SECTION 11.2 Systems of Linear Equations: Matrices 783 Find the currents I I , , 1 2 and I .3 4V 3V I3 I1 5V 12 V I2 1 2 1 2 24 V 1V 2V Source: Ibid., Prob. 36, p. 861. 87. Fruit Smoothie Friends sent you a text message with information on how to make a large fruit smoothie.They used voice dictation to send the message, and some information was missing. Since they are on a cruise and can’t be reached, you will have to figure out the missing information. The smoothie is made of bananas, oranges, and papayas. You don’t know how many of each to put in the smoothie, but you know that the recipe calls for seven pieces of fruit in the mix and that there are twice as many oranges as bananas. You also know that the seven pieces of fruit cost a total of $10.50, where bananas cost $1.00 each, oranges $1.50 each, and papayas $2.50 each. (a) Determine the number of each kind of fruit in the recipe. (b) Suppose your sister sends you a recipe and the same issue happens with voice dictation! In your sister’s message, the differences are that her recipe calls for 12 pieces of fruit, she paid $0.50 for each banana, and her total bill was $18.00; everything else is the same. Determine the number of each kind of fruit in your sister’s recipe. 88. Catapult Giancarlo is working on a school project for a physics class. The assignment is to build a protective casing for an egg that will be launched using a catapult. The goal is for the casing to keep the egg from breaking when it hits the ground.When the egg is released from the catapult it will follow a parabolic path until it hits the ground. Giancarlo knows that the vertical height of the egg is modeled by projectile motion with height ( ) = + + h t at bt c, 2 where ( ) h t represents the height in feet and t represents time in seconds. (a) Find the height function ( ) h t for an egg that is thrown from the catapult with a height of 24 feet after 1 2 second, a height of 37 feet after 1 second, and a height of 9 feet after 3 seconds. (b) What is the initial height of the egg when it is released from the catapult? (c) How much time elapses before the egg hits the ground? 89. Pharmacy A doctor’s prescription calls for a daily intake of a supplement containing 40 milligrams (mg) of vitamin C and 30 mg of vitamin D. Your pharmacy stocks three supplements that can be used: one contains 20% vitamin C and 30% vitamin D; a second, 40% vitamin C and 20% vitamin D; and a third, 30% vitamin C and 50% vitamin D. Create a table showing the possible combinations that could be used to fill the prescription. 90. 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 three powders that can be used to make these pills: one contains 20% vitamin B12 and 30% vitamin E; a second, 40% vitamin B12 and 20% vitamin E; and a third, 30% vitamin B12 and 40% vitamin E. Create a table showing the possible combinations of these powders that could be mixed in each pill. [Hint: 10 units of the first powder contains ⋅ = 10 0.2 2 units of vitamin B . 12 ] Explaining Concepts 91. Write a brief paragraph or two outlining your strategy for solving a system of linear equations using matrices. 92. When solving a system of linear equations using matrices, do you prefer to place the augmented matrix in row echelon form or in reduced row echelon form? Give reasons for your choice. 93. Make up a system of three linear equations containing three variables that have: (a) No solution (b) Exactly one solution (c) Infinitely many solutions Give the three systems to a friend to solve and critique. Problems 94–103 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. Retain Your Knowledge 94. Solve: − < + x x x 3 6 2 2 95. Graph: ( ) = − − + + f x x x x x 2 1 2 1 2 2 96. State the domain of ( ) = − − + f x e 6. x 5 97. Use a calculator to approximate ( ) − − cos 0.75 1 in radians, rounded to two decimal places. 98. Simplify: ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ x y x y 18 27 4 5 3 9 3 99. Find an equation of the hyperbola with vertices ( ) 4, 1 and ( ) 4, 9 and foci ( ) 4, 0 and ( ) 4, 10 . 100. Write π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ i 6 cos 5 12 sin 5 12 4 in rectangular form +x yi and in exponential form θ re . i 101. What is the amount that results if $2700 is invested at 3.6% compounded monthly for 3 years? 102. Find the average rate of change of ( ) = − f x x sin 1 from = − x 1 to = x 1. 103. Find the difference quotient for ( ) = − f x x 1 . 2 Express the answer as a single fraction.
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