Linear Programming Questions and Answers

13 Use the list of properties on the right to indicate which property is illustrated in each equation on the left Note Not all properties will be used and they may be used more than once Please label to the left of the left column a x y z xy xz b 2 5 5 2 c 0 4 4 d 3 2 5 3 2 5 e 1 1 8 1981 f 1 3 5 15 1 1 1 3 3 A Commutative Property of Addition B Commutative Property of Multiplication C Associative Property of Addition D Associative Property of Multiplication E Closure of Addition MOI d F Closure of Multiplication G Additive Identity Property H Multiplicative Identity Property 1 Distributive Property of Multiplication over Addition

Which of the following is the correct value for the force in member BF T Tension C Compression 4 m 4 m 4 m O 10 kN C O 10 kN T O OKN O 5 kN C O 5 kN T 5 kN 10 kN 10 kN B D 4 m 5 kN E F

Find the horizontal support reaction at point C 13 3 kN 8 75 KN O 14 6 KN 10 5 KN 3 m B 3 m 10 KN 3 m 15 kN 3 m C

4 In a right triangle one leg measures 4 inches and the other leg measures 6 inches What is the length of the hypotenuse in inches OA 52 B 10 O C 13 2 OD 2 13

The hyperbolic cosine of the argument given in radians can be calculated by the series cosh z 1 n 1 2n T 2n P 1 cosh r 1 N 2 n 1 I I 6 a Write a Matlab function myCosh that takes as input r and the number of terms N to use to estimate cosh z i c 7 2n 2n 1 2 and returns the estimate for cosh r together with the corresponding true relative error You may use the Matlab function factorial to calculate the factorial in the formula For the true value of cosh r use Matlab s build in function cosh x Do not print out any results to screen or do any plotting within the function b Calculate and report with at least 15 digits the estimate for cosh 20 and the true relative error for i N 5 ii N 20 iii N 30 c In a semilogarithmic plot use semilogy plot the absolute of the true relative error vs N for N 1 40 d Explain why the absolute of the true relative error does not decrease for N 32

Question 9 Points 2 What musical structure governed composition of the twentieth century No single musical structure that governed compositions of the twentieth century The musical structure of the European Court governed compositions of the twentieth century The musical structure of Neoclassicism governed compositions of the twentieth century The musical structure of American jazz governed compositions of the twentieth century

P7 27 Write a Boolean expression for the output of each of the logic circuits shown in Figure P7 27 A B C A B A B C a BA Figure P7 27 b c F

3 Find the domain of the following functions a f x 2 0 0 X b g x c h x 1 00 1 R 0 0 x v HOC23 A2 3 0 2 1 x2 5x 6 x 7 x x 55046 son 32

Using the given 1 f x x 2x and g x 5 x a f g x c fog x 2 f x 2 and g x x a g f x c f g x 4x 3 5 3 f x x 7 and g x 2x a f g x c g f x b g f x d x b g f x d 114 x b g f x d g x

Price indexes can be used to compare prices across different periods Suppose that a year of tuition for college at public institutions averaged a cost of 1 958 in 1989 and that the CPI index was 121 in 1989 If the CPI index was 212 in 2009 then the cost of tuition in 2009 as the result of inflation would equal Enter your response rounded to the nearest whole number

Click the following link to view GDP GNP data from FRED Then use that data to answer the following questions Real time data provided by Federal Reserve Economic Data FRED Federal Reserve Bank of Saint Louis For this exercise you will need to enter data from FRED for Gross Domestic Product GDP and Gross National Product GNP Using the data from FRED enter the values for the first quarter of 2023 shown as 2023 01 01 in FRED for the following series IDS Enter your responses exactly as they appear in FRED Value Title Gross Domestic Product Gross National Product Series ID GDP GNP billion billion

Given a floating point format with a k bit exponent and an n bit fraction write formulas for the exponent E significand M the fraction f and the value V for the quantities that follow In addition describe the bit representation a The number 15 0 b The reciprocal of the smallest positive normalized value c The largest odd integer that can be represented exactly

a Use the simplex method to maximize x y 4z subject to the constraints x 2y 32 50 2xy z 10 a 0 y 0 How would your answer change if the 50 in the question were replaced b Write down a starting tableau for the problem of minimizing 8x 5y to the constraints 2 0 y 0 by 55 z subjec x y z 4 3x 215 x 0 Identify the first pivot element but do not go further c Determine the optimal strategy for R for the game whose payoff matrix is 25 Which player does the game favour and by how much z 0

Problem 5 Solve the linear programming problem You may find it useful to use desmos com to graph the constraints Maximize P 3x 2y subject to x y 3 2x y 8 T 0 y 0 The maximum is P at x and y

Integrate 13 6x 21x dx a 6 126x5 b 13x 3x 3x7 OC 6 126x5 C d 13x 3x 3x C

Convert 95 degrees Celsius to its equivalent measure on the Fahrenheit scale Round to the nearest tenth degree Use one of the formulas shown below 9 C F 32 F C 32 1 228 6 degrees F 2 35 degrees F

le TV company has 5400 customers paying 130 each month If each 1 reduction in price attracts 50 new customers find the price that yields maximum revenue the maximum revenue

uate the integral Check your answer by differentiating Remember the constant of integration 34 x 4 dx

1 Tristan plans to pay back a loan of 230 000 in 5 years He makes quarterly payments at the end of each quarter The bank charges 7 8 per year compounded quarterly K4 Determine Tristan s quarterly payment using a TVM solver N 1 PV PMT FV P Y C Y DMT END EGIN

The diameter of a pipe is normally distributed with a mean of 0 8 inch and a variance of 0 0004 What is the probability that the diameter of a randomly selected pipe will exceed 0 84 inch You may need to use the standard normal distribution table Round your answer to three decimal places

State the hypotheses for the simple linear regression tests Ho H K Using the accompanying Concert Sales data on sales dollars and the number of radio TV and newspaper ads promoting the concerts for a group of cities Develop simple linear regression models for predicting sales as a function of the number of each type of ad Compare these results to a multiple linear regression model using both independent variables State the model and explain R square Significance F and p values with an alpha of 0 05 Click the icon to view the Concert Sales data at least one B is not 0 B B Bk 0 B 0 O Points 0 of 5 B 0 Save 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Sales 1000 Thousands of radio and TV ads 1064 997 919 626 869 950 946 1199 881 1011 1614 1562 1067 934 1395 1336 1401 1496 1561 1715 1774 1786 0 0 25 25 32 30 34 33 38 41 47 43 48 48 53 54 62 62 68 64 70 68 Thousands of newspaper ads 39 41 26 25 29 30 34 33 25 24 45 47 48 52 20 21 32 32 33 33 39 42

Solve the following optimization model using Excel Solver and generate a sensitivity report Upload the final excel fil min z 8x 11x 9x3 1 15x 13x 19x 450 16x 15x 18x32 490 1 5x 4x 10x 200 2 3 2x 5x 50 3 3x 45 1 2 30

Find the general solution of t 2 y 4ty 6y t 7 t 0 given that y1 t 3 Find the general solution to xy 4x 1 y 4x 2 y 0 x 0 given that y1 e 3x is a solution Solve the differential equation x 2 y 6xy 10y 0

A volunteer has been asked to drop off some supplies at a facility housing victims of a hurricane evacuation The volunteer would like to bring at least 132 bottles of water 82 first aid kits and 45 security blankets on his visit The relief organization has a standing agreement with two companies that provide victim packages Company A can provide packages of 6 water bottles 4 first aid kits and 3 security blankets at a cost of 3 Company B can provide packages of 9 water bottles 5 first aid kits and 1 security blanket at a cost of 2 50 How many of each package should the volunteer pick up to minimize the cost What amount does the relief organization pay Answer Enter the value in the first box and the ordered pair in the second box Minimum Cost of S Minimum Cost at Tables Keypad Keyboard Shortcuts

Find the maximum value of f x x2 4x x2 subject to the following constraints Write the exact answer Do not round If the function has no maximum value write None Answer x 2x 3 3x x 2 4x1 5x 5 Maximum value Tables Keyp Keyboard Short

Find the maximum value of f x1 x2 x3 2x 4x2 6x3 subject to the following constraints Write the exact answer Do not round If the function has no maximum value write None Answer 2x x x3 x 5x x3 x1 x2 4x3 11 20 14 Keypa Keyboard Shorte Tables

Integer Encoding Convert the given expressions to 4 bit decimal 4 bit binary 6 bit decimal and 6 bit binary Use C convention for type casting Example Expression x 8 Example solution 4 bit decimal 8 4 bit binary 1000 6 bit decimal 8 6 bit binary 11 1000 a x 5 b x 15 c x 16 d x 37

For the data in the 0 22 range in the table which of the Newton Cotes Integral calculations made for the h values and their integral boundaries is are correct X y 0 2 6 8 156 230 282 310 364 10 12 14 396 422 445 16 480 18 500 for h 4 A 1 5 156 2 422 3 282 364 480 510 541 for h 6 A 3 156 2 310 422 500 for h 4 A 4 3 156 4 282 422 510 2 364 480 541 541 20 510 10 22 54

Suppose you toss a coin and will win 1 if it comes up heads If it comes up tails you toss again This time you will receive 2 if it comes up heads If it comes up tails toss again This time you wil receive 4 if it is heads Continue in this fashion for a total of 1 000 flips of the coin after which you receive nothing if it comes up tails What is the mathematical expectation for this game X

A function and the graph of a feasible region are given f 7x 8y x 2y 19 3x 2y 29 Find both the maximum and minimum values of the function if they exist and the point at which each occurs If an answer does not exist e The maximum value of the objective function is X which occurs at x y The minimum value of the objective function is 91 which occurs at x y 5 7 X

Maximize f Form the dual maximization problem Use x and x as the variables and f as the function and use the simplex method to solve the minimization problem Assume that all variables are nonnegative Minimize g 4y 3y2 subject to the following 5y1 2y 2 16 3y1 71 2 27 S4 3 subject to X1 X 0 The minimum value of the objective function is MY NOTES which occurs at y y ASK YOUR TEACHE

1 0 0 Step 2 Corner 0 3 0 0 1 Step 1 We want to find the maximum and minimum values of the objective function C 6x 8y given the feasible region determined by the constraint inequalities We know that the optimal values of the objective function will occur at a corner a corner of the feasible region Thus we need to test the coordinates of the corner points in our objective function 7 0 42 6 2 52 C 6x 8y 0 52 4 4 56 4 4 56 3 5 7 0 42 52 6 2 56 7 0 0 3 24 24 The maximum value of the objective function is which occurs at x y

12 168 The pin follows the path described by the equation r 0 2 0 15 cos 0 m At the instant 0 30 8 0 7 rad s and 0 5 rad s Determine the magnitudes of the pin s velocity and acceleration at this instant Neglect the size of the pin 0 7 rad s r 0 2 0 15 cos

f 7x 8y x 2y 19 3x 2y 29 Find both the maximum and minimum values of the function if they exist and the point at which each occurs If an answer does not exist enter D The maximum value of the objective function is which occurs at x y The minimum value of the objective function is C which occurs at x y

The problem below involves three variables Solve it with the simplex method Excel or some other technology A contractor builds three types of houses the Aries the Belfair and the Wexford Each house requires one lot and the following table gives the number of labor hours and the amount of capital needed for each type of house as well as the profit on the sale of each house There are 13 lots 47 800 labor hours and 3 398 400 available for the contractor use Labor hours Capital Profit Aries 3 000 205 000 20 000 Belfair 3 700 279 600 25 000 Wexford 5 000 350 000 30 000 a Building how many of each type of house will maximize his profit Aries houses Belfair houses Wexford houses b What is the maximum possible profit

sec 780 Determine the reference angle for 780 ing expression

For each problem find the equation of the secant line that intersects the given points on the function You may use the provided graph to sketch the function 9 y 2x 2 1 4 0 2 10 y x x 2 1 0 0 2 Ay y 6 x 44

2 Suppose that consumers demand for each of the differentiated goods is Qd 10 p p where Qa is the quantity of demand n is the number of differentiated goods p is the price of the differentiated good and p is the average price of differentiated goods The cost function to produce each of the differentiated good is C 10 Qs where C is the total cost and Q is the quantity of supply Answer the following questions a 8 points Calculate the equilibrium level of n and p b 8 points Suppoe the cost function is C 5 Q Should n increase or decrease Explain

The spinner below is spun P even shaded 9 8 5 7 2 A 6

3 For each of the following matrices write down the quadratic form and decide with proof whether the matrix is positive definite a 4 marks b 4 marks B A 9

The grade point averages of 20 college seniors selected at random from a graduating class are in the accompanying data set available below Calculate the standard deviation Click the icon to view the grade point averages The standard deviation is s Round to two decimal places as needed Grade Point Averages 1 7 1 9 3 4 3 7 1 9 2 4 3 5 2 1 3 3 2 0 3 6 3 2 3 2 3 5 2 2 2 8 2 9 3 5 2 5 2 2 D X

A company produces three types of shoes formal casual and athletic at its two factories Factory I and Factory II Daily production of each factory for each type of shoe is listed below Formal Casual Athletic Z Factory 100 100 300 The company must produce at least 5800 pairs of formal shoes 8300 pairs of casual shoes and 9900 pairs of athletic shoes If the cost of operating Factory I is 1800 per day and the cost of operating Factory II is 1850 how many days should each factory operate to complete the order at a minimum cost and what is the minimum cost a Letting a be the first of the variables listed in the problem statement and y the second write the objective function Factory II 100 200 100 b Graph the feasible region on paper then list the corner points of the feasible region Enter ordered pairs x y separated by commas c Test the corner points to find the optimum value of the objective function The winning corner point is The optimum value of the objective function is

Graph the function over a two period interval 3x Choose the correct graph below OA Q o OB 41 OC

4 2 9 7 6 4 5 3 8 Is b in the range of linear transformation x Ax If so find x b Let A 5 5 0 3 4 8 5 and b 5 9 7

You may need to use the appropriate appendix table or technology to answer this question Given are five observations for two variables x and y Round your answers to two decimal places 123 4 7 474 11 14 a Use Sp x 1 n X x c Use Spred x b Use af to develop a 95 confidence interval for the expected value of y when x to to estimate the ate the standard deviation of p SO ate the standard deviation of an individual value of w to estimate th 0 Used to develop a 95 prediction interval for y when x 5 x s

Let p be a prime number and w be the roots of p x x x 1 If either p 5 mod 12 or p 11 mod 12 then Z w is a field of order p Remark Zp w Zp 3

Describe the the feature of the graph of the following function f x Domain 0 0 U 0 Range 0 0 U 0 0 The graph does not cross the x and y axis Domain 0 00 Range 0 00 The graph is a straight line with slope of 1 and y intercept 0 0 Domain 0 00 Range 0 00 The graph is half of a parabola lying on its side Domain 0 00 Range 00 00 The graph is an increasing function As x values gets smaller and smaller y values gets smaller and smaller As x values gets larger and larger y values gets larger

r 6 puy on suomun uavis air asn x 4 and g z x 3 Use sqrt x to enter a square root of x

Use the graphing tool at the right to create a scatter plot for the data in the table Use the shape of the scatter plot to determine if the data are best modeled by a linear function an exponential function a logarithmic function or a quadratic function X 4 3 2 1 0 ATTOTT y 4 Plot each point one at a time Using the graphing tool start by plotting 4 4 COLL 10 B B 10 B B B 10 10

Use the given feasible region determined by the constraint inequalities to find the maximum and minimum of the given objective function if they exist If an answer does not exist enter C 3x 9y 7 5 y 3 0 3 1 0 0 HH 1 maximum C 48 minimum C 3 4 4 5 X 6 2 7 0 7