WIR Math 141-copyright Joe Kahlig, 11B Page 1Sample Problems For Exam 1Compiled by Joe KahligThis collection of questions is intended to give an idea of different types of question that might be asked on theexam. This is not intended to represent an actual exam.These questions cover chapters 1 and 2 in the Applied Finite Mathematics, 9thedition by S. T. Tan.Video solutions can be found at this link:http://www.math.tamu.edu/∼kahlig/141WIRpage.htmlNote: The questions start with number 5 since questions 1-4 were f rom material that is no longer covered and Idid not want to redo all of the recorded solutions.5. As a person descends into the ocean, pressure increases linearly. The pressure is 15 pounds per square inchon the surface and 30 pounds per square inch 33 feet below the surface. If y is the pressure in pounds persquare inch and x is the d ep th below the surface in feet, write an equation that expresses the pr essure interms of the dep th .6. Auto-time, a manufacturer of 24-variable timers, has a monthly fixed cost of $48,000 and a production costof $8 for each timer manufactured.(a) Find the Cost function.(b) What is the selling price of the timers, if the company has a profit of $112,000 when selling 5,000 timers.(c) Find the Revenue function.(d) Find the Profit function.(e) How many items should be sold to break even?7. The supply function for a product is given by 3x − 11p + 45 = 0 and the demand f unction for this productis given by 2x + 7p − 56 = 0.(a) What is the equilibrium price?(b) What is the equilibrium quantity?8. The number of milk cows in the United States is given in the table. (Source: US Dept. of Agriculture andfrom Finite Mathematics: Practical Applications by John son/Mowry).Year 1970 1975 1980 1985 1990Millions of Cows 12.091 11.220 10.758 10.777 10.153For This problem let the time start with zero in 1970.(a) Determine the equation of the least-squares (regression) line for this data.(b) Sketch a scatter diagram and the least-squares line for the data.(c) Pred ict the number of cows in the year 1985.(d) In what year would we expect to have 8 million milk cows?(e) Pred ict the number of cows in the year 2150.(f) In what year would we expect to have 15 million milk cow s?9. Give an example of a matrix in row-reduced form (reduced row-echelon form) that describes a system withan in finite number of solutions.10. Give an example of a matrix in row-reduced form (reduce row-ech elon form) with exactly one solution.11. Give an example of a matrix in row-reduced form (reduce row-ech elon form) with no solution.WIR Math 141-copyright Joe Kahlig, 11B Page 212. For the next two word problems do the following.I) Define the variables that are us ed in setting up the system of equations.II) Set up the system of equations that represent this problem.III) Solve for the solution.IV) If the solutions is parametric, then tell what r estrictions can be placed on the parameter. Also givethree specific solutions.(a) The management of a private investment club has a fund of $300,000 earmarked for investment instocks. To arrive at an acceptable overall level of risk, th e stocks that management is considering havebeen classified into three categories: high-risk, medium-risk, and low-risk. Management estimates thathigh-risk stocks will have a rate of return of 16 percent per year; medium-risk stocks, 10 percent peryear; and low-risk stocks, 4 percent per year. The investment in medium-risk stocks is to be twice theinvestment in stocks of the other two categories combined. If the investment goal is to have an averagerate of r eturn of 11 percent per year on the total investment, determine how much the club shouldinvest in each type of stock.(b) A chemical manufacturer wants to purchase a fleet of 24 railroad tank cars with a combined carryingcapacity of 250,000 gallons. Tank cars with th ree different carrying capacities are available: 6,000gallons, 8,000 gallons, and 18,000 gallons. How many of each type of tank car should be pu rchased?13. Fill in the missin g entries by performing the indicated row operations.3 6 1597 12 39252 6 543 0 61R1(13) → R1∗ ∗ ∗∗7 12 39252 6 543 0 61R2+ (−7)R1→ R23R3+ (−2)R4→ R31 2 53∗ ∗ ∗∗∗ ∗ ∗∗3 0 6114. Solve for the variables x, y, z, and u. If this is not possible, then explain why1 −1 00 1 01 −2 3−2x 03 4x + 2 3− 3y − 1 21 24 2z + 1=−7 −2u0 −28 1015. Find the matrix K that makes the following true. If this is not possible, then explain why.0 8 17 −6 03 0 1+122 0 05 2 06 6 1K =7 0 60 1 43 7 016. Find a m atrix A and a matrix B s uch that AB can be computed but BA can not be computed.17. Use the following matrices for this problem. Compute the following operations. If it is not possible, thenexplain why.A =1 0−1 −2B =1 −1 30 2 1C =1 −20 24 −1D =1 −2 0−1 3 2E =1 −1 30 1 01 −2 3D + C = D − 3B = DC =DA = B + CT= B−1=A−1= E−1=WIR Math 141-copyright Joe Kahlig, 11B Page 318. John and Matt have a roofing company and they each have a crew that they work with to roof houses. Johnand Matt have given each worker a designation based on the number of years of experience. A worker whohas less than one year of experience is classified as N. A worker with at least one and less than 3 years ofexperience is classifies as B. A worker with at least 3 and less than 7 years of experience is classified as E.A worker with more than 7 years of experience is classified as VE. The breakup of John and Matt’s crewscan be sh own in this matrix W.P =NBEVE5.156.257.509.00W = JohnMattN B E VE2 3 4 14 2 0 3The matrix P represents the pay per hour of each worker designation.(a) Find WP(b) Explain the meaning of the entries in th e matrix WP.19. Solve the systems of equations by using inverses.2x + y + z = b15x + 2y + z = b23x + 2y + 4z = b3(a) b1= 2, b2= −1, b3= 0(b) b1= 3, b2= 4, b3=
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