2 62-194-91 Due date: Tuesday, February 24, 2016 Instructions: You must submit your assignment through regular mail and it must be postmarked by the due date given at the top of the assignment page. I am not responsible for loss of any assignment in mail. If you wish you can use registered mail or express post for your security. Late assignments will not be accepted. You must show all your work for the word problems to receive full credit, however for multiple choice and true/false questions the answers are sucient, you don't need to show your work. Keep a copy of each of your submitted assignments. Include your name (last/rst) and student number in each assignment submitted. DO NOT EMAIL ME YOUR ASSIGNMENT. Multiple Choice Instructions. For each question pick the single best answer. Each correct answer is worth 1 point and each incorrect answer is worth 0 points. 1. Suppose that the xed costs of producing calculators is $2250 and the cost of producing each calculator is $10. What is the total cost of producing 4 calculators? (a) $40 (b) $2290 (c) $9010 (d) $2254 2. The TitleWave company has the total cost C(x) = 30x + 18700 and the total revenue R(x) = 250x, where x is the number of surf boards produced and sold. What is the break even point? (a) (85, 21250) (b) (2125, 80) (c) (748, 18700) (d) (30, 18700) 3. The prot function of a company is P (x) = 20x 340. For what value of x is prot equal to $1,000? (a) 17 (b) 33 (c) 500 (d) 67 1 4. How many solutions does the following system of equations have? x + 3y = 1 2x + 6y = 2 (a) The system is inconsistent (b) The system has only one solution (c) The system has innite solutions 5. How many solutions does the following system of equations have? x 3y = 2 2x + 6y = 3 (a) The system is inconsistent (b) The system has only one solution (c) The system has innite solutions 6. How many solutions does the following system of equations have? 2x 10y = 2 x + 2y = 2 (a) The system is inconsistent (b) The system has only one solution (c) The system has innite solutions 7. The solution of the following system of equations is 4x 2y = 2 12x 6y = 6 (a) x R and y = 1 2x (b) x R and y = x 2 (c) x R and y = 4x 2 (d) x R and y = 2x 1 8. The interest rates paid on an investment over the last three years are 2.2%, 2.4%, and 2.6%. What is the expected value of the interest rate for the next year? (a) 2.2% (b) 2.4% (c) 2.6% (d) 2.8% 2 9. The solution set of the inequality 3x + 9 0 can be described as (a) Solution set = {x : x 3} (b) Solution set = {x : x > 3} (c) Solution set = {x : x 3} (d) Solution set = {x : x < 3} 10. The inequality 2x + 5y + 300 0.6(x + 3z) + 650 can be rewritten as (a) 1.4x + 5y 1.8z 350 (b) 2.6x + 5y 1.8z 350 (c) 1.4x + 5y + 1.8z 350 (d) 1.4x + 5y 0.6z 350 True/False Instructions. Determine whether each of the following statements is true or false. Each correct answer is worth 1 point and each incorrect answer is worth 0 points. 1. The equation x + 5 y z = 1 is linear in the variables x, y and z. 2. In a prot graph the portion above the x-axis represents loss and the portion below the x-axis represents prot. 3. One way to nd the break-even points given a cost function C(x) and revenue function R(x) is to set C(x) = R(x) and solve for x. 4. The supply of a commodity is directly related to its price (all other things equal). 5. The solution set of the equation 2x + y = 3 contains the point (x, y) = (1, 1). 6. A system of 2 equations in 2 variables will always have a solution. 7. A system of 2 equations in 2 variables cannot have more than two solutions. 8. x = 2 is a part of the solution set of the inequality 3x 6 0. 9. The point (1, 2) satisfy the inequality x + 2y 5. 10. The feasible region is always bounded. 11. Every linear programming problem has optimal solutions. 12. If the feasible region is nonempty, closed and bounded, then the linear programming problem must have an optimal solution. 3 Word Problems Instructions. In each question you need to use the method discussed in Section 2.3 to 4.4 of the course textbook. Some questions ask for a typical method to be used for the solution of the problem. You will not be given any credit if you use some other method instead. For the graphs of feasible regions, you can draw them on graph paper or on the same paper where you do your assignment. The graph should clearly show the feasible region in a shaded form. Do not use any computer program to draw the graphs. All graphs submitted should be hand drawn. SHOW ALL YOUR WORK TO RECEIVE FULL CREDIT. 1. (9 marks) Solve each of the following equations. 3 2x + 1 x + 13 7 4x 4= (b) = (a) x+1 x x2 3(x 2) 3 2. (8 marks) A company will use straight-line depreciation to determine the value v, in dollars, of a $75, 000 machine after t years. It is estimated that the machine will have a residual value of $9, 000 after 30 years. (a) What is the depreciation schedule? (b) What is the value of the machine after 10 years? (c) When is the value of the machine equal to $28, 800? 3. (8 marks) A book store owner will buy 10 books if the price is $75 per book, and 30 books if the price is $15. The supplier is willing to provide 45 books if the price is $40 per book but only 15 books if the price is $25. Given that the supply and demand functions are linear, (a) nd the demand and supply functions. (b) nd the equilibrium price and the equilibrium quantity. 4. (4 marks) Jessica wants to invest $40,000 in two accounts. The annual returns of the accounts are 8% and 6% of her investment. Jessica would like to have an annual return of $2,500 from her two investments. Set up, but do not solve, the set of linear equations that will determine how much she should invest in each account. Be sure to include a well written denition of the variables. 5. (6 marks) (Set up a system of linear equations, but do not solve. Be sure to include a well written denition of the variables.) A fast food restaurant sells 3 kinds of burgers: cheese burger, chicken burger, and hamburger. The prices are $1.50 for a cheese burger, $2 for a chicken burger, and $1.30 for a hamburger. On a certain day, the restaurant sold 110 burgers and earned $176. If the number of cheese burgers sold was twice the number of chicken burgers sold, how many burgers of each type were sold on that day? 4 6. (7 marks) Graph the feasible region dened by the constraints given below. Shade the feasible region and nd the coordinates of all the corner points. 3x 2y 3y x y 2 9 0 0 7. (11 marks) Solve the following linear programming problem using the graphical method. maximize 2x + y = z subject to 2x + 2y 36 x + 3y 36 2x + y 32 x 0 y 0 5