Math 141:512 Due: February 6, 2019 Practice Exam 1 (extra credit) This is an open book, extra credit practice exam which covers the material that Exam 1 will cover (Sections 1.3, 1.4, 2.1, 2.2, 2.3, 2.4, 2.5, 3.2). Please print off this exam and bring it in 5-10 minutes before class on Wednesday, February 6, 2019 for extra credit. Name: 1. Plot the line y = 2x + 4 y x 2. Plot the line 6x 2y + 2 = 0 y x
Math 141:512 Practice Exam 1, Page 2 of 16 February 6, 2019 3. Consider the line that passes through the points (1, 5) and (3, 1) (a) What is the equation of the line? (b) What is the y-intercept? (c) What is the x-intercept? 4. What is the equation of the horizontal line that passes through the point (3, 5)? 5. What is the equation of the vertical line that passes through the point (3, 5)? 6. Conisder the line y = 2x 999. If the value of x increases by 2, find the corresponding change in y.
Math 141:512 Practice Exam 1, Page 3 of 16 February 6, 2019 7. Find the point of intersection of these straight lines, using any method you like 2x + 4y = 11 5x + 3y = 5
Math 141:512 Practice Exam 1, Page 4 of 16 February 6, 2019 8. Melanie bought a brand-new car in 2008. In 2013 the car was worth $14, 500 and in 2028 she sold it to for $5, 500. Assume a linear depreciation model. (a) Determine the original amount that Melanie paid for the car in 2008. (b) In what year was the car worth $12,700? (c) In what year will the car be worth nothing?
Math 141:512 Practice Exam 1, Page 5 of 16 February 6, 2019 9. I pay $5,000 for my own personal ice cream machine. I m able to buy the ingredients for a single ice cream cone for a mere $0.25. In an attempt to pay off the debt I took on buying my ice cream machine, I sell an ice cream cone for $1.00. (a) What is the cost function? (b) What is the revenue function? (c) What is the profit function? (d) What is the break-even point? (e) If I sell 5,000 ice cream cones, how much of a profit have I made? (f) How many ice cream cones do I need to sell if I want to buy a second $5,000 ice cream machine without going into debt?
Math 141:512 Practice Exam 1, Page 6 of 16 February 6, 2019 10. At a unit price of $50, consumers will demand 40 units. If the price is raised to $650, only 10 dedicated fans will demand the product. At a unit price of $163, suppliers will supply only 1 unit. At a unit price of $655, suppliers are willing to sell 165 units. (a) What is the demand function for this product? (b) What is the supply function for this product? (c) What is the market equilibrium for this product?
Math 141:512 Practice Exam 1, Page 7 of 16 February 6, 2019 11. You re going to the mall with your friends and you have $200 to spend. You discover a store that has all jeans for $25 and all dresses for $50. You really, really want to take home 6 items of clothing because you need that many new things and you want to spend all your money. How many pairs of jeans and how many dresses should you buy? 12. In a classroom, 3 times as many students were watching the teacher as were watching the spider in the teacher s hair. After 7 more students noticed the spider, there were an equal number of students watching the teacher and the spider. How many students were originally watching the teacher, and how many were originally watching the spider?
Math 141:512 Practice Exam 1, Page 8 of 16 February 6, 2019 13. Wanting to stay warm for the winter, two giraffes bought 300 scarves. The first giraffe, which had a slightly shorter neck, bought 36 fewer scarves than the second giraffe. How many scarves did the first giraffe buy? 14. What is the augmented matrix corresponding to the following system of equations? 3x 5y + 1z w = 10 x + 3y 5z + 10w = 0 24x y + 3z + 7w = 35 y + w = 4
Math 141:512 Practice Exam 1, Page 9 of 16 February 6, 2019 15. Circle the matrices below which are in row reduced echelon form (RREF)? [4] For those that are in row reduced echelon form, indicate how many solutions there are. 1 2 3 4 1 2 3 4 0 2 3 4 1 0 0 0 0 1 0 0 0 0 1 0 1 0 8 5 2 0 1 1 0 3 0 0 0 0 0 1 0 8 5 2 0 1 1 0 3 0 0 0 1 0 1 0 0 2 1 0 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 0 1 3 1 0 8 2 0 1 1 3 0 0 0 2 1 0 8 2 0 1 3 2 0 0 1 6 16. A word problem has a solution (x, y, z, w) = (2t 5, 10, 33 4t, t) where x, y, z, and w represent the number of freshmen, sophomores, juniors, and seniors in a math class respectively. What are the possible values for the parameter, t? (a) 0 t 8 (b) t = 0, 1,..., 8 (c) 3 t 8 (d) t = 3, 4,..., 8 (e) None of these. The answer is:
Math 141:512 Practice Exam 1, Page 10 of 16 February 6, 2019 17. Solve the following system of equations. (indicate explicitly if there is a unique solution, infinitely many solutions, or no solutions. If there is a unique solution, state it. If there are infinitely many solutions, write them in terms of some parameters). x + y + z = 4 2x 3y + z = 2 x + 2y z = 1
Math 141:512 Practice Exam 1, Page 11 of 16 February 6, 2019 18. The following matrices are in RREF form. Assume the variables represented are x 1, x 2, x 3 (and then x 4 and x 5 if necessary). Find the solutions in each case. (a) 1 0 0 1 0 1 0 0 0 0 1 3 (b) 1 0 0 0 1 0 1 0 0 0 0 0 1 0 3 (c) 1 0 1 4 1 0 1 2 10 0 0 0 0 0 3 (d) 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
Math 141:512 Practice Exam 1, Page 12 of 16 February 6, 2019 Consider the matrices 1 2 a b c A = 3 4 B = d e f C = 5 6 x y z [ 0 ] 1 1 3 Determine which of the following algebra calculations are possible. If it is possible, then do the calculation. (a) 4C (b) B + A T (c) (BA)C (d) AB 3C T (e) AC
Math 141:512 Practice Exam 1, Page 13 of 16 February 6, 2019 19. Given the matrix equation below, find the correct value of a + b [ ] 2 0 3b 2 [ ] T 2 4 = 2 3 a [ ] 0 3 2 b 6 20. Given the matrix equation below, find the correct values of x, y, and z 1 x 0 4 2 1 4 8 5 29 3 7 = 22 24 5 10 2 y 6 50 z
Math 141:512 Practice Exam 1, Page 14 of 16 February 6, 2019 21. A cruise company operates three different cruises (Cruise I, Cruise II, Cruise II). The number of cruises departing to each destination in 2019 is given by matrix A: A = Cruise I Cruise II Cruise III ( 3 10 23 ) For each cruise, the classes of cabins are classified into three categories (Small, Medium, and Large). The number of cabins in each category for each cruise are given by matrix B: B = Small M ed Large ( ) Cruise I 10 30 40 Cruise II 5 20 55 Cruise III 15 35 65 Compute the matrix multiplication AB and explain the meaning of the product AB.
Math 141:512 Practice Exam 1, Page 15 of 16 February 6, 2019 22. Formulate but do not solve the following problem: A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits? 23. Formulate but do not solve the following problem: A nutritionist at the Medical Center has been asked to prepare a special diet for certain patients. She has decided that the meals should contain a minimum of 500mg of calcium, 10mg of iron, and 30mg of vitamin C. She has further decided that the meals are to be prepared from Foods A and B. Each ounce of Food A contains 30mg of calcium, 2mg of iron, 3mg of vitamin C, and 3mg of cholesterol. Each ounce of Food B contains 25mg of calcium, 0.5mg of iron, 3mg of vitamin C, and 4mg of cholesterol. How many ounces of each type of food should be used in a meal so that the cholesterol content is minimized and the minimum requirements of calcium, iron, and vitamin C are met?
Math 141:512 Practice Exam 1, Page 16 of 16 February 6, 2019 You may use this page for scrap work.