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 minimum and a maximum value as the elevator moves up the building. This pattern is called a standing wave pattern. Calculate the ratio of the maximum to the minimum signal strength that the pager sees. This is called the standing wave ratio (SWR). 2. Show that the distance between two consecutive minima is 1 meter. 3. Show that the distance between two consecutive maxima is 1 meter. 4. Show that the distance between a minimum and its nearby maximum is 0.5 meter. 5. What should the sensitivity of the pager be to guarantee reception of the signal regardless of the distance of the elevator from the ground? 6. Draw the graph in the complex plane as h varies. Explain all your answers to Questions 1 through 5 using this figure. E norm Signal Fades Due to Interference Consider an individual wearing a pager in an exterior el- evator on a high-rise building.The pager receives direct signals from a transmitting tower far away, as well as their reflections from the flat ground (see the figure). Oversimplifying things, it can be shown that the normalized signal received by the pager is given by where h is the height of the pager (elevator) from the ground expressed in meters. 1. Show that the magnitude of the normalized sig- nal received by the pager oscillates between a E(h) norm L 1 + 0.25 C cos (2 p h) + i sin (2 p h) D h 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?