Practice Final Exam Answers

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1 Practice Final Exam Answers 1. AutoTime, a manufacturer of electronic digital timers, has a monthly fixed cost of $48,000 and a production cost $8 per timer. The timers sell for $14 apiece. (a) (3 pts) Write the cost function. C(x) = 8x (b) (2 pts) Write the revenue function. R(x) = 14x (c) (5 pts) Plot both functions on the same set of axes. (d) (3 pts) Find the number of digital timers AutoTime needs to manufacture in order to break even. Based on the graph, AutoTime should manufacture 8000 timers in order to have revenue and costs balance out. 2. A man invests $100 in an account that earns 6.5% annual interest, compounded continuously. How much time must pass before the acount contains $100,000?

2 A = P e rt = 100e.065t 1000 = e.065t ln 1000 =.065t ln = t years = t 3. A couple are considering buying a home for $300,000. If they make a down-payment of 20% and obtain a 20-year mortgage with an annual rate of 4.5%, what will their monthly mortgage payment be? The monthly mortgage is $1, P V = = i = = n = = 240 P V = P MT 1 (1 + i) n i = P MT 1 ( ) = P MT = P MT 4. A woman is saving for retirement. Her plan is to make a deposit every month into an account that earns 6.8% annual interest. After 25 years, she will retire and stop making the deposits. Instead, she will withdraw $1500 per month from the account. If her retirement lasts 20 years, what is the monthly deposit she should make? This problems has two parts: for the first 25 years, the woman is saving for the

3 retirement; for the next 20 years, she is living off her savings. Deal with the second part first: i = = n = = 240 P MT = 1500 P V = P MT 1 (1 + i) n i P V = ( ) = $ To support her retirement, the woman needs to save up $196, over 25 years: i = = n = = 300 F V = i P MT = F V (1 + i) n P MT = ( ) P MT = The woman should save about $ per month. 5. A dietitian in a hospital is to arrange a special diet composed of three basic foods. The diet is to include exactly 340 units of calcium, 180 units of iron, and 220 units of vitamin A. The number of units per ounce of each special ingredient for each of the foods is Food A Food B Food C Calcium Iron Vitamin A (a) Write a system of equations that represents this situation. 30x x x 3 = x x x 3 = x x x 3 = 220 (b) Find how many ounces of each food must be used to meet the dietary requirements. Write the system as an augmented matrix:

4 Convert to reduced-row echelon form: to meet the nutritional requirements, the dietician should use 8 units of Food A, 2 units of Food B, and 4 units of Food C. 6. Use the encoding matrix to decode the message The plaintext matrix is Translated, the message is mashed potatoes =

5 7. (8 pts) Solve the equation. log(x + 5) + log(x + 2) = 1 log(x + 5) + log(x + 2) = 1 log ((x + 5)(x + 2)) = 1 10 log((x+5)(x+2)) = 10 1 (x + 5)(x + 2) = 10 x 2 + 7x + 10 = 10 x 2 + 7x = 0 x(x + 7) = 0 x = 0, 7 8. Solve the polynomial inequality. Write your answers in interval notation. Solve the associated equality: Write and test the candidate intervals: The solution is (, 6] [0, ). x 3 (x 2) 2 (x + 6) 0 x 3 (x 2) 2 (x + 6) = 0 x = 0, x = 2, x = 6 Candidate Intervals: (, 6] [ 6, 0] [0, 2] [2, ) Test Values: Test Conclusion: > 0 < 0 > 0 > 0 9. A political scientist received a grant to fund a research project on voting trends. The budget includes $3,200 for conducting door-to-door interviews on the day before an election. Undergraduate students, graduate students, and faculty members will be hired to conduct the interviews. Each undergraduate student will conduct 18 interviews for $100. Each graduate student will conduct 25 interviews for $150. Each faculty member will conduct 30 interviews for $200. Due to limited transportation facilities, no more than 20 interviewers can be hired. How many undergraduate students, graduate students, and faculty members should be hired in order to maximize the number of interviews? What is the maximum number of interviews? (a) Write a system of constraints and an objective function that represents this problem. P = 18x x x 3 x 1 + x 2 + x x x x x 1, x 2, x 3 0

6 (b) Introduce slack variables and write the initial tableau. Introduce slack variables: Write the initial tableau P 18x 1 25x 2 30x 3 = 0 x 1 + x 2 + x 3 + s 1 = x x x 3 + s 2 = 3200 x 1, x 2, x 3 0 x 1 x 2 x 3 s 1 s 2 P Value s s P (c) Select the pivot column and pivot row. The pivot column is x 3, and the pivot row is s A student performing simplex maximization arrives at the following tableau: x 1 x 2 x 3 s 1 s 2 s 3 P s x s P (a) What basic solution does this tableau represent? x 1 = 150, x 2 = 0, x 3 = 0, s 1 = 0, s 2 = 70, s 3 = 60, P = (b) Identify the pivot row and pivot column and then perform a pivot operation. The pivot column is x 2 and the pivot row is s 3. After performing the pivot operation the tableau is x 1 x 2 x 3 s 1 s 2 s 3 P x x s P (c) Is the tableau obtained after the pivot operation the final tableau? If so, identify the basic solution. If not, identify the pivot column and pivot row. This is not the final tableau. The pivot column is s 1 and the pivot row is s For the following system of inequalities: 3y x 6 2x + 3y 15 x, y 0

7 (a) Graph the system. (b) List the corner points. (c) Maximize the objective function P = 4x + 7y over the feasible region. Corner Point Objective Function Value (0, 0) 0 (7.5, 0) 30 (3, 3) 33 (0, 2) 14 The maximum value of the objective function is 33, and it occurs at the point (3, 3). 12. Perform the multiplication, simplify, and write your answer in radical form. x 3/5 ( x 7/5 x 1/5) x 3/5 ( x 7/5 x 1/5) x 3/5 x 7/5 x 3/5 x 1/5 x x x 2 x 4/5 x 2 5 x 4

8 13. Solve the equation. Remember to check for extraneous solutions. x + 1 x 1 = 8 x x 2 + 2x 3 x + 1 x 1 = 8 x x 2 + 2x 3 x + 1 x 1 = 8 x (x + 3)(x 1) (x + 1)(x + 3) = 8(x 1) + 8 x 2 + 4x + 3 = 8x x x 13 = 0 (x + 13)(x 1) = 0 x = 13, x = For the function f(x) = (x 1) do the following: (a) Sketch the graph. (b) Identify the domain and the range. The domain is all real numbers, or (, ) in interval notation. The range is [2, ).

9 15. Given the following linear programing problem: Minimize: C = 2x 1 + 3x 2 6x 1 + 4x x 1 + 5x 2 30 subject to: x 1 + x 2 8 x 1, x 2 0 Form the dual problem. Write the matrix that represents the linear program and transpose: The dual problem is Transpose Maximize: P = 40y y 2 + 8y 3 6y 1 + 3y 2 + y 3 2 subject to 4y 1 + 5y 2 + y 3 3 y 1, y 2, y (5 pts) Add the rational expressions and simplify as much as possible. 2x + 6 x x 3 2x + 6 x x 3 = 2x + 6 (x + 3)(x 3) + 7 x 3 2x (x + 3) = (x 3)(x + 3) 9x + 27 = (x + 3)(x 3) 9(x + 3) = (x + 3)(x 3) = 9 x 3 (x + 3) (x + 3)

10 17. (5 pts) Solve the following equation. 3(5 x ) 10 = 98 3(5 x ) 10 = 98 3(5 x ) = x = 36 ln(5 x ) = ln 36 x ln 5 = ln 36 ln 36 x = ln 5 x = For the linear programming problem do the following: Maximize: P = 2x 1 + 3x 2 4x 3 x 1 + 2x 2 x 3 8 x 1 2x 2 + 2x 3 10 subject to: 2x 1 + 4x 2 3x 3 = 12 x 1, x 2, x 3 0 (a) (3 pts) Introduce slack, surplus, and artificial variables. (b) (2 pts) Write the initial simplex tableau. x 1 x 2 x 3 s 1 s 2 a 1 P s s a P x 1 + 2x 2 x 3 + s 1 = 8 x 1 2x 2 + 2x 3 s 2 = 10 2x 1 + 4x 2 3x 3 + a 1 = 12 (c) (3 pts) Identify a pivot column and pivot row. The pivot column is x 1 and the pivot row is a 1.