with the spilled oil forming a circle around the tanker. Theradius of the circle is increasing at the rate of 2.2 feet per hour.(a) Write the area of the circle as a function of theradiusr.(b) Write the radius of the circle as a function of timet.(c) What is the radius of the circle after 2 hours? What isthe radius of the circle after 2.5 hours?(d) Use the result of part (c) to determine the area of thecircle after 2 hours and 2.5 hours.(e) Determine a function that represents area as a func-tion of timet.(f)Use the result of part (e) to determine the area of thecircle after 2 hours and 2.5 hours.(g) Compute the average rate of change of the area ofthe circle from 2 hours to 2.5 hours.(h) Compute the average rate of change of the area ofthe circle from 3 hours to 3.5 hours.(i)Based on the results obtained in parts (g) and (h),what is happening to the average rate of change ofthe area of the circle as time passes?

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(j)If the oil tanker is 150 yards from shore, when willthe oil spill first reach the shoreline? (1 yard3 feet)(k) How long will it be until 6 miles of shoreline is cont-aminated with oil? (1 mile5280 feet)==

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Digital Transmission over the AirDigital communications is a revolutionary technol-ogy of the century. For many years, Motorola hasbeen one of the leading companies to employ digi-tal communication in wireless devices, such as cellphones.Figure 1 shows a simplified overview of a digitalcommunication transmission over the air. The infor-mation source to be transmitted can be audio, video,or data. The information source may be formattedinto a digital sequence of symbols from a finite setEanF={0, 1}. So 0110100 is an example of a digitalsequence.The period of the symbols is denoted byT.The principle of digital communication systems isthat, during the finite interval of timeT, the infor-mation symbol is represented by one digital wave-form from a finite set of digital waveforms before itis sent. This technique is calledmodulation.Modulation techniques use a carrier that is modu-lated by the information to be transmitted.The modu-lated carrier is transformed into an electromagneticfield and propagated in the air through an antenna.Theunmodulated carrier can be represented in its generalform by a sinusoidal functions(t)=AsinAv0t+fB,whereAis the amplitude,v0is the radian frequency,andfis the phase.Let’s assume thatA=1,f=0, andv0=2pf0radian, wheref0is the frequency of the unmodulatedcarrier.1.Writes(t)using these assumptions.2.What is the period,T0, of the unmodulated carrier?P r o j e c ta tM o t o r o l a3.Evaluates(t)fort=0,1/A4f0B,1/A2f0B,3/A4f0B,and1/f0.4.Graphs(t)for 0t12T0. That is, graph 12cycles of the function.5.For what values oftdoes the function reach itsmaximum value?[Hint:Expresstin terms off0].Three modulation techniques are used for trans-mission over the air: amplitude modulation, fre-quency modulation and phase modulation. In thisproject, we are interested in phase modulation.Figure 2 illustrates this process. An informationsymbol is mapped onto a phase that modulatesthe carrier. The modulated carrier is expressedbySi(t)=sinA2pf0t+ciB.Let’s assume the following mapping scheme:EanFSEcnF016.Map the binary sequenceM=010 into a phasesequenceP.7.What is the expression of the modulated carrierS0(t)forci=c0andS1(t)forci=c1?8.Let’s assume that in the sequenceMthe periodof each symbol isT=4T0. For each of the threeintervalsC0, 4T0D,C4T0, 8T0D, andC8T0, 12T0D, in-dicate which ofS0(t)orS1(t)is the modulatedcarrier. On the same graph, illustrateM,P, andthe modulated carrier for 0t12T0.c1=pc0=0Figure 1Simplified Overview of a Digital Communication TransmissionFigure 2Principle of Phase ModulationDigitalsymbols{n}Phase{n}Digital waveformsSi(t )sin (2f0t i)Carriers (t )sin (2f0t )MappingModulateInformationsourceDigitalsymbolsDigitalwaveform(Modulatedcarrier)Si(t )Carriers (t )FormatModulateTransmit

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