Pr. 1 Newlyweds Nick and Dana each work two part-time jobs in order to make ends...

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Pr. 1 Newlyweds Nick and Dana each work two part-time jobs in order to make ends meet as they finish college. Nick's college schedule allows him to work a total of 30 hours per week both for a department store that pays him $9.50 per hour and for a fast food restaurant that pays him $8.50 per hour. Dana's college schedule allows her to work a total of 25 hours per week for a grocery store that pays her $10.00 per hour and for a novelty shop that pays her $8.00 per hour. The couple must earn $500 per week in order to eat and pay bills. (a) Write a system of equations that describes Nick and Dana's earnings opportunities. Be sure to name all variables. (b) Solve the system by writing its augmented matrix in reduced row echelon form. (c) Prepare a table showing various work options for Nick and Dana. (d) What would you recommend to them? EXAMPLE 9 Financial Planning A couple has $60,000 to invest. They wish to earn an average of $5000 per year on the investment over a 5 year period. Based on the yearly average return on mutual funds for 5 years ending December 31, 2009, they are considering the following funds: Franklin High Income at 5%, Royce 100K at 6%, TCW Small Capital Growth at 7%, and Franklin Natural Resources at 10%. High Income: 0.05 x₁ (a) As their financial advisor, prepare a table showing the various ways the couple can achieve their goal. (b) Comment on the various possibilities and the overall plan. SOLUTION Begin by naming the variables: Let x₁ be the amount invested in the High Income Fund, x₂ the amount invested in the Royce 100 K Fund, x3 the amount invested in the Small Capital Growth Fund, and x4 the amount in the Natural Resources Fund. Then the aver- age annual amount earned by each investment choice would be Royce 100 K: 0.06 x₂ Small Capital Growth: 0.07 x3 Natural Resources: 0.01x4 Since the couple requires $5000 from these investments, we have the equation 0.05x₁ +0.06x₂ + 0.07x3 + 0.10x = 5000 The total amount available to invest is $60,000, which leads to the equation x₁ + x₂ + x3 + x4 = 60,000 These two equations form the system [0.05x1 + 0.06x2 + 0.07x3 + 0.10x4 = 5000 (1) x₁ + x₂ + x3 + x4 = 60,000 (2) Write the augmented matrix of this system and proceed to row reduce. 0.05 0.06 0.07 0.10 5000 1 1 1 160,000 Interchange rows. R₂ = -0.05r₁ + 1₂ R₂ = 100r₂ 1 1 1 1 60,000 0.05 0.06 0.07 0.10 5000 - 1 0 1 1 1 60,000 2000 0.01 0.02 0.05 1 1 1 1 60,000 0 1 2 5200,000 1 0 -1 -4-140,000 R₁ = −1₂+1₂ 01 2 5 200,000 This matrix is in reduced row echelon form. The solution of the system is x₁ = x3+4x4 140,000 x₂ = -2x35x4 + 200,000 where x3 and x, are parameters. Now, each of the variables must be nonnegative and each must be less than or equal to 60,000. Set up the table shown in Table 3. Pr. 1 Newlyweds Nick and Dana each work two part-time jobs in order to make ends meet as they finish college. Nick's college schedule allows him to work a total of 30 hours per week both for a department store that pays him $9.50 per hour and for a fast food restaurant that pays him $8.50 per hour. Dana's college schedule allows her to work a total of 25 hours per week for a grocery store that pays her $10.00 per hour and for a novelty shop that pays her $8.00 per hour. The couple must earn $500 per week in order to eat and pay bills. (a) Write a system of equations that describes Nick and Dana's earnings opportunities. Be sure to name all variables. (b) Solve the system by writing its augmented matrix in reduced row echelon form. (c) Prepare a table showing various work options for Nick and Dana. (d) What would you recommend to them? EXAMPLE 9 Financial Planning A couple has $60,000 to invest. They wish to earn an average of $5000 per year on the investment over a 5 year period. Based on the yearly average return on mutual funds for 5 years ending December 31, 2009, they are considering the following funds: Franklin High Income at 5%, Royce 100K at 6%, TCW Small Capital Growth at 7%, and Franklin Natural Resources at 10%. High Income: 0.05 x₁ (a) As their financial advisor, prepare a table showing the various ways the couple can achieve their goal. (b) Comment on the various possibilities and the overall plan. SOLUTION Begin by naming the variables: Let x₁ be the amount invested in the High Income Fund, x₂ the amount invested in the Royce 100 K Fund, x3 the amount invested in the Small Capital Growth Fund, and x4 the amount in the Natural Resources Fund. Then the aver- age annual amount earned by each investment choice would be Royce 100 K: 0.06 x₂ Small Capital Growth: 0.07 x3 Natural Resources: 0.01x4 Since the couple requires $5000 from these investments, we have the equation 0.05x₁ +0.06x₂ + 0.07x3 + 0.10x = 5000 The total amount available to invest is $60,000, which leads to the equation x₁ + x₂ + x3 + x4 = 60,000 These two equations form the system [0.05x1 + 0.06x2 + 0.07x3 + 0.10x4 = 5000 (1) x₁ + x₂ + x3 + x4 = 60,000 (2) Write the augmented matrix of this system and proceed to row reduce. 0.05 0.06 0.07 0.10 5000 1 1 1 160,000 Interchange rows. R₂ = -0.05r₁ + 1₂ R₂ = 100r₂ 1 1 1 1 60,000 0.05 0.06 0.07 0.10 5000 - 1 0 1 1 1 60,000 2000 0.01 0.02 0.05 1 1 1 1 60,000 0 1 2 5200,000 1 0 -1 -4-140,000 R₁ = −1₂+1₂ 01 2 5 200,000 This matrix is in reduced row echelon form. The solution of the system is x₁ = x3+4x4 140,000 x₂ = -2x35x4 + 200,000 where x3 and x, are parameters. Now, each of the variables must be nonnegative and each must be less than or equal to 60,000. Set up the table shown in Table 3.

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a System of equations Let x hours Nick works at department store earning 950hr ... View the full answer