782 CHAPTER 11 Systems of Equations and Inequalities 72. − + = − − = − + = − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ x y z x y z x y z x y 3 1 2 4 0 3 2 1 2 5 73. + + − = − + + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y z w x y z w 4 4 2 3 3 74. − + = − + − = + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y x y z w z w 4 5 2 5 4 83. Production To manufacture an automobile requires painting, drying, and polishing. Epsilon Motor Company produces three types of cars: the Delta, the Beta, and the Sigma. Each Delta requires 10 hours (h) for painting, 3 h for drying, and 2 h for polishing. A Beta requires 16 h for painting, 5 h for drying, and 3 h for polishing, and a Sigma requires 8 h for painting, 2 h for drying, and 1 h for polishing. If the company has 240 h for painting, 69 h for drying, and 41 h for polishing per month, how many of each type of car are produced? 84. Production A Florida juice company completes the preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes (min) for sterilizing, 6 min for filling, and 1 min for labeling. Each case of grapefruit juice requires 10 min for sterilizing, 4 min for filling, and 2 min for labeling. Each case of tomato juice requires 12 min for sterilizing, 4 min for filling, and 1 min for labeling. If the company runs the sterilizing machine for 398 min, the filling machine for 164 min, and the labeling machine for 58 min, how many cases of each type of juice are prepared? 85. Electricity: Kirchhoff’s Rules An application of Kirchhoff’s Rules to the circuit shown results in the following system of equations: I I I I I I I I I I 200 40 80 0 360 20 80 40 0 80 70 200 0 1 2 4 3 1 2 3 2 4 1 + + = − − + = − − + = − + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ Find the currents I I I , , , 1 2 3 and I .4 I1 I4 200V 80V 1 2 40 V I2 20V 1 2 360 V 70V 1 2 80 V I3 Source: Based on Raymond A. Serway and John W. Jewett, Jr. Physics for Scientists and Engineers with Modern Physics, 9th ed. (Boston: Brooks/Cole, Cengage Learning, 2014) Prob. 30, pp. 860–861. 86. Electricity: Kirchhoff’s Rules An application of Kirchhoff’s Rules to the circuit shown results in the following system of equations: I I I I I I I 24 6 3 0 12 24 6 6 0 1 3 2 1 3 1 2 = + − − = + − − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ 75. Curve Fitting Find the function = + + y ax bx c 2 whose graph contains the points ( ) ( ) − − 1,2 , 2, 7 ,and ( ) − 2, 3. 76. Curve Fitting Find the function = + + y ax bx c 2 whose graph contains the points ( ) ( ) − − 1, 1 , 3, 1 ,and ( ) −2, 14 . 77. Curve Fitting Find the function ( ) = + + + f x ax bx cx d 3 2 for which ( ) ( ) ( ) − = − − = − = f f f 3 112, 1 2, 1 4, and ( ) = f 2 13. 78. Curve Fitting Find the function f x ax bx cx d 3 2 ( ) = + + + for which ( ) ( ) ( ) − = − − = = f f f 2 10, 1 3, 1 5, and ( ) = f 3 15. 79. Nutrition A dietitian at Palos Community Hospital wants a patient to have a meal that has 78 grams (g) of protein, 59 g of carbohydrates, and 75 milligrams (mg) of vitamin A. The hospital food service tells the dietitian that the dinner for today is salmon steak, baked eggs, and acorn squash. Each serving of salmon steak has 30 g of protein, 20 g of carbohydrates, and 2 mg of vitamin A. Each serving of baked eggs contains 15 g of protein, 2 g of carbohydrates, and 20 mg of vitamin A. Each serving of acorn squash contains 3 g of protein, 25 g of carbohydrates, and 32 mg of vitamin A. How many servings of each food should the dietitian provide for the patient? 80. Nutrition A dietitian at General Hospital wants a patient to have a meal that has 47 grams (g) of protein, 58 g of carbohydrates, and 630 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is pork chops, corn on the cob, and 2% milk. Each serving of pork chops has 23 g of protein, 0 g of carbohydrates, and 10 mg of calcium. Each serving of corn on the cob contains 3 g of protein, 16 g of carbohydrates, and 10 mg of calcium. Each glass of 2% milk contains 9 g of protein, 13 g of carbohydrates, and 300 mg of calcium. How many servings of each food should the dietitian provide for the patient? 81. Student Loans Carletta took out $10,000 in student loans. She was able to find three different banks that provide loans with simple interest. Bank One offers school loans at an annual interest rate of 2%, Bank Two provides an annual interest rate of 3% on school loans, and Bank Three gives loans at 4%.While in school, Carletta will pay a total of $330 in annual interest, and the loan amount from Bank One is half of the amount borrowed from Bank Three. Find the amount borrowed from each bank. 82. Landscaping A landscape company is hired to plant trees in three new subdivisions. The company charges the developer for each tree planted, an hourly rate to plant the trees, and a fixed delivery charge. In one subdivision it took 166 labor hours to plant 250 trees for a cost of $7520. In a second subdivision it took 124 labor hours to plant 200 trees for a cost of $5945. In the final subdivision it took 200 labor hours to plant 300 trees for a cost of $8985. Determine the cost for each tree, the hourly labor charge, and the fixed delivery charge. Source: www.bx.org Applications and Extensions
RkJQdWJsaXNoZXIy NjM5ODQ=