SECTION 11.1 Systems of Linear Equations: Substitution and Elimination 767 69. Computing a Refund The grocery store we use does not mark prices on its goods. My partner went to this store, bought three 1-pound packages of bacon and two cartons of eggs, and paid a total of $13.45. Not knowing that she went to the store, I also went to the same store, purchased two 1-pound packages of bacon and three cartons of eggs, and paid a total of $11.45. Now we want to return two 1-pound packages of bacon and two cartons of eggs. How much will be refunded? 70. Finding the Current of a Stream Pamela requires 3 hours to swim 15 miles downstream on the Illinois River. The return trip upstream takes 5 hours. Find Pamela’s average speed in still water. How fast is the current? (Assume that Pamela’s speed is the same in each direction.) 71. Pharmacy A doctor’s prescription calls for a daily intake containing 40 milligrams (mg) of vitamin C and 30 mg of vitamin D. Your pharmacy stocks two liquids that can be used: One contains 20% vitamin C and 30% vitamin D, the other 40% vitamin C and 20% vitamin D. How many milligrams of each compound should be mixed to fill the prescription? 72. Pharmacy A doctor’s prescription calls for the creation of pills that contain 12 units of vitamin B12 and 12 units of vitamin E. Your pharmacy stocks two powders that can be used to make these pills: One contains 20% vitamin B12 and 30% vitamin E, the other 40% vitamin B12 and 20% vitamin E. How many units of each powder should be mixed in each pill? 73. Curve Fitting Find real numbers a, b, and c so that the graph of the function = + + y ax bx c 2 contains the points ( ) ( ) −1, 4 , 2, 3 , and ( ) 0, 1 . 74. Curve Fitting Find real numbers a, b, and c so that the graph of the function = + + y ax bx c 2 contains the points ( ) ( ) − − − 1, 2 , 1, 4 ,and ( ) 2, 4 . 75. IS–LM Model in Economics In economics, the IS curve is a linear equation that represents all combinations of income Y and interest rates r that maintain an equilibrium in the market for goods in the economy. The LM curve is a linear equation that represents all combinations of income Y and interest rates r that maintain an equilibrium in the market for money in the economy. In an economy, suppose that the equilibrium level of income (in millions of dollars) and interest rates satisfy the system of equations − = + = ⎧ ⎨ ⎪⎪ ⎩⎪⎪ Y r Y r 0.06 5000 240 0.06 6000 900 Find the equilibrium level of income and interest rates. 76. IS–LM Model in Economics In economics, the IS curve is a linear equation that represents all combinations of income Y and interest rates r that maintain an equilibrium in the market for goods in the economy. The LM curve is a linear equation that represents all combinations of income Y and interest rates r that maintain an equilibrium in the market for money in the economy. In an economy, suppose that the equilibrium level of income (in millions of dollars) and interest rates satisfy the system of equations − = + = ⎧ ⎨ ⎪⎪ ⎩⎪⎪ Y r Y r 0.05 1000 10 0.05 800 100 Find the equilibrium level of income and interest rates. 62. Mixing a Solution A chemist wants to make 14 liters of a 40% acid solution. She has solutions that are 30% acid and 65% acid. How much of each must she mix? 63. Presale Order A wireless store owner takes presale orders for a new smartphone and tablet. He gets 340 preorders for the smartphone and 250 preorders for the tablet. The combined value of the preorders is $486,000. If the price of a smartphone and tablet together is $1665, how much does each device cost? 64. Financial Planning Bryce’s friend bought him a lottery ticket for his birthday that won $10,000. Bryce decided to give half to his friend and invest the other half. He wants to earn $500 in extra income each year from his investments. His financial adviser recommends investing in B-rated bonds paying 15% per year and in a certificate of deposit (CD) paying 7% per year. (a) How much money should Bryce invest in each account to realize exactly $500 in interest per year? (b) After 2 years, Bryce decides he wants to earn $650 per year from his investments. How should he reallocate his investments to achieve the new amount? 65. Computing Wind Speed With a tail wind, a small Piper aircraft can fly 600 miles in 3 hours. Against this same wind, the Piper can fly the same distance in 4 hours. Find the average wind speed and the average airspeed of the Piper. 3 hours 4 hours 600 mi. 66. Computing Wind Speed The average airspeed of a singleengine aircraft is 150 miles per hour. If the aircraft flew the same distance in 2 hours with the wind as it flew in 3 hours against the wind, what was the wind speed? 67. Restaurant Management A restaurant manager wants to purchase 200 sets of dishes. One design costs $25 per set, and another costs $45 per set. If she has only $7400 to spend, how many sets of each design should she order? 68. Cost of Fast Food One group of people purchased 10 hot dogs and 5 soft drinks at a cost of $35.00. A second bought 7 hot dogs and 4 soft drinks at a cost of $25.25. What is the cost of a single hot dog? A single soft drink? We paid $35.00. How much is one hot dog? How much is one soda? HOT DOGS SODA We paid $25.25. How much is one hot dog? How much is one soda? HOT DOGS SODA
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