Math324 - Test Review 2 - Fall 206 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the vertex of the parabola. ) f(x) = x 2-0x + 33 ) (0, 5) (8, 0) (8, 5) (5, 8) 2) y = 5 (x + 4)2-2 2) (-4, -2) (2, 4) (-2, -4) (4, -2) Find the x- and y-intercepts. If no x-intercepts exist, state so. 3) f(x) = x 2-3x - 28 3) x-intercepts: (-4, 0), (8, 0); y-intercept: (0, 28) x-intercepts: (-5, 0), (7, 0); y-intercept: (0, 28) x-intercepts: (4, 0), (-7, 0); y-intercept: (0, -28) x-intercepts: (-4, 0), (7, 0); y-intercept: (0, -28) Solve the problem. 4) A ball is thrown vertically upward at an initial speed of 80 ft/sec. Its height (in feet) after t seconds 4) is given by h(t) = t(80-6t) What is the maximum height attained by the ball? 200 feet 00 feet 33.3 feet 88.9 feet 5) John owns a hotdog stand. His profit is represented by the equation P = -x 2 + 2x + 4, with P being profits and x the number of hotdogs. What is the most he can earn? $36 $83 $65 $77 5) Give the equation of the vertical asymptote(s) of the rational function. x + 8 6) g(x) = x 2 + 2x - 24 x = 6, x = -4 x = -8 x = -6, x = 4, y = 0 x = -6, x = 4 6) Give the equation of the horizontal asymptote of the rational function. 7) g(x) = 8-2x 2x + y = - y = 4 y = y = 0 7)
Solve the system of two equations in two variables. 8) 8x + 7y = 36 3x - 4y = -3 8) (, 4) (0, 5) No solution (, 5) Determine whether the given ordered set of numbers is a solution of the system of equations. 9) (6, -2) 9) x + y = 4 x - y = 8 Yes No Solve the system of two equations in two variables. 0) 5x - 2y = 8 0) 5x - 6y = 6 (0, -4) No solution (, 0) (, -.5) Solve the system by back substitution. ) x + 4y+ 4z = - ) 2y + 5z = -2 2z = - 0 (, -5, 2) No solution (-6, 2, -5) (, 2, -5) Write an augmented matrix for the system of equations. 2) 9x + 5y = 49 2) 2y = -2 49 5 9 9 5 49 2 0-2 9 5 49-2 0 2 0 2-2 9 5 5 2-2 0 Use the Gauss-Jordan method to solve the system of equations. 3) x - y + 3z = -8 x + 5y + z = 40 5x + y + 3z = 0 3) (8, 0, 8) (0, 8, 0) (8, 8, 0) No solution Solve the problem. 4) What is the size of the matrix? 4) -5 5 - -4-4 2 3 3 3 2 6 Perform the indicated operation where possible. 5) - 0 - - 4 4 3 0 4-3 - 0 0-4 0-2 4 7 2 5) 2
Perform the indicated operation. 6) Let C = -3 2 and D = -4 2-8 - 3-2. Find C - 3D. 6) 4-2 8 4-6 4-2 6-4 Given the matrices A and B, find the matrix product AB. 7) A = 0 -, B = -2 0 Find AB. 7) 2 2 - -4-4 2 3 - -6 2 0 2-2 2 - -2-6 Determine whether the two matrices are inverses of each other by computing their product. 8) 5 3 and 2-3 3 2-3 5 No Yes 8) Find the inverse, if it exists, of the given matrix. 9) 5 6 4 5 5-6 -4 5-5 4 6-5 5 4 6 5-5 -6-4 -5 9) 20) A = -4 2 0 3 20) - 4-6 - 4 6 3 6 0 3 0 3 0 3 0-4 - 4 6 3
Graph the feasible region for the system of inequalities. 2) 2x + y 4 2) x - 0 4
22) 22) x + 2y 2 x + y 0 5
A manufacturer of wooden chairs and tables must decide in advance how many of each item will be made in a given week. Use the table to find the system of inequalities that describes the manufacturer's weekly production. 23) Use x for the number of chairs and y for the number of tables made per week. The number of 23) work-hours available for construction and finishing is fixed. Hours per chair Hours per table Total hours available Construction 2 4 48 Finishing 2 3 42 2x + 4y + 48 0 2x + 3y + 42 0 2x + 4y 48 2x + 3y 42 x 0 y 0 2x + 4y 48 2x + 3y 42 x 0 y 0 2x + 4y 48 2x + 3y 42 Find the value(s) of the function on the given feasible region. 24) 24) Find the maximum and minimum of z = 20x + 4y. 220, 200 200, 2 220, 2 20, 2 6
Use graphical methods to solve the linear programming problem. 25) Maximize z = 6x + 7y 25) subject to: 2x + 3y 2 2x + y 8 x 0 y 0 Maximum of 32 when x = 2 and y = 3 Maximum of 24 when x = 4 and y = 0 Maximum of 32 when x = 3 and y = 2 Maximum of 52 when x = 4 and y = 4 State the linear programming problem in mathematical terms, identifying the objective function and the constraints. 26) A breed of cattle needs at least 0 protein and 8 fat units per day. Feed type I provides 5 protein 26) and 3 fat units at $5/bag. Feed type II provides 4 protein and 4 fat units at $3/bag. Which mixture fills the needs at minimum cost? Minimize 3x + 5y Minimize 5x + 3y Subject to: 5x + 3y 0 Subject to: 5x + 4y 0 4x + 4y 8 3x + 4y 8 x, y 0. x, y 0. Minimize 5x + 3y Subject to: 5x + 4y 8 3x + 4y 0 x, y 0. Minimize 5x + 3y Subject to: 5x + 4y 8 3x + 4y 0 x, y 0. Convert the constraints into linear equations by using slack variables. 27) 27) Maximize z = 2x + 8x2 Subject to: x + 6x2 5 7x + 5x 2 25 x 0, x2 0 x + 6x2 = s + 5 7x + 5x2 = s2 + 25 x + 6x2 + s 5 7x + 5x2 + s2 25 x + 6x2 + s 5 7x + 5x 2 + s 2 25 x + 6x2 + s = 5 7x + 5x2 + s2 = 25 7
Introduce slack variables as necessary and write the initial simplex tableau for the problem. 28) Maximize z = 4x + x2 28) subject to: 2x + 5x2 3x + 3x2 9 x 0, x2 0 x x2 s s2 z 2 5 0 0 3 3 0 0 9 4 0 0 0 x x2 s s2 z 2 5 0 0 3 3 0 0 9-4 - 0 0 0 x x2 s s2 z 2 5 0 0 9 3 3 0 0 4 0 0 0 x x2 s s2 z 2 5 0 0 9 3 3 0 0-4 - 0 0 0 Find the pivot in the tableau. 29) 29) 2 in row 2, column 4 in row 2, column 2 2 in row, column 4 in row, column 3 Use the indicated entry as the pivot and perform the pivoting once. 30) 30) 8
Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to 0. 3) 3) x x2 x3 x4 x5 z 3 4 0 3 0 8 5 7 0 0 25-3 4 0 0 9 x = 0, x2 = 0, x3 = 25, x4 = 0, x5 = 8, z = 9 x = 25, x2 = 0, x3 = 0, x4 = 0, x5 = 8, z = 9 x = 0, x2 = 0, x3 = 8, x4 = 0, x5 = 25, z = 9 x = 25, x2 = 0, x3 = 8, x4 = 0, x5 = 0, z = 9 A bakery makes sweet rolls and donuts. A batch of sweet rolls requires 3 lb of flour, dozen eggs, and 2 lb of sugar. A batch of donuts requires 5 lb of flour, 3 dozen eggs, and 2 lb of sugar. Set up an initial simplex tableau to maximize profit. 32) The bakery has 270 lb of flour, 220 dozen eggs, 250 lb of sugar. The profit on a batch of sweet 32) rolls is $6.00 and on a batch of donuts is $9.00. x x2 s s2 s3 s4 3 5 0 0 0 270 3 0 0 0 220 2 2 0 0 0 250 6 9 0 0 0 0 x x2 s s2 s3 s4 3 5 0 0 0 270 3 0 0 0 220 2 2 0 0 0 250-6 9 0 0 0 0 x x2 s s2 s3 s4 3 5 0 0 0 270 3 0 0 0 220 2 2 0 0 0 250-6 - 9 0 0 0 0 x x2 s s2 s3 s4 3 5 0 0 0 270 3 0 0 0 220 2 2 0 0 0 250-9 - 6 0 0 0 0 9
Answer Key Testname: UNTITLED ) D 2) A 3) D 4) B 5) D 6) D 7) A 8) A 9) A 0) B ) D 2) B 3) D 4) A 5) C 6) B 7) B 8) B 9) A 20) B 2) A 22) D 23) B 24) C 25) C 26) B 27) D 28) C 29) B 30) D 3) A 32) C 0