Chapter Review 847 of each type of seat should an aircraft be configured to maximize revenue? (b) If management decides that the ratio of first class to coach seats should never exceed 1:8, with how many of each type of seat should an aircraft be configured to maximize revenue? (c) If you were management, what would you do? [Hint: Assume that the airline charges C$ for a coach seat and $F for a first-class seat; > > C F C 0, .] 32. Challenge Problem Maximize = + z x y 10 4 subject to the constraints ≥ ≥ − ≥− − ≥− x y x y x y 0, 0, 4 9, 2 25, + ≤ + ≤ + ≤ − ≤ x y x y x y x y 2 31, 19, 4 43, 5 38, − ≤ x y2 4 30. Animal Nutrition Kevin’s dog Amadeus likes two kinds of canned dog food. Gourmet Dog costs $1.40 per can and has 20 units of a vitamin complex; the calorie content is 75 calories. Chow Hound costs $1.12 per can and has 35 units of vitamins and 50 calories. Kevin likes Amadeus to have at least 1175 units of vitamins a month and at least 2375 calories during the same time period. Kevin has space to store only 60 cans of dog food at a time. How much of each kind of dog food should Kevin buy each month to minimize his cost? 31. Airline Revenue An airline has two classes of service: first class and coach. Management’s experience has been that each aircraft should have at least 8 but no more than 16 firstclass seats and at least 80 but no more than 120 coach seats. (a) If management decides that the ratio of first class to coach seats should never exceed 1:12, with how many Explaining Concepts 33. Explain in your own words what a linear programming problem is and how it can be solved. Retain Your Knowledge Problems 34–43 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 34. Solve: − = m m 2 1 2 5 1 5 35. Graph π ( ) = − − y x tan 2 for at least two periods. Use the graph to determine the domain and range. 36. Radioactive Decay The half-life of titanium-44 is 63 years. How long will it take 200 grams to decay to 75 grams? Round to one decimal place. 37. Find the equation of the line that is parallel to = + y x3 11 and passes through the point ( ) −2, 1 . 38. The sum of two numbers is 16. If the larger number is 4 less than 3 times the smaller number, find the two numbers. 39. What amount must be invested at 4% interest compounded daily to have $15,000 in 3 years? 40. Find an equation of the ellipse that has vertices ( ) ± 0, 5 and foci ( ) ± 0, 3. 41. If ( ) = − + f x x x 2 7 5 1 , find ( ) −f x . 1 42. Factor completely: ( ) ( ) − + − x x x x 30 7 15 7 2 3/2 3 1/2 43. Consider the functions ( ) = − + + f x x x x 1 3 3 5 7 3 2 and ( ) ′′ = − f x x2 6. Given that f is concave up where ( ) ′′ > f x 0 and f is concave down where ( ) ′′ < f x 0, find where f is concave up and where f is concave down. Chapter Review Things to Know Systems of equations (pp. 755–757) Systems with no solutions are inconsistent. Systems with a solution are consistent. Consistent systems of linear equations have either a unique solution (independent) or an infinite number of solutions (dependent). Matrix (p. 770) Rectangular array of numbers, called entries Augmented matrix (p. 770) Row operations (p. 771) Row echelon form (p. 773) Reduced row echelon form (p. 776) Determinants and Cramer’s Rule (pp. 784–792) Matrix Algebra (pp. 795–807) m by n matrix (p. 796) Matrix with m rows and n columns Identity matrix In (p. 803) An n by n square matrix whose diagonal entries are 1’s, while all other entries are 0’s Inverse of a matrix (p. 803) −A 1 is the inverse of A if = = − − AA A A I . n 1 1 Nonsingular matrix (p. 804) A square matrix that has an inverse
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