MATH 1108 MIDTERM EXAM MODEL SOLUTION

Transcription

1 MATH 1108 MIDTERM EXAM MODEL SOLUTION FALL MOON Name: FIDN#: Write your answer neatly and show steps. Except calculators, any electronic devices including a cell phone are not allowed. 1) Quick survey. a) 1pt) This class is: Too easy Moderate Too difficult b) 1pt) Write any suggestion for improving this class. For instance, give more examples in class, explain proofs of formulas in detail, give more homework, slow down the tempo,...) ) 5 pts) Let Compute A [ ] [ 3 1, B 3AB C. ] [ 1 0, C 3 ] ) 1 + 1) 4 0 AB [ ) ) 1 0 C 3 3 1) + 3) 0 + 3) 3) AB C Calculating AB : pts Evaluating C : 4 pts Obtaining 3AB C : 5 pts ] Date: October 10,

2 3) A dietitian in a hospital is to arrange a special diet composed of three basic foods. The diet is to include exactly 34 units of calcium, 18 units of iron, and units of vitamin A. The number of units per ounce of each special ingredient for each of the foods is indicated in the table. Units per Ounce Food A Food B Food C Calcium 3 1 Iron 1 1 Vitamin A 1 3 a) pts) Set up a system of linear equations and corresponding augmented matrix for this problem. x number of units of A, y number of units of B, z number of units of C System of linear equations: Augmented matrix: 3x + y + z 34 x + y + z 18 x + 3y + z Writing the system of linear equations: 1 pt. Finding the augmented matrix: pts. b) 7 pts) Find the reduced row echelon form for the augmented matrix in a) R 1 +R 3 R 3 R +R 1 R R 3 R 3 R 1 R R 1+R R 1 R R R +R 3 R 3 R 3+R R Finding a correct reduced row echelon form: 7 pts. Performing an incorrect row operation: - pts for each.

3 c) pts) How many ounces of each food must be used to meet the diet requirements? Food A: 8 units, Food B: units, Food C: 4 units 4) a) 4 pts) Find the inverse of the matrix R 1 R R 1 +R R R R R +R 1 R Setting up an augmented matrix : 1 pt Finding the inverse matrix : 4 pts. 5 Performing an incorrect row operation: -1 pt for each. b) 3 pts) By using the inverse matrix, solve the system of linear equations x + y 1 5x + 3y 1 x y 1 x y ) x 1, y 1 1 x 1 Setting up a matrix equation : 1 pt. 5 3 y Using the inverse matrix, find the solution x 1, y 1: 3 pts. If you doesn t use the inverse matrix, you can t get any credit. 3

4 5) A furniture manufacturing company manufactures dining-room tables and chairs. The relevant manufacturing data are given in the table below. Labor-Hours per Unit Maximum Labor-Hours Department Table Chair Available per Day Assembly Finishing 1 0 Profit per unit $90 $5 a) pts) Identify the variables and write the constraints as inequalities. x number of produced tables in a day, y number of produced chairs in a day x 0 y 0 8x + y 400 x + y 0 b) pts) Write the objective function. p 90x + 5y 4

5 c) 5 pts) Graph the solution region and find all corner points. 8x + y 400, x + y 0 8x + y 400, y x + 0 8x + x + 0) 400 4x x 40, y Corner points: 0, 0), 50, 0), 0, 0), 40, 40) Sketching graph: 3 pts. Finding all corner points 0, 0), 50, 0), 0, 0), 40, 40): 5 pts. d) 3 pts) Find the maximum profit and determine how many tables and chairs should be manufactured each day to maximize the profit. Corner points profit 0, 0) 0 50, 0) , 0) , 40) 4600 Maximum profit: $4, 600 It occurs when 40 tables and 40 chairs are made. Comparing the objective function at all corner points and finding the maximum $4, 600: pts. Indicating the numbers of tables 40) and chairs 40): 3 pts. 5

6 6) 5 pts) If you paid $40 to a loan company for the use of $1, 000 for 40 days, what annual rate of interest did they charge? Answer by using simple interest. Use a 365-day year and answer in percentage. Round to three decimal places. 40 I P rt r 365 r r 36.5% 40 Setting up a formula r to find the annual interest: 3 pts. 365 Finding the annual interest 0.365: 4 pts. Writing the answer in percentage and getting 36.5%: 5 pts. 7) A newborn child Jimmy receives a $0, 000 gift toward college from his grandparents. His mother Amy wants to invest the $0, 000 for 17 years. a) 5 pts) Bank of America offers a money market account that earns 4.3% compounded monthly. How must will it worth in 17 years? Round to the nearest cent. A P 1 + i) n P 0000, i 0.043, n A ) $41, Finding relevant information P 0000, i 0.043, n 04: pts. Setting up a formula ) 04 for the future value: 4 pts. Obtaining the correct answer $41, : 5 pts. 6

7 b) 5 pts) There are two additional investment options. Chase offers an account that earns 4.31% compounded quarterly. Wells Fargo offers an account that earns 4.9% compounded continuously. Find APY of all of three choices. Answer in percentage and round to three decimal places. APY for Bank of America: APY 1 + r m APY for Chase: APY 1 + r m APY for Wells Fargo: r 0.043, m ) % ) m r , m 4 ) % ) m r APY e % Finding all APY s: 5 pts. Evaluating two of three APY s: 3 pts. Getting one of three APY s: 1 pt. c) pts) Which one is the best investment among them? Bank of America d) 6 pts) Amy decides to choose Bank of America. How long will it take $0, 000 to grow to $30, 000? A 30000, P 0000, i ) n ln 3 ln ) n n ln ) n It takes 114 months. ln 3 ln ) Setting up the equation of periods n: 3 pts. By using logarithm finding n : 5 pts. Getting 114 months: 6 pts. You have to write the unit for the time. ) n ) n for the number 7