# Mix problems

Problem #1

The starting salaries for teachers in Broward County for selected years are given in the table. Use this

information to answer the questions below. Show work for all calculations and give clear concise

explanations to all questions.

School Year Starting Salary

1995

2000

2005

*2014

*Projected

$27,910

$33,427

$37,000

$41,400

a) Find and interpret the average rate of change of starting salaries for teachers in Broward County

between 1995 and 2014. Please write as a decimal rounded to the nearest penny.

b) Assume starting salaries in the future increase at the same rate, use the years 1995 and 2014 to find a

linear model (equation) that can be used to predict future starting salaries (let x = 0 in year 1995). Use

your answer from part a for the slope and use function notation with s(x) representing starting salary.

c) Use your model from part b to predict starting salaries for teachers in Broward County for the year

2025 (let x = 0 in year 1995). Please write as a decimal rounded to the nearest penny.

d) Use your model from part b and algebra to predict the year in which starting salaries for teachers in

Broward County will reach $50,000 (let x = 0 in year 1995).

Problem #2

Justin started a lawn maintenance business. His basic rate to maintain an average size lawn is $30. It cost

Justin $1,250 to start the business (trailer, equipment, etc.) and he estimates that his cost per lawn is $1.25

(gas, oil, blades, etc.). Use this information to answer the questions below.

a) Write the cost function in terms of c(x) and the revenue function in terms of r(x) for Justin's business.

b) Set up an equation and use algebra to find the number of lawns that Justin has to maintain for his

business to break-even (cost equal to revenue)?

c) The profit function p(x) can be found by subtracting the cost function from the revenue function

p(x) = r(x) - c(x). Write the profit function in simplest form for Justin's business in terms of p(x).

d) Use the profit function from part c to evaluate and interpret p(500).

e) Set up an equation and use algebra to find when Justin's business will make a profit of $25,000?

f) Sketch the cost and revenue functions on the same graph.

Problem #3

Use the shortcut as demonstrated in class to perform the following conversion between number bases.

Please show all work and use the proper notation in your final answer.

A4B3C2D1sixteen to Binary

Problem #4

First choose a five digit number without repeating digits and a three digit number without repeating digits

(ex. 15,294 and 205). Then in the space below, show how the lattice method would be used to multiply

these numbers. Please show all work as if you were teaching someone how to use this method for the

first time (clearly identify the answer). Also, briefly explain why this method was discussed in this class.

Problem #5

First choose a two digit number greater than 32 and not equal to a power of 2 and a three digit number

greater than 100 and not equal to a power of two (ex. 43 and 105). Then in the space below, demonstrate

how the ancient Egyptian's would multiply these numbers. Please show all work as if you were teaching

someone how to use this method for the first time (clearly identify the answer).

Problem #6

First choose and write a three digit Babylonian number (use Babylonian numeral) without repeated digits.

Then in the space below, show step by step how to convert the Babylonian number to Hindu-Arabic.

Then convert the Hindu-Arabic to Mayan numerals (make sure to use appropriate spacing for the

Babylonian and Mayan numeral as demonstrated in class).

Problem #7

A local sandwich store has a fixed weekly cost of $525 and a variable cost of $.55 per sandwich. Use this

information to answer the questions below.

a) Write the cost function in terms of c(x) for this sandwich store.

b) If the weekly revenue function is given by the function ( )

, write the stores

weekly profit function in simplest form and in terms of p(x). hint: p(x) = r(x) - c(x)

c) Use the profit function p(x) from part b to find the number of sandwiches the store should make and

sell each week to maximize profits. Also, what is the maximum weekly profit in dollars to the nearest

penny?

d) Find and interpret p(1000), p(1200), p(1300), and p(1500). Why is it not better for this sandwich

shop to sell more sandwiches?

Problem #8

The risk of having a car accident increases exponentially as the concentration of alcohol in the blood

increases. The risk can be modeled by

where x is the blood alcohol concentration and R,

given as a percentage, is the risk of having a car accident.

a) In the state of Florida, the legal blood alcohol level (BAL) is 0.08. At this level, what is the risk of a

car accident? Give the answer rounded to two decimal places.

b) At what blood alcohol level will the risk of having a car accident increase to 25%? Give the answer

rounded to three decimal places.

Your blood alcohol level can be approximated by the formula (

) where D =

number of drinks consumed with alcohol content equivalent to one 12 oz beer, W is the weight of the

person, H = number of hours since last drink, and r = .79 for men and r = .71 for women.

c) Based on your gender and your weight, would you over the legal limit immediately after having 4

drinks (H=0)? Give the answer rounded to three decimal places. If you are over the limit, how many

hours would you have to wait before you can legally drive?

d) Based on your BAL immediately after 4 drinks (H=0), if you did drive, what is the risk of you having

a car accident? Give the answer rounded to two decimal places.

Problem #9

Xanax is a tranquilizer used in the short-term relief of symptoms of anxiety. Its half-life in the

bloodstream is 12 hours.

a) Find an exponential decay function that models the decay of Xanax in the bloodstream. Round the

decay constant k to six decimal places.

b) Use the model from part a to find what percentage of the original dose of Xanax is present in the

bloodstream after 18 hours. Round the percent to the nearest hundredth.

c) Use the model from part a to find out how long it will take for Xanax to decay to 15% of its original

dosage. Round to the nearest hour.

Problem #10

A ball is propelled upward from ground level. After t seconds, its height in feet is defined by the equation:

h(t) = height in feet and t = time in seconds

a) What is the maximum height that the ball will reach and how long will it take to reach maximum

height (please include the appropriate units of measure)?

b) Find h(5). Interpret these results. Is the ball on the way up or on the way down at this time?

c) Find h(t) = 112. Interpret these results. Is the ball on the way up or on the way down at this time?

d) How many seconds will it take for the ball to hit the ground? Use algebra to justify your answer.

https://brainmass.com/math/basic-algebra/561060

#### Solution Preview

1)

Year (x) Starting Salary(y)

1995 (0) $27,910

2000 (5) $33,427

2005 (10) $37,000

2014 (19) $41,400

a)

mean:

year:

X = (0+5+10+19)/4 = 8.5

Starting salaray:

Y = (27910+33427+37000+41400)/4 = $34,934.25

Linear relation:

y = s(x) = Bo + B1.x

Rate of change,

B1 = sum((xi-X)(yi-Y))/sum((xi-X)^2)

=> B1 = sum((0-8.5)*(27910-34934.25) + (5-8.5)*(33427-34934.25) + (10-8.5)*(37000-34934.25) + (19-8.5)*(41400-34934.25))/sum((0-8.5)^2+(5-8.5)^2+(10-8.5)^2+(19-8.5)^2)

=> B1 = $690.21

Hence, rate of change of starting salary = $690.21/year --Answer

b)

s(x) = Bo + B1.x

Bo = Y - B1.X = 34934.25 - 690.21*8.5 = 29067.50

As 1995 == 0

Hence, starting salary (y) as function of year (x) equation:

s(x) = 29067.50 + 690.21(year - 1995) --Answer

c)

year = 2025, starting salary

s(x) = 29067.50 + 690.21(2025 - 1995) = $49,773.80 --Answer

d)

satrting salary = $50,000

s(x) = 50000 = 29067.50 + 690.21(year - 1995)

=> year = 1995 + (50000-29067.50)/690.21 = 2025.33 --Answer

2)

Fixed cost, fc = $1,250

Variable cost, vc = $1.25/lawn

Basic rate, br = $30/lawn

a)

Cost funtion in terms of number of lawns, x:

c(x) = 1250 + 1.25x --Answer

Revenue function,

r(x) = 30x

b)

For break-even point,

c(x) = r(x)

1.25x + 1250 = 30x

=> 28.75x = 1250

=> x = 1250/28.75 = 43.48 --Answer

c)

Profit function,

p(x) = r(x) - c(x)

=> p(x) = 30x - (1250 + 1.25x)

=> p(x) = 28.75x - 1250

d)

p(500) = 28.75*500 - 1250 = ...

#### Solution Summary

A few problems of finance and number systems, from linear algebra are solved.

Statistics: 23 problem exam

1. Distinguish between descriptive statistics and inferential statistics.

2. Distinguish between and independent and a dependent variable - give example.

3. The annual incomes of the five vice presidents of Erlen industries are

75000 78000 72000 83000 90000

a. What is the range?

b. What is the arithmetic mean income?

c. What is the population variance? The standard deviation?

4. The ages of a sample of Canadian tourists flying to Hong Kong were

32 21 60 47 54 17 72 55 33 41

What is the standard deviation of the sample?

5. A report by the Department of Justice on rape-victims reports on interviews with 3721 victims. The attacks were classified by the age of the victim and the relationship of the victim to the rapists. The results of the study are given in the table below.

Relationship of Rapist

Age of Victim Family Acquaintance or Friend Stranger

under 12 153 167 13

12 to 17 230 746 172

over 17 269 1232 739

a. What is the probability that a victim was under 12 years of age?

b. What is the probability that a victim was between 12 and 17 and that the rapist was a member of the family?

c. What is the probability that a victim was under 12 or that the rapist was an acquaintance or a friend?

d. What is the probability the victim was not under 12 years of age?

e. What is the probability the rapist was not a family member, acquaintance, or friend?

6. The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds.

a. What is the area between 415 pounds and the mean of 400 pounds?

b. What is the area between the mean and 395 pounds?

c. What is the probability of selecting a value at random and discovering it has a value of less than 395 pounds?

7. The mean score of a college entrance test is 500; the standard deviation is 75. The scores are normally distributed.

a. What percent of the students scored below 320?

b. Twenty percent of the students had a test score above what score?

c. Ten percent of the students had a test score below what score?

8. Ms. Maria Wilson is considering running for mayor of the town of Bono, Ohio. Before completing the petitions, she decides to conduct a survey of voters in Bono. A sample of 400 voters revealed that 300 would support her in the November election.

a. What proportion of the voters in Bono do you estimate would support Ms. Wilson?

b. Develop a 99 percent confidence interval for the proportion of voters in the population that would support Ms. Wilson.

9. Past surveys revealed that 30 percent of the tourists going to Atlantic City to gamble during a weekend spent more than $1,000. Management wants to update that percentage.

a. Using the .90 degree of confidence, management wants to estimate the percentage of the tourists spending more than $1,000 within 1 percent. What sample size should be employed?

b. Management said that the sample size suggested in part a is much too large. Suggest something that could be done to reduce the sample size. Based on your suggestion, recalculate the sample size.

10. A new industrial oven has just been installed at the Piatt Bakery. To develop experience regarding the oven temperature, an inspector reads the temperature at four different places inside the oven each half hour. The first reading taken at 8:00 AM was 340 F. (Only the last two digits are given in the following table).

Reading

Time 1 2 3 4

8:00 AM 40 50 55 39

8:30 AM 44 42 38 38

9:00 AM 41 45 47 43

9:30 AM 39 39 41 41

10:00 AM 37 42 46 41

10:30 AM 39 40 39 40

Based upon this initial experience, determine the control limits for the mean temperature.

Determine the grand mean. Plot the experience on a QC chart.

11. Seiko purchases watch stems in lots of 10,000. Seiko's sampling plan calls for checking 20 items, and if 3 or fewer are defective, the lot is accepted. Based upon their sampling plan, what is the probability that a lot of 10 percent defective will be accepted?

12. The Board of Realtors of a small city reports that 80% of the houses that are sold have been on the market for more than 6 months. The Board takes a random sample of 15 homes that have recently been sold and counts the numbers that were on the market for more than 6 months. What is the Probability that of 15 homes in the sample:

a. less than 12 have been on the market for more than 6 months?

b. between 8 and 13 have been on the market for more than 6 months?

c. at least 10 homes have been on the market for more than 6 months?

d. at most 4 have been on the market for more than 6 months?

13. Hugger Polls contends that an agent conducts 53 in-depth home surveys every week. A streamlined survey form has been introduced and Hugger wants to evaluate its effectiveness. The number of in-depth surveys conducted during a week by a random sample of agents is:

53 57 50 55 58 54 60

52 59 62 60 60 51 59 56

At the .05 level of significance, what do you conclude about the number of in-depth surveys completed during a week using the new form?

14. The scores of two groups of inmates at Southard Prison on a rehabilitation test are:

First offenders Repeat offenders

Mean score 300 305

Standard variance 20 18

Sample size 16 13

Test at the .05 level that there is no difference between the mean scores of the two groups.

15. Samples of efficiency ratings of employees at Allied Chemicals in plant number 1 and plant number 2 are:

Plant no.1 Plant no.2

160 163

158 161

162 160

161 162

160 163

160 162

161 164

159 163

159 165

160 162

159

160

At the .02 level test is there a difference in the mean(s) of the employees.

16. Coppersfield, a nationwide advertising firm, wants to know if the size of an advertisement and the color of the advertisement make a difference in the response of magazine readers. A random sample of readers are shown ads of four different colors and three different sizes. Each reader is asked to give the particular combination of size and color a rating between 1 and 10. The rating for each combination is shown in the following table (for example, the rating for a small, red ad is 2).

Color of ad

Size of ad Red Blue Orange Green

Small 2 3 3 8

Medium 3 5 6 7

Large 6 7 8 8

Is there a difference in the effectiveness of an advertisement by color and by size?

17. Sabin Motorcycle Works plans to develop a brochure for its new revolutionary X2B cycle. One of the facets to be explored and reported on is the speed-mileage question: Is there a linear relationship between the cycle's speed and miles per gallon? Tests on their track revealed the following:

Constant speed (miles per hour) Miles per gallon

X Y

40 54

30 60

70 37

50 46

60 48

Compute the coefficient of correlation, and evaluate its strength.

18. The University of Winston has five scholarships available for the women's basketball team. The coach provided two scouts with the names of 10 high school players with potential. Each scout attended at least three of their games and then ranked the players with respect to potential.

Rank by scout

Player Jean Cann John Cannelli

Cora Jean Seiple 7 5

Bette Jones 2 4

Jeannie Black 10 10

Norma Tidwell 1 3

Kathy Marchal 6 6

Candy Jenkins 3 1

Rita Rosinski 5 7

Anita Lockes 4 2

Brenda Towne 8 9

Denise Ober 9 8

a. Determine Spearman's rank correlation coefficient.

b. Evaluate the benefit of having the two scouts rank the player's potential.

19. Using the selected prices per 100 pounds for hogs and cattle (beef):

Hogs Cattle Beef

$47.10 $57.30

15.3 20.4

41.8 66.1

49.3 52.6

44 53.7

22.7 27.1

46.1 37.2

38 62.4

44 53.7

46 55.5

Let cattle prices be the dependent variable.

a. Compute the regression equation.

b. The estimated hog price this year is $45.00 per pounds. What is the predicted cattle price?

20. A national study with conducted with respect to the major leisure indoor activity of males. The percent of the total for each activity is shown in the center column of the following table. The results of a similar study of a sample of males older than 60 living in the Rocky Mountain area are given in the right column.

National Rocky Mountain

results study

Major indoor activity % 0f total (Number)

Photography 22 337

Stamp & Coin Colecting 19 293

Needlework, crocheting, sewing 6 82

Greenhouse & indoor gardening 9 128

Metalworking and wood 12 182

Gourmet cooking 4 54

Painting & sculpture 7 99

Chess, checkers, & others 21 325

Test at the .05 level that there is no difference between the national results and those of males older than 60 in the Rocky Mountain area.

21. A mortgage department in a large bank is studying it's recent loans. Of particular interest is how such factors as the value of the home, educational level of the head of household, age of the head of household, current monthly mortgage payment, and se (male = 1, female =0) relate to family income. Are these variables predictors of household income?

Value Years Mortgage Sex

Income in ,000 Educ Age Payment

40,300 190 14 53 230 1

49,600 121 15 49 370 1

40,800 161 14 44 397 1

40,300 161 14 39 181 1

40,000 179 14 53 378 0

38,100 99 14 46 304 0

41,400 114 15 42 285 1

40,700 202 14 49 551 0

40,800 184 13 37 370 0

37,100 90 14 43 135 0

39,900 181 14 48 332 1

41,400 143 15 54 217 1

38,000 132 14 44 490 0

39,000 127 14 37 220 0

39,500 153 14 50 270 1

40,600 145 14 50 279 1

41,300 174 15 52 329 1

41,100 177 15 47 274 0

42,700 188 15 49 433 1

40,100 153 15 53 333 1

45,600 150 16 58 148 0

40,400 173 13 42 390 1

40,900 163 14 46 142 1

40,100 150 15 50 343 0

39,500 139 14 45 373 0

a. Determine the regression equation.

b. What is the value of R2. Comment on the value.

c. What variables could be drop.

d. If the value is 111; the years of education 14; age is 51, mortgage payment 383; and the sex is a male, what is the predicated Income?

22. Forecast sales for the next 4 quarters (2004). [Use deseasonalization]

Year Quarter Sales

1997 1 210

2 180

3 60

4 246

1998 1 214

2 216

3 82

4 230

1999 1 246

2 228

3 91

4 280

2000 1 258

2 250

3 113

4 298

2001 1 279

2 267

3 116

4 304

2002 1 302

2 290

3 114

4 310

2003 1 321

2 291

3 120

4 320

23. Using Age, make a Frequency Distribution, a Histogram, and a Box Pot of the data. Compute the Mean, Median, Mode, Standard deviation, Quartile 1 & 3 of age.

57 51 30 41 61 34 61 38 29 43

57 28 49 50 20 63 32 42 37 42

49 36 52 57 64 21 22 36 49 42

28 36 24 32 22 57 31 58 22 44

40 28 26 18 60 25 26 52 27 28

48 55 57 27 34 43 42 31 35 56

43 43 32 24 35 27 28 47 32 37

27 41 59 44 26 36 43 33 54 33

62 53 56 19 21 35 32 31 60 29

25 46 25 48 26 42 23 33 54 42

See attached file.

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