Math 141 Final Review

A new car sells for $24,000. It depreciates linearly to a value of $16,500 in 3 years.

a) Find the value of the car, V(t), as a function of t=age in years of the car.

b) Find the value of the car when it is 5 years old.

A manufacturer has a monthly fixed cost of $30,000. Each unit he produces costs an additional $5. The selling price per unit is $8.

a) Find the cost, revenue and profit functions, C(x), R(x) and P(x).

b) How many units should be produced and sold per month to break even?

3.a) Suppliers of a certain brand of refrigerator will supply none if the unit price is $500 or less. They will supply 300 when the unit price is $710. Find the supply equation.

b) Demand for the refrigerator is 200 units when the price is $600 and demand decreases by 50 units for each $25 increase in price. Find the demand equation.

c) Find the equilibrium quantity and price.

4. The table shows the production, in billions, of aluminum cans from 1983 to 1989. Let x=0 represent 1983. Use the least squares line to estimate the production for 1991.

1983 1985 1987 1989

57 65.8 74.2 83.3

5. a) Compute 2A+3B b) Compute (2A+3B)

6. A manufacturer has 2 kinds of products at 2 different locations. Matrix A shows the number in each location at the beginning of December. Matrix B shows the number sold in December. If he expects to sell 20% of his remaining stock in January, what matrix shows how much he will have left at the beginning of February?

7. Find the product AB for

8. Two salesmen travel in 2 different cars. Salesman 1 uses a regular civic which has an average cost of 4 cents per mile in the city and 3 cents per mile in the country. Salesman 2 uses a civic hybrid which has an average cost of 2 cents per mile in the city and 1.5 cents per mile in the country. They both average 150 miles in the city and 50 miles in the country per day. What matrix product shows the daily cost for each car.

9. A company produces 2 kinds of candy. Chocolate chunk uses 3 lb of butter, 1 lb of chocolate and 2 lbs of sugar per batch. Fudge uses 2 lbs of butter, 2 lbs of chocolate, and 1.5 lbs of sugar per batch. Butter costs $3 per lb, chocolate costs $2.50 per lb and sugar costs $1 per lb. They have an order for 5 batches of chocolate chunk and 6 batches of fudge. What matrix product gives the total cost of the order?

10. Set up the inequalities and the objective function but do not solve.

A couple is investing $100,000 in a money market fund, gold and a stock fund. The returns for the money market, gold and stocks are 5%, 6% and 10% respectively. They have been advised that the sum of the amounts in gold and the money market should be no more than 30% of the total. The amount in stocks should be at least three times the amount in gold. How much should be invested in each to maximize return?

11. Solve each system of equations or state there is no solution. If infinite, give 2 particular solutions.

a) x+2y+3z=10, x – y - 2z=15, 5x - y=35

b) x+y+z=1, 2x – 3y+2z=0, 4x – y+4z=5

c)x+2y – z=3, 2x – 3y+z=5, x – 5y+2z=2

12. Which set of equations in 11 can be solved using an inverse matrix? Set up this system as AX=B. Find AC for .

13. What row operations will perform the first pivot in the row reduction of

14. The economy of a country is based on two sectors, agriculture and oil. Production of a dollar’s worth of agriculture requires an input of $0.40 from agriculture and $0.35 from oil. Production of a dollar’s worth of oil requires an input of $0.20 from agriculture and $0.05 from oil. Find the output from each sector that is needed to satisfy a demand of $40

million for agriculture and $250 million for oil.

15. Use the graphical method of corners to find the maximum and minimum of 4x+5y subject to 3x+y<12, x+3y<12, x+y<5, 3x+2y>6, x>0, y>0.

16. A pet store is purchasing puppies and kittens. Each puppy requires 12 square feet of living space and each kitten requires 8 square feet of living space. They have available 120 square feet for these animals. Each puppy costs them $12 and each kitten costs them $10. They want to spend no more than $126 for animals.

Weekly care costs $3 for each animal. They want to spend no more than $36 per week. Revenues are $80 per puppy and $70 per kitten. How many of each should they purchase to maximize their revenue? Are there any leftover resources and if so, what and how much?

17. U is the universal set. A, B, and C are subsets of U. Describe with unions, intersections and complements:

a) {x in U: x is in at least one of A, B, and C}

b) {x in U: x is in all of A, B, and C}

c) {x in U: x is in A and B but not C}

d) {x in U: x is in A or B but x is not in C}

e) {x in U: x is not in A or B but x is in C}

18. To get a flu shot, anyone under 60 must have a special condition. Anyone else can get a flu shot. A is the set of people under 60. B is the set of people with a special condition.

a) Describe in words the set of people who can get a flu shot.

b) Describe the set in a) with unions, intersections and complements.

c) Describe with unions, intersections and complements those who cannot get a flu shot.

19. Some students were asked if they had ever taken a course in accounting, engineering or psychology. From a total of 250 students, it was found that

85 had taken a course in accounting only.

5 had taken at least one course in each area.

5 had taken accounting and engineering but not psychology.

25 had taken accounting and psychology.

150 had taken accounting or psychology but had not taken engineering.

20 had taken engineering and psychology.

15 had had taken none of the three.

How many had taken courses in:

a) exactly one of the three subjects?

b) exactly two of the three subjects?

20. Seven plays are nominated for three different prizes. How many ways can the prizes be awarded if:

a) no play gets more than one prize?

b) any play can get any number of prizes?

c) a single play can get at most 2 prizes?

21. How many distinguishable arrangements of 5 red, 4 blue and 3 yellow balls are there?

Think of Mississippi.

22. How many ways can 1 1^st prize, 5 2^nd prizes, and 10 3^rd prizes be awarded among 70 people if no-one gets more than one prize?

23. Twelve athletes are to line up for a photo. There are 5 football players, 3 swimmers and 4 runners. How many ways can they line up if athletes of the same sport must be kept together?

24. A box contains 50 apples. 17 are honeycrisp, 20 are gala, and 13 are cameo. 5 apples are chosen at random. What is the probability of getting:

a) at least one honeycrisp?

b) at least 2 honeycrisp?

c) all of the same type?

d) exactly 2 honeycrisp or exactly 3 gala?

25. A 5 sided die is tossed 15 times. The random variable X is the number of times the die lands with either 1 or 2 on top probability that:

a) X is exactly 6?

b) X is at least 4 and at most 8.

c) Find the normal approximation to the probability in b.

26. A store survey asked 150 people to rate the sales service on a scale of 0=awful to 5=excellent. The results were rating 5 4 3 2 1 0

# of respondents 55 40 25 15 5 10

If X is a randomly selected rating find:

a) P(X>3) b) E(X) c) d) median of X e) mode of X.

27. P(E)=.7 P(F)=.8 P()=.06 a) Find P(). b) Are E and F independent?

28. Six people choose from 12 varieties of trees to plant. What is the probability :

a) at least 2 people will choose the same variety?

b) they will all choose the same variety?

c) exactly 3 people will choose the same variety and there are no other repeats?

29. A disease has a 15% rate of occurrence. A test for the disease is negative in 5% of people who have the disease and positive in 7% of people who do not have the disease. Let + be the event the test is positive and D the event the person has the disease. Find:

a) the test is positive b) the test is positive and the person has the disease c) a person who has the disease gets a positive test. d) a person who gets a positive test actually has the disease.

Review tree diagrams and be able to use one with 3 stages.

30. Use the answer to 29 d rounded to 4 decimal places to answer the following:

Ten people are randomly chosen from those who tested positive.

a) What is the probability at least 5 of them have the disease?

b) What is the expected number who have the disease?

31. A fictitious study tested to see if caffeine could improve memory. 100 people drank coffee and were asked to memorize a list. 100 people consumed no caffeine and memorized the same list. 110 people remembered at least 90% of the list. 75 of the

people who drank coffee remembered at least 90% of the list. Let C represent those who drank coffee and M those who remembered at least 90% of the list.

a) Are C and M independent? Why or why not?

b) Does the study show coffee helps memory? Why or why not?

c) What is the probability someone who did not drink coffee remembered at least 90% of the list?

32. a) The odds in favor of a certain team winning a game are 3:4. What is the probability this team will win?

b) The chance of rain is 40%. What are the odds in favor of rain?

33. Three security checkpoints 1, 2, and 3 have independent probabilities of letting an intruder pass through of .05, .06, and .04 respectively. What is the probability an intruder

a) will be detected by at least one checkpoint?

b) will be detected by checkpoints 1 or 2?

34. A city has 3 hospitals A, B, and C. 40% of new babies in the year were born at hospital A and had an average weight of 7.2 lbs. 35% were born at hospital B and had an average weight of 6.9 lbs. 25% were born at hospital C and had an average weight of 6.5 lbs. What is the average weight of babies born in the city hospitals that year?

35. A stock is watched for 14 days with the following results:

price/share $45 $47 $49 $50 $52 $53

# days 1 3 2 2 2 4

a) What is the expected price per share?

b) What is the standard deviation of the price per share?

36. A student takes 14 credits and receives the following grades.

Grade A B C D F

# credits 3 4 4 3 0

a) Find the students gpa for the 14 credits.

b) If he previously had a gpa of 3.0 for 70 credits, find the new gpa.

37. A set of exam grades is normally distributed with a mean of 75 and a standard deviation of 12.

a) Find the probability that a randomly selected score is better than 80. Round to 2 decimal places.

For b) and c) : Ten students are selected at random. X is the number who score better than 80. Then X~B(10,p) where p is the answer to a.

b)What is the probability at least three will score better than 80?

c) What is the expected number to score better than 80?

d) What is the probability a score is between 70 and 80 rounded to four decimal places?

e) Find a so the probability a randomly selected score is less than a is 0 .7.

f) Find b so the probability that a randomly selected score is greater than b is 0 .7.

40. The lengths of babies born at a certain hospital are normally distributed with a mean of 20 inches and a standard deviation of .75 inches. Twenty babies are selected at random and X is the number who are between 19 and 21 inches long.

a) What is the expected value of X?

b) What is the probability that X is at least 14?

c) Find P(15<X<18).

d) Find a length L so that 75% of babies are longer than L.

Key:

1.a) V(t)=24000-2500t b) $11,500

2 a) C(x)=30000+5x R(x)=8x P(x)=3x-30000 b) 10,000 units

3. a) y=.7x+500 b) y=-.5x+700 c) 167 (rounded) d) $616.90

4. 91.9 billion

5. a) b)

6. .8(A-B)=

7. AB=

8. =

9. or the transpose of this,

10. x+y+z<100000, .7x+.7y-.3z<0, -3x+z>0, x,y,z>0, Maximize R=.05x+.06y+.1z

11. a) x=10, y=15, z= -10 b) no solution c) x=19/7 + 1/7 t, y=1/7 +3/7 t, z=t, t any real number At t=1, x=20/7, y=4/7, z=1 At t=0, x=19/7, y=1/7, z=0

12. only a can be solved with an inverse.