C HAPTER P ROJECTS (D OWNLOAD O NLY ) T RIGONOMETRY : A U NIT C IRCLE A PPROACH E LEVENTH E DITION Michael Sullivan Chicago State University Platinum Plan: $100/month for up to 3000 K-bytes of service plus $0.04 for each addi- tional K-byte of service You have been requested to write a report that answers the following questions in order to aid em- ployees in choosing the appropriate pricing plan. (a) If C is the monthly charge for x K-bytes of serv- ice, express C as a function of x for each of the three plans. (b) Graph each of the three functions found in part (a). (c) For how many K-bytes of service is the Silver Plan the best pricing option? When is the Gold Plan 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 usage of wireless Internet services have increased. The in- dustry has developed a number of pricing proposals for such services. Marketing data have indicated that subscribers of wireless Internet services have tended to desire flat fee rate structures as compared with rates based totally on usage.The Computer Resource Department of Indigo Media (hypothetical) has en- tered into a contractual agreement for wireless Internet services. As a part of the contractual agree- ment, employees are able to sign up for their own wireless services. Three pricing options are available: Silver Plan: $20/month for up to 200 K-bytes of service plus $0.16 for each addi- tional K-byte of service Gold Plan: $50/month for up to 1000 K-bytes of service plus $0.08 for each addi- tional K-byte of service P r o j e c t a t M o t o r o l a 3. Cost of Cable You work for the Silver Satellite & Cable TV Company in the Research & Development Department. You have been asked to come up with a formula to determine the cost of running cable from a connection box to a new cable household. The first ex- ample that you are working with involves the Steven family, who own a rural home with a driveway 2 miles long extending to the house from a nearby highway. The nearest connection box is along the highway but 5 miles from the driveway. It costs the company $100 per mile to install cable along the highway and $140 per mile to install cable off the highway. Because the Steven’s house is surrounded by farmland that they own, it would be possible to run the cable overland to the house directly from the connection box or from any point between the connection box to the driveway. (a) Draw a sketch of this problem situation, assuming that the highway is a straight road and the driveway is also a straight road perpendicular to the highway. Include two or more possible routes for the cable. (b) Let x represent the distance in miles that the cable runs along the highway from the connection box be- fore turning off toward the house. Express the total cost of installation as a function of x . (You may choose to answer part (c) before part (b) if you would like to examine concrete instances before creating the equation.) (c) Make a table of the possible integral values of x and the corresponding cost in each instance. Does one choice appear to cost the least? (d) If you charge the Stevens $800 for installation, would you be willing to let them choose which way the cable would go? Explain. (e) Using a graphing calculator, graph the function from part (b) and determine the value of x that would make the installation cost minimum. (f) Before proceeding further with the installation, you check the local regulations for cable companies and find that there is pending state legislation that says that the cable cannot turn off the highway more than 0.5 mile from the Steven’s driveway. If this legislation passes, what will be the ultimate cost of installing the Steven’s cable? (g) If the cable company wishes to install cable in 5000 homes in this area, and assuming that the figures for the Steven’s installation are typical, how much will the new legislation cost the company overall if they cannot use the cheapest installation cost, but instead have to follow the new state regulations? 4. Oil Spill An oil tanker strikes a sand bar that rips a hole in the hull of the ship. Oil begins leaking out of the tanker with the spilled oil forming a circle around the tanker. The radius 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 the radius r . (b) Write the radius of the circle as a function of time t . (c) What is the radius of the circle after 2 hours? What is the radius of the circle after 2.5 hours? (d) Use the result of part (c) to determine the area of the circle after 2 hours and 2.5 hours. (e) Determine a function that represents area as a func- tion of time t . (f) Use the result of part (e) to determine the area of the circle after 2 hours and 2.5 hours. (g) Compute the average rate of change of the area of the circle from 2 hours to 2.5 hours. (h) Compute the average rate of change of the area of the 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 of the area of the circle as time passes? (j) If the oil tanker is 150 yards from shore, when will the oil spill first reach the shoreline? (1 yard 3 feet) (k) How long will it be until 6 miles of shoreline is cont- aminated with oil? (1 mile 5280 feet) = = Digital Transmission over the Air Digital communications is a revolutionary technol- ogy of the century. For many years, Motorola has been one of the leading companies to employ digi- tal communication in wireless devices, such as cell phones. Figure 1 shows a simplified overview of a digital communication transmission over the air. The infor- mation source to be transmitted can be audio, video, or data. The information source may be formatted into a digital sequence of symbols from a finite set E a n F ={0, 1} . So 0110100 is an example of a digital sequence.The period of the symbols is denoted by T . The principle of digital communication systems is that, during the finite interval of time T , the infor- mation symbol is represented by one digital wave- form from a finite set of digital waveforms before it is sent. This technique is called modulation. Modulation techniques use a carrier that is modu- lated by the information to be transmitted.The modu- lated carrier is transformed into an electromagnetic field and propagated in the air through an antenna.The unmodulated carrier can be represented in its general form by a sinusoidal function s(t)=A sin A v 0 t+ f B , where A is the amplitude, v 0 is the radian frequency, and f is the phase. Let’s assume that A=1 , f =0 , and v 0 =2 p f 0 radian, where f 0 is the frequency of the unmodulated carrier. 1. Write s(t) using these assumptions. 2. What is the period, T 0 , of the unmodulated carrier? P r o j e c t a t M o t o r o l a 3. Evaluate s(t) for t=0 , 1/ A 4 f 0 B , 1/ A 2 f 0 B , 3/ A 4 f 0 B , and 1/ f 0 . 4. Graph s(t) for 0  t  12 T 0 . That is, graph 12 cycles of the function. 5. For what values of t does the function reach its maximum value? [ Hint: Express t in terms of f 0 ]. Three modulation techniques are used for trans- mission over the air: amplitude modulation, fre- quency modulation and phase modulation. In this project, we are interested in phase modulation. Figure 2 illustrates this process. An information symbol is mapped onto a phase that modulates the carrier. The modulated carrier is expressed by S i (t)= sin A 2 p f 0 t+ c i B . Let’s assume the following mapping scheme: E a n F S E c n F 0 1 6. Map the binary sequence M= 010 into a phase sequence P . 7. What is the expression of the modulated carrier S 0 (t) for c i = c 0 and S 1 (t) for c i = c 1 ? 8. Let’s assume that in the sequence M the period of each symbol is T =4 T 0 . For each of the three intervals C 0, 4 T 0 D , C 4 T 0 , 8 T 0 D , and C 8 T 0 , 12 T 0 D , in- dicate which of S 0 (t) or S 1 (t) is the modulated carrier. On the same graph, illustrate M , P , and the modulated carrier for 0  t  12 T 0 . c 1 = p c 0 = 0 Figure 1 Simplified Overview of a Digital Communication Transmission Figure 2 Principle of Phase Modulation Digital symbols {  n } Phase {  n } Digital waveforms S i ( t )  sin (2  f 0 t   i ) Carrier s ( t )  sin (2  f 0 t ) Mapping Modulate Information source Digital symbols Digital waveform (Modulated carrier) S i ( t ) Carrier s ( t ) Format Modulate Transmit