1070 UNIFORM FINAL EXAM DECEMBER 12, 1998

PRINT YOUR NAME CIRCLE YOUR SECTION:

INSTRUCTIONS: READ THE PROBLEMS CAREFULLY. IF YOU DO NOT UNDERSTAND A QUESTION, ASK FOR CLARIFICATION. YOU HAVE 3 HOURS TO COMPLETE THE EXAM. YOU MAY USE AN INCH CHEAT SHEET AND A CALCULATOR. TO RECEIVE PARTIAL CREDIT, YOU MUST SHOW ALL WORK. THE MAXIMUM SCORE FOR THIS EXAM IS 100 POINTS: 50 POINTS FOR PART 1 AND 50 POINTS FOR PART 2. GOOD LUCK.! DO NOT WRITE BELOW THIS LINE

1. WHAT IS THE VALUE OF ?
2. LET , , AND F(X) = . WHAT IS F(3)?
3. WHAT ARE THE COORDINATES OF THE VERTEX OF THE PARABOLA DESCRIBED BY THE FUNCTION ?
4. WHAT ARE THE X-INTERCEPT(S) OF THE FUNCTION H(X) = ?
5. WHAT ARE THE VERTICAL ASYMPTOTE(S) OF THE FUNCTION H(X) = ?
6. WHAT IS THE RANGE OF THE EXPONENTIAL FUNCTION?
7. A SMALL COMBINATION LOCK ON A SUITCASE HAS 3 WHEELS, EACH LABLED WITH 10 DIGITS FROM 0 TO 9. HOW MANY 3-DIGIT COMBINATIONS ARE POSSIBLE IF NO DIGIT IS REPEATED.
8. EVALUATE .
9. EVALUATE .
10. EVALUATE .
11. EVALUATE .
12. SUPPOSE YOU INVEST $100 IN AN ACCOUNT THAT OFFERS 17% INTEREST COMPOUNDED QUARTERLY. WHAT IS THE BALANCE IN THE ACCOUNT AFTER TWO YEARS?
13. HOW MUCH SHOULD YOU INVEST IN AN ACCOUNT EARNING 15% COMPOUNDED MONTHLY IF YOU WANT $1600 IN 2 YEARS?
14. IF , AND I = 0.03, FIND PMT.
15. IF N = 20, I = 0.0175, AND , FIND PV.
16. USE ROW OPERATIONS TO CHANGE THIS MATRIX TO REDUCED FORM.

    

17. EVALUATE AB WHEN

    

    REFER TO THE FOLLOWING SET OF EQUATIONS FOR THE NEXT PROBLEMS.

    

18. CONSTRUCT THE AUGMENTED MATRIX FOR THIS SYSTEM.
19. STATE IF THE SYSTEM HAS NO SOLUTION, EXACTLY ONE SOLUTION, OR INFINITELY MANY SOLUTIONS.
20. IS (1,1,4,4) A PARTICULAR SOLUTION OF THIS SYSTEM? IF NOT, FIND ONE!
21. THE CORNER POINTS OF A BOUNDED REGION FOR A SYSTEM OF LINEAR INEQUALITIES ARE

    

    WHAT IS THE MAXIMUM VALUE OF P = X + 2Y ON THIS REGION? REFER TO THE FEASIBLE REGION AND GRAPH BELOW FOR THE FOLLOWING PROBLEMS.

22. FIND THE COORDINATES OF THE CORNER AT A.
23. FIND THE COORDINATES OF THE POINT B.
24. COULD A LINEAR OBJECTIVE FUNCTION HAVE A MAXIMUM AT THE POINT (3,2)?
25. WHICH OF THE FOLLOWING POINTS ARE NOT IN THE FEASIBLE REGION? (5,5), (2,7), (6,3).
26. THE CONSUMER PRICE INDEX (CPI) IS A MEASURE OF THE COST OF LIVING IN THIS COUNTRY. ITS VALUE WAS SET AT 100 IN 1984; IT STOOD AT 160 IN 1997. ASSUME THAT THE CPI INCREASES AT A RATE OF 3% PER YEAR FOR ALL YEARS AFTER 1997.
    1. LETTING T=0 REPRESENT 1997, FIND AN EXPONENTIAL FUNCTION THAT GIVES THE CPI FOR ALL YEARS AFTER 1997.
    2. WHAT IS THE DOUBLING TIME FOR THE CPI ASSUMING A 3% ANNUAL GROWTH RATE?
    3. IN WHAT YEAR WILL THE CPI REACH 200 ASSUMING A 3% ANNUAL GROWTH RATE?
27. A TOWN COUNCIL HAS 11 MEMBERS, 6 DEMOCRATS AND 5 REPUBLICANS.
    1. IF THE PRESIDENT AND VICE PRESIDENT ARE SELECTED AT RANDOM, WHAT IS THE PROBABILITY THAT THEY ARE BOTH DEMOCRATS?
    2. A TOWN COUNCIL HAS 11 MEMBERS, 6 DEMOCRATS AND 5 REPUBLICANS. IF A 3-PERSON COMMITTEE IS SELECTED AT RANDOM, WHAT IS THE PROBABILITY THAT REPUBLICANS MAKE UP THE MAJORITY?
28. SUPPOSE YOU BOUGHT A CAR 25 MONTHS AGO FOR $9000 AND FINANCED IT AT 12% COMPOUNDED MONTHLY FOR 48 MONTHS.TO KEEP THINGS SIMPLE, ASSUME NO TAXES OR FEES OF ANY KIND!!
    1. WHAT ARE YOUR MONTHLY PAYMENTS?
    2. IF YOU MAKE 25 MONTHLY PAYMENTS, HOW MUCH IS STILL OWED ON THE CAR?
    3. IF YOU SELL THE CAR AS-IS FOR $6000, HOW MUCH OF THIS WOULD ACTUALLY GO INTO YOUR POCKET?
29. 1. JJ HAS A LITTLE BUSINESS OF SELLING BEANY BABIES. THEY COME IN TWO DIFFERENT SIZES, SMALL AND LARGE, AND TWO DIFFERENT COLORS, RED AND TEAL; SO THERE ARE FOUR DISTINCT TYPES OF BEANY BABIES. ALL TEAL BEANY BABIES SELL FOR 30% MORE THAN THE RED BEANY BABIES. THE SMALL RED BEANY BABY SELLS FOR $20 AND THE LARGE RED BEANY BABY SELLS FOR $35. WRITE A MATRIX, WITH COLUMNS REPRESENTING COLOR AND ROWS REPRESENTING SIZE, THAT GIVES THE PRICE OF THE FOUR TYPES OF BEANY BABIES.
    2. JJ ALSO SELLS T-SHIRTS. HE HAS A $5 PROFIT ON SMALL T-SHIRTS AND A $7 PROFIT ON LARGE T-SHIRTS. HIS GOAL NEXT WEEK IS TO SELL 50 PACKAGES AND MAKE $300. DEFINE THE VARIABLES AND SET UP THE SYSTEM OF EQUATIONS FOR THIS PROBLEM. HOW MANY T-SHIRTS OF EACH SIZE DOES HE NEED TO SELL?
30. A VINEYARD PRODUCES TWO SPECIAL WINES, A WHITE AND A RED. A BOTTLE OF THE WHITE WINE REQUIRES 4 POUNDS OF GRAPES AND 1 HOUR OF PROCESSING TIME. A BOTTLE OF THE RED WINE REQUIRES 5 POUNDS OF GRAPES AND 2 HOURS OF PROCESSING TIME. THE VINEYARD HAS ON HAND 420 POUNDS OF GRAPES AND CAN ALLOT 150 HOURS OF PROCESSING TIME TO THE PRODUCTION OF THESE WINES. A BOTTLE OF THE WHITE WINE SELLS FOR $11.00, WHILE A BOTTLE OF THE RED WINE SELLS FOR $20.00. THE GOAL IS TO DETERMINE THE NUMBER OF BOTTLES OF EACH TYPE THAT SHOULD BE PRODUCED IN ORDER TO MAXIMIZE GROSS SALES.
    1. SET UP THE LINEAR PROGRAMMING PROBLEM THAT NEEDS TO BE SOLVED TO ANSWER THE QUESTION.
    2. GRAPH THE FEASIBLE REGION FOR THIS PROBLEM.

       

    3. GIVE THE COORDINATES OF THE CORNER POINTS OF THE FEASIBLE REGION.
    4. FIND THE NUMBER OF BOTTLES OF RED WINE AND THE NUMBER OF BOTTLES OF WHITE WINE NEEDED TO MAXIMIZE THE PROFIT. WHAT IS THE MAXIMUM PROFIT?