Exam 1 Review (Sections Covered: 1.3, 1.4, 2.1, 2.2, 2.3,2.4, 2.5, 3.2)

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1 Exam 1 Review (Sections Covered: 1.3, 1.4, 2.1, 2.2, 2.3,2.4, 2.5, 3.2) 1. Find the slope of the line that passes through the points ( 2, 6) and (10, 6). 2. Find the slope of the line that passes through the points ( 2, 6) and ( 2, 1). 3. Find the equation of the line that passes through the points (2, 9) and (2, 8). Is the line vertical or horizontal? 4. Find the equation of the line that passes through the points (1, 5) and ( 2, 5). Is the line vertical or horizontal? 5. Given that the slope of line L is 6 and the change in y is 12, what is the corresponding change in x? 6. Given that the slope of line L is 10 and the change in x is 6, what is the corresponding change in y? 7. The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $132. The quantity demanded each month is 1000 when the unit price is $102. The suppliers will market 700 espresso makers when the unit price is $66. At a unit price of $96, they are willing to market 2200 units. Both the supply and demand equations are known to be linear. (a) Find the demand equation. Find the supply equation. Find the equilibrium quantity. 8. The quantity demanded x of a certain brand of DVD player is 3000/week when the unit price p is $485. For each decrease in unit price of $20 below $485, the quantity demanded increases by 250 units. The suppliers will not market any DVD players if the unit price is $300 or lower. But at a unit price of $525, they are willing to make available 2500 units in the market. The supply equation is also known to be linear. (a) Find the demand equation. Find the supply equation. Find the equilibrium price. 9. If the cost function is C(x) =.50x and the revenue function is R(x) = 24.50x. (a) Find the profit equation. Find the break-even point.

2 10. If the fixed cost of producing a certain commodity is $10, 000 and the cost per item is $50 and the sell price is 150. (a) Find the profit equation. Find the break-even point. 11. Formulate but do not solve the problem. Michael Perez deposited a total of $2000 with two savings institutions. One pays interest at a rate of 5%/year, whereas the other pays interest at a rate of 8%/year. If Michael earned a total of $136 in interest during a single year, how much did he deposit in each institution? (Let x and y denote the amount of money, in dollars, invested at 5% and 8%, respectively.) 12. Formulate but do not solve the problem. The management of a private investment club has a fund of $100, 000 earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high risk (x), medium risk (y), and low risk (z). Management estimates that high risk stocks will have a rate of return of 15%/year; medium risk stocks, 10%/year; and low risk stocks, 6%/year. The investment in low risk stocks is to be twice the sum of the investments in stocks of the other two categories. If the investment goal is to have a rate of return of 9% on the total investment, determine how much the club should invest in each type of stock. (Assume that all the money available for investment is invested.) 13. Formulate but do not solve the problem. The management of Hartman Rent-A-Car has allocated $2.24 million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full-size cars. Compacts cost $16, 000 each, intermediate-size cars cost $24, 000 each, and full-size cars cost $32, 000 each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100, determine how many cars of each type will be purchased. (Assume that the entire budget will be used. Let x, y, and z denote the number of compact, intermediate-sized, and full-size cars purchased, respectively.) 14. What are the row operations needed to pivot the following matrices about the indicated entry? (a) Row 2, column Row 1, column Fall 2016, Maya Johnson

3 Row 3, column Determine the resulting matrix from performing the indicated row operations. (a) R 2 + ( 10)R ( 1/52)R /26 R 1 + ( 6)R Solve the following system of equations. (If there are infinitely many solutions, give your answer as a parametric solution using t and/or s.) (a) 4x + 20y = 18 6x + 8y = 30 2x y z = 6 3x + 2y + z = 23 x + 2y + 2z = 13 2x 12y 10z = 35 4x + 24y + 20z = 48 (d) 2x + 2y = 8 62x + 10y = 40 32x 4y = 16 (e) 3y + 2z = 2 2x y 3z = 2 2x + 2y z = 4 (f) 3x 2y + 4z = 14 2x + y 2z = 7 x + 4y 8z = A person has 6 times as many dimes as quarters. If the total face value of these coins is $10.20, how many of each type of coin does this person have? 3 Fall 2016, Maya Johnson

4 18. Find the values of a, b, c, and d in the matrix equation below. T a b = c d Find the transpose of the following matrices (a) y 6 2 z 2 0 x x z y w 20. Find the indicated entry of the matrix products below. (a) The entry in the second row, first column The entry in the first row, second column. 9 2x 3 9 5y The entry in the second row, third column. 10 x y 0 z 9 2z 0 2x Find the values of x, y, and z. x y z 4 2 = Fall 2016, Maya Johnson

5 22. Find the values of x and y and z in the matrix equation below. T 2 2 x = 0 z y Three network consultants, Alan, Maria, and Steven, each received a year -end bonus of $10, 000, which they decided to invest in a 401(k) retirement plan sponsored by their employer. Under this plan, employees are allowed to place their investments in three funds: an equity index fund (I), a growth fund (II), and a global equity fund (III). The allocations of the investments (in dollars) of the three employees at the beginning of the year are summarized in the matrix A = I II III Alan Maria Steven If the returns for each of the funds are listed in the matrix below, then what matrix product will give the total returns for Alan, Maria and Steven respectively? B = I II III Returns Matrix A shows the number of servings of each food that each person eats per day. Matrix B shows the number of units of each nutrient in a serving of each food. A = pork carrots rice Arthur Fred Jo A = Cu D B12 pork carrots rice What does the product matrix AB represent? (a) The total amount of each nutrient for each food. The total amount of each food each person has daily. The product is meaningless. (d) The total daily amount of each nutrient for each person. (e) The total daily amount of each food each person eats. 5 Fall 2016, Maya Johnson

6 25. Formulate but do not solve the problem. A company manufactures x units of product A and y units of product B, on two machines, I and II. It has been determined that the company will realize a profit of $2 on each unit of product A and $4 on each unit of product B. To manufacture a unit of product A requires 6 min on machine I and 4 min on machine II. To manufacture a unit of product B requires 10 min on machine I and 4 min on machine II. There are 138 min available on machine I and 92 min available on machine II in each work shift. How many units of a product should be produced in each shift to maximize the company s profit P? 26. Formulate but do not solve the problem. A farmer plans to plant two crops, A and B. The cost of cultivating crop A is $35/acre whereas that of crop B is $60/acre. The farmer has a maximum of $6820 available for land cultivation. Each acre of crop A requires 18 labor-hours, and each acre of crop B requires 28 labor-hours. The farmer has a maximum of 3276 labor-hours available. If she expects to make a profit of $160/acre on crop A and $210/acre on crop B, how many acres of each crop, x and y, respectively, should she plant in order to maximize her profit, P? 27. Formulate but do not solve the problem. Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-a vessel has 60 deluxe cabins and 165 standard cabins, whereas a type-b vessel has 70 deluxe cabins and 130 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 330 deluxe and 720 standard cabins for their 15-day cruise in May. It costs $47, 000 to operate a type-a vessel and $56, 000 to operate a type-b vessel for that period. How many of each type of vessel (x type-a and y type-b) should be used in order to minimize cost, C? 28. Determine if the following matrices are in row-reduced form (a) Fall 2016, Maya Johnson

7 Answers: Undefined 3. x = 2 Vertical 4. y = 5 Horizontal (a) D(x) =.04x S(x) =.02x units 8. (a) D(x) =.08x S(x) =.09x $ (a) P (x) = 24x 216 (9, 220.5) 10. (a) P (x) = 100x (100, 15000) 11. x + y = x +.08y = x + y + z = z = 2x + 2y.15x +.10y +.06z = x y z = x = 2y x + y + z = (a) ( 1/6)R 2, R 1 + ( 8)R 2, and R 3 + ( 10)R 2 (1/4)R 1, R 2 + ( 2)R 1, and R 3 + (3)R 1 7 Fall 2016, Maya Johnson

8 (1/10)R 3, R 1 + ( 4)R 3, and R 2 + (6)R (a) / / / (a) (-3, 3/2) (5,4,0) No solution (d) (0, -4) (e) (4/3 + 7t/6, 2/3-2t/3,t) (f) (4, t,t) dimes and 12 quarters. 18. a = 6, b = 0, c = 8, and d = (a) z y x x y z w 20. (a) x 2 + 2yz 21. x = 2, y = 4, and z = x = 2, y = 6, and z = 63 8 Fall 2016, Maya Johnson

9 23. AB T 24. (d) The total daily amount of each nutrient for each person. 25. Maximize: P = 2x + 4y Subject to: 6x + 10y 138 4x + 4y 92 x 0, y Maximize: P = 160x + 210y Subject to: 35x + 60y x + 28y 3276 x 0, y Minimize: C = 47000x y Subject to: 60x + 70y x + 130y 720 x 0, y (a) Yes No No 9 Fall 2016, Maya Johnson