A66 APPENDIX Review NOTE The icon is a Model It! icon. It indicates that the discussion or problem involves modeling. j Figure 27 The modeling process Real problem Verbal description Language of mathematics Mathematical problem Solution Check Check Check In Problems 79–84, without solving, determine the character of the solutions of each equation in the complex number system. Verify your answer using a graphing utility. 79. − + = x x 3 3 4 0 2 80. − + = x x 2 4 1 0 2 81. + = x x 2 3 4 2 82. + = x x 6 2 2 83. − + = x x 9 12 4 0 2 84. + + = x x 4 12 9 0 2 85. + i 2 3 is a solution of a quadratic equation with real coefficients. Find the other solution. 86. −i 4 is a solution of a quadratic equation with real coefficients. Find the other solution. In Problems 87–90, = − z i 3 4 and = + w i 8 3 . Write each expression in the standard form +a bi. 87. +z z 88. −w w 89. zz 90. −z w Applications and Extensions 91. Electrical Circuits The impedance Z , in ohms, of a circuit element is defined as the ratio of the phasor voltage V , in volts, across the element to the phasor current I , in amperes, through the element. That is, = Z V I . If the voltage across a circuit element is +i 18 volts and the current through the element is − i 3 4 amperes, find the impedance. 92. Parallel Circuits In an ac circuit with two parallel pathways, the total impedance Z , in ohms, satisfies the formula = + Z Z Z 1 1 1 , 1 2 where Z1 is the impedance of the first pathway and Z2 is the impedance of the second pathway. Find the total impedance if the impedances of the two pathways are = + Z i 2 1 ohms and = − Z i 4 3 2 ohms. 93. Use = + z a bi to show that + = z z a2 and − = z z bi 2 . 94. Use = + z a bi to show that = z z. 95. Use = + z a bi and = + w c di to show that + = + z w z w. 96. Use = + z a bi and = + w c di to show that ⋅ = ⋅ z w z w. A.8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications OBJECTIVES 1 Translate Verbal Descriptions into Mathematical Expressions (p. A67) 2 Solve Interest Problems (p. A67) 3 Solve Mixture Problems (p. A69) 4 Solve Uniform Motion Problems (p. A70) 5 Solve Constant Rate Job Problems (p. A71) Applied (word) problems do not come in the form “Solve the equation . . . .” Instead, they supply information using words, a verbal description of the real problem. So, to solve applied problems, we must be able to translate the verbal description into the language of mathematics. This can be done by using variables to represent unknown quantities and then finding relationships (such as equations) that involve these variables. The process of doing all this is called mathematical modeling . An equation or inequality that describes a relationship among the variables is called a model . Any solution to the mathematical problem must be checked against the mathematical problem, the verbal description, and the real problem. See Figure 27 for an illustration of the modeling process .
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