937 9.5 Nonlinear Systems of Equations 71. (Modeling) Equilibrium Demand and Price The supply and demand equations for a certain commodity are given. supply: p = 20.1q + 9 - 2 and demand: p = 225 - 0.1q (a) Find the equilibrium demand. (b) Find the equilibrium price (in dollars). 72. (Modeling) Circuit Gain In electronics, circuit gain is modeled by G= Bt R + Rt , where R is the value of a resistor, t is temperature, Rt is the value of R at temperature t, and B is a constant. The sensitivity of the circuit to temperature is modeled by S = BR 1R + Rt22. If B = 3.7 and t is 90 K (kelvins), find the values of R and Rt that will result in the values G= 0.4 and S = 0.001. Round answers to the nearest whole number. 73. (Modeling) Revenue for Public Colleges The percents of total revenue for public colleges in the United States from tuition/fees and state sources are modeled in the accompanying graph. (a) Interpret this graph. How are the sources of funding for public colleges changing with time? (b) During what time period was the revenue from state sources increasing? (c) Use the graph to estimate the year and the percent when the amounts from both sources were equal. 74. (Modeling) Revenue for Public Colleges The following equations model the percents of revenue from both sources in Exercise 73, where x = 0 represents fiscal year 2008, x = 1 represents fiscal year 2009, and so on. Use the equations to determine the year and percent, to the nearest tenth, when the amounts from both sources were equal as shown on the graph. S = 0.2259x2 - 2.652x + 25.59 State sources T = 0.4052x + 18.11 Tuition / Fees ’08 ’10 ’12 ’16 0 10 5 20 15 25 30 Fiscal Year Percent of Total Revenue Public College Revenue from Tuition/Fees and State Sources Data from National Center for Education Statistics. ’09 ’11 ’13 ’14 ’15 Tuition/Fees State sources Relating Concepts For individual or collaborative investigation (Exercises 75–80) Consider the following nonlinear system. Work Exercises 75–80 in order. y = x - 1 y = x2 - 4 75. How is the graph of y = x - 1 obtained by transforming the graph of y = x ? 76. How is the graph of y = x2 - 4 obtained by transforming the graph of y = x2? 77. Use the definition of absolute value to write y = x - 1 as a piecewise-defined function. 78. Write two quadratic equations that will be used to solve the system. (Hint: Set both parts of the piecewise-defined function in Exercise 77 equal to x2 - 4.) 79. Solve both equations from Exercise 78. Pay close attention to the restriction on x. 80. Use the values of x found in Exercise 79 to find the solution set of the system.
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