Carnegie Mellon University - BUSINESS 21270sol1

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OLUTIONS FOR PART I 1. NUMBERS, SETS, AND FUNCTIONS 1.1. “We have at least four times as many chairs as tables.” The number of chairs (c) is at least (≥) four times the number of tables (t). ... Hence c ≥ 4t. 1.2. Fill in the blanks. The equation x 2Cbx Cc D 0 has exactly one solution when b2 D 4c, and it has no solutions when b2 < 4c. These statements follow from the quadratic formula. 1.3. Given that x C y D 100, the maximum value of x y is 2500. By the AGM Inequality, x y ≤ . xC2 y/2 D . 1000 2 /2 D 2500. This is achieved by x D y D 50. 1.4. The square has the largest area among all rectangles with a given perimeter. With side›lengths x; y and perimeter p, we have x C y D p=2. By the AGM Inequality, x y ≤ . xC2 y/2 D .p=4/2. The bound is achieved with equality when x D y, which is the case of a square. 1.5. Translation of “The temperature was 10◦ C and increased by 20◦ C”. “The temperature was 50◦ F and increased by 36◦ F”. (One converts a change of 20 degrees C to a change of 36 degrees F, not to a temperature of 68 degrees.) 1.6. Temperature scales. If f denotes the current temperature in Fahren› heit degrees and c denotes the current temperature in Celsius degrees, then we always have f D .9=5/c C 32. a) Equality in the values occurs at −40 degrees Fahrenheit, since −40 is the solution to f D .9=5/ f C 32. b) Equal magnitude with opposite signs occurs at 80=7 degrees Fahren› heit, since 80=7 is the solution to f D .9=5/.− f / C 32. c) The Fahrenheit value is twice the Celsius value at [Show More]

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