Mathematics: Preparing for College and Apprenticeship 12

 

Chapter 4  Single-Variable Statistics

 

4.1  Collecting Data: Sampling Techniques

 

Question 10, page 162

 

http://www.statcan.gc.ca/
On the English page of the Statistics Canada web site, choose The Daily in the top menu and then The Daily Archives in the side menu. Find the issue for October 12, 2000, and read the report on The Waste Management Industry, 1998. Or you can go directly to Search and type in part of the sentence you are looking for.

 

Page 167, Question 9

 

http://www.statcan.gc.ca/
On the English page of the Statistics Canada web site, choose The Daily in the top menu and then Previous Issues in the side menu. Find the issue for October 12, 2000, and read the report on The Waste Management Industry, 1998. Or you can go directly to Search and type in part of the sentence you are looking for.

 

4.5  Measures of Dispersion

 

Page 189, Question 8

 

http://www40.statcan.ca/z01/cs0002_e.htm
Use "Internet" as your search term to find data on households with at least one regular Internet user. Some of the tables are available broken down by province - use the menu on the left to select the specific data you're looking for. Take some time to explore these tables - there's a lot of interesting information here.

 

Chapter 5   Two-Variable Statistics

 

5.1 Discover E-Stat

 

Page 208, Question 1

 

Use the first word on page 269 of your student book as both password and User ID to access E-STAT.

 

5.2   Methods for Collecting, Storing, and Retrieving Data
 
Page 219, Question 16

 

http://www.ibc.ca/
On the English home page, select Home & Auto Insurance, then Auto Insurance, for some very clear explanations of what you need to know about automobile insurance. Try Insurance Concepts for explanations of terms, and Buying Car Insurance to learn what facts are taken into consideration when your premiums are established.

 

5.3  Scatter Plots
 
Page 228, Question 9
The Boston Marathon

 

http://www.bostonmarathon.org/
The inspiration for modern marathons comes from a famous story from ancient Greece, in which a soldier ran from the Plains of Marathon to Athens to announce an astounding victory over the Persian army. The annual Boston Marathon has been held since the late nineteenth century. Choose Boston Marathon, then History in the side panel, and then Records, for runners' winning times.

 

 

 

Chapter 6  Interpreting and Analysing Data

 

Section 6.1  Retrieving Information

 

Page 267, Question 6

 

www.movieweb.com/movie/alltime.html
MovieWeb lists more than 50 of the most popular movies, with a summary/review, names of the actors, dates, rating, and Quicktime trailers for each.

 

Page 269, Question 9
http://www.msc-smc.ec.gc.ca/

On the English page under Topics, choose Canadian Climate And Water Information and Data Site, and then Canadian Climate Normals. You can search for precipitation and other climate information by province or city.

 

 

Page 270, Question 10

 

http://www.tctrail.ca/
Get all the background, learn how the trail is being built, and pick up some trail trivia at this official TransCanada Trail site. Select Trail Locator in the top menu to find descriptions of trail segments in the various provinces.

 

Page 288, Example 3

http://www.stats.org/
Intriguing titles and revelations of faulty logic make these webzine articles fun and interesting to read. Check out the recent articles in the News column on the main page. If you prefer to browse by subject, follow the links on the side panel to such subjects as "gender issues" and "media factoids."

 

Section 6.4  Validity and Bias

 

Page 291, Question 11

 

http://www.injuryresearch.bc.ca/
Click on Publications in the menu on the left, and then select the Ice Hockey Related Injuries report, which is based on data collected in 2002.

 

 

Cumulative Review Chapters 4-6

 

Page 301, Question 6 
Old Faithful data

 

http://www.stat.sc.edu/~west/javahtml/Histogram.html
This histogram applet is designed to teach students how bin widths (or the number of bins) affect a histogram. The histogram shown is for the Old Faithful data set. The observations are the duration (in minutes) for eruptions of the Old Faithful geyser in Yellowstone National Park. You can interactively change the bin width by dragging the arrow underneath the bin width scale.

 

http://www.yellowstone.net/geysers/geyser11.htm
Old Faithful was named by the first official expedition to Yellowstone in 1870. The visitors were impressed by its size and frequency. Old Faithful has been erupting in nearly the same fashion throughout the recorded history of the park. Read about other facts, and some misconceptions about Old Faithful's eruptions, at this site, and watch a video of an eruption.

 

 

This applet allows you to vary the launch speed and angle of a projectile, and then observe the resulting trajectory, as described in McGraw-Hill Ryerson Mathematics 12 Preparing for College & Apprenticeship, Section 7.4, Page 334.

 

 

Chapter 7  Exploring Quadratic Models

 

There are no web links in Chapter 7

 

Chapter 8  Interpreting Graphical and Algebraic Models

 

Section 8.2  Evaluating Formulas for Any Variable

 

Page 373, Question 16
Gemmology

 

www.pbs.org/wgbh/nova/diamond
Diamonds in the heavens? Diamonds slowing the speed of light? This companion web site to the Nova film "The Diamond Deception" discusses the optical effects of diamonds, the crystal structure of diamonds, and the making of synthetic diamonds.

 

 

 

 

 

 

 

 

Bachelor of Science in Applied Mathematics
Applied mathematics students focus on mathematical models as applied to physical systems, such as nuclear reactors, fluid dynamics, theoretical physics, computational methods, and industrial processes. Graduates find careers in government, industry, education, or business administration.

 

Bachelor of Science in Mathematics
Pure mathematics students focus on algebra, the mathematics of abstract systems, analysis, the mathematics of change and topology, the mathematics of transformation. Graduates find careers in government, industry, education, research, or business administration.

 

Bachelor of Mathematics
Students may choose to major in one or two of actuarial science, applied mathematics, bioinformatics, business administration, combinatorics and optimization, computational mathematics, computer science, accountancy, teaching, operations research, pure mathematics, software engineering, or statistics. Graduates find careers in insurance, research, business administration, quality control, software development, accountancy, government, and industry.

 

Bachelor of Engineering Math
Engineering mathematics students focus on applications of mathematics to engineering. Their program includes both mathematics courses and engineering electives. Graduates find careers in industry, research, government, education, and business administration.

 

Bachelor of Operations Research
Students in this program focus on the application of statistics to business and industrial systems including mathematical simulation of industrial systems, quality control, and production analysis. Graduates find careers mainly in industry.

 

Disclaimer: This page is intended as a general guide only. Universities vary in their requirements, and sometimes prerequisites are changed. Check the specific web site for your desired university for detailed and updated information.

 

Accounting: Calculus & Advanced Functions or Geometry & Discrete Mathematics

 

Applied Science: One Grade 12 mathematics course

 

Architecture: See Science

 

Aviation Management: Calculus & Advanced Functions, one other Grade 12U mathematics course

 

Commercial Studies: Calculus & Advanced Functions, one other Grade 12U mathematics course

 

Computer Science: Geometry & Discrete Mathematics, one other Grade 12 mathematics course

 

Dentistry: See Science

 

Earth Science: One Grade 12U mathematics course

 

Engineering: Calculus & Advanced Functions, Geometry & Discrete Mathematics

 

Environmental Science: One Grade 12U mathematics course

 

Health Science: Calculus & Advanced Functions or Geometry & Discrete Mathematics

 

Human Ecology: Functions and Relations Grade 11U

 

Information Technology: One Grade 12U or 12M mathematics course

 

Justice Studies: One Grade 12U or 12M mathematics course

 

Kinesiology: Calculus & Advanced Functions

 

Landscape Architecture: One Grade 12 mathematics course

 

Mathematics: Calculus & Advanced Functions, Geometry & Discrete Mathematics

 

Music Administration: Calculus & Advanced Functions

 

Nuclear Science: Calculus & Advanced Functions

 

Nursing: Functions and Relations Grade 11U or Functions Grade 11U/C

 

Oenology & Viticulture: One Grade 12U mathematics course

 

Optometry: See Science

 

Science: Calculus & Advanced Functions, one other Grade 12U mathematics course

 

Veterinary Medicine: See Science

 

Software Design Engineer
A software design engineer typically holds a bachelor's degree with a heavy emphasis on computer science and mathematics. The daily job may include designing software for specific tasks, testing software that has already been written, and computer-controlled automation of industrial or commercial processes.

 

Aerospace Scientist
A mathematician working in the aerospace industry typically holds a master's degree or adoctorate. Samples of tasks include constructing mathematical simulations of aerodynamic and orbital conditions of space flight with a view to predicting problems before they occur, optimization of spaceborne systems, and the logistics of integrating a wide variety of mechanical and electronic systems such that control can be maintained by a limited flight crew.

 

Biostatistician
A biostatistician typically holds a bachelor's or a master's degree in mathematics with supporting courses in biology, primarily genetics. Fields of endeavour include integrating a variety of statistics collected from crop and animal farming for the purpose of combing genetic traits that will boost yield, improve resistance to blights and disease, and decrease the need for chemical pesticides and fertilizers. The pharmaceutical industry employs biostatisticians to investigate the effectiveness of new drugs, and determine the probability of harmful side effects.

 

Trademark Lawyer
A trademark lawyer holds a law degree, with a strong background in mathematics. The logic of mathematics is necessary when building up a case for trademark protection, or prosecuting a defendant for alleged trademark infringement. Statistics are used to show the level of harm produced by the infringement.

 

Electronic Commerce Specialist
An electronic commerce specialist holds a bachelor's degree with a strong background in mathematics, specifically combinatorics and optimization, and systems analysis, as well as computer science. A typical job focus is managing systems of electronic data interchange for a large corporation, or among corporations. It is important to transfer business information and data in a timely and accurate manner while taking into account that a variety of standard file formats may be in use.

 

Commercial Marine and Fishing Analyst
A researcher working in the marine and fishing industry holds a bachelor's degree or an advanced degree with an emphasis on ecology, biology, statistics, dynamic systems, and linear algebra. The focus is on managing the marine environment and fishing practices to create a sustainable harvest of fish and shellfish. The recent collapse of several of the world's major fisheries underscores the importance of this line of research.

 

Laser Optics Design
A mathematician working in this industry typically holds an advanced degree, with emphasis on vector analysis and optimization. Tasks include working out optimal mathematics models for computer control of laser-guided targeting and ranging applications for both civil and military clients. Civil applications include autopilot systems for commercial aircraft, which can guide the aircraft from gate to gate in zero visibility. Military systems include terrain-following missiles, as well as self-guided targeting for various weapon systems.

 

Digital Camera Design
A mathematician working in this field typically holds an advanced degree with an emphasis on creating mathematical models and systems design. A strong background in computer science is also needed. Modern digital cameras require complex algorithms to process the raw data from the image sensor electronics into a photograph that accurately reflects the luminance and chrominance of the original subject. As new technologies are developed, it is important to develop algorithms that exploit the strengths of the new technology in an optimal fashion.

 

Computer Chip Mask Design
A mathematician working in this field holds an advance degree with an emphasis on network optimization as well as knowledge of solid-state physics. Modern computer chips are etched, layer-by-layer, on silicon wafers using a laser or an electron beam. Each layer is represented by a mask, which guides the beam, such that the desired circuit pathways are etched into the silicon. Mathematics is used to ensure that no undesired connections (short circuits) occur, and that the maximum density of components is achieved for optimal miniaturization.

 

Actuary
An actuary holds a bachelor's degree or an advanced degree in mathematics, and has passed the required actuarial exams. The most common application is insurance. Insurance companies employ actuaries to predict the claim payouts to be expected for different kinds of insurance so that realistic fee schedules can be drawn up. Other subjects useful to a mathematician working in this field include psychology, sociology, economics, and political science.

 

Add and then subtract the polynomials 2x + 3y and 3x - 5y.

 

Solution:

 

To add the polynomials, add like terms.

 

 

To subtract the polynomials, subtract like terms.

 

 

 

Find the measure of   in the triangle shown.

 

 

Solution:

 

The interior angles of a triangle always add up to 180o. To find , subtract the measures of the other two angles from 180o.

 

 

 

 

Find the degree of the polynomial  .

 

Solution:

 

The degree of a polynomial is equal to the degree of the term with the greatest degree in the polynomial. To find the degree of each term, add the exponents of all of the variables in that term. In the given polynomial, the degree of the first term is 3. The degree of the second term is 2 + 2 = 4. The degree of the third term is 1. Therefore, the degree of the polynomial is 4.

 

Draw a diagram to represent each of the following situations.

 

a) In order to enhance stability, the wing of an aircraft, which is 10.2 m long, is angled upward so that the outer end is 0.5 m higher than the root end. (This is called dihedral).

 

b) A grain hopper built in the shape of an inverted cone with a slant height of 5 m and a radius of 3 m.

 

Solution:

 

a) The wing forms the hypotenuse of a triangle, as shown in the diagram.

 

 

b) The slant height is the hypotenuse of a right-angled triangle, as shown in the diagram.

 

 

If x = 2, y = -1, and z = 3, evaluate the expression  .

 

Solution:

 

Substitute 2 for x, -1 for y, and 3 for z. Then, simplify using order of operations (BEDMAS) rules.

 

 

Graph the exponential function

.

Solution:

 

Make a table of values for selected values of x. For example, if
x = 1, then

 

 

X	Y
-2	62.5
-1	12.5
0	2.5
1	0.5
2	0.1

 

 

 

Factor the polynomial

by finding the greatest common factor.

 

 

Solution:

 

First, inspect the coefficients. There is a greatest common factor of 12 among the three coefficients. Next, inspect the variable x. The highest power of x common to all three terms is x1. Finally, inspect the variable y. The highest power of y common to all three terms is y2. Therefore, the greatest common factor is 12xy2. Factor this from each term to obtain the other factor.

 

 

 

Find all of the factors of 120.

 

Solution:

 

First, find all of the prime factors of 120. The first few prime numbers are:

 

2, 3, 5, 7, 11,...

 

120 is divisible by 2, with a quotient of 60.

 

60 is divisible by 2, with a quotient of 30.

 

30 is divisible by 2, with a quotient of 15

 

15 is divisible by 3, with a quotient of 5.

 

5 is a prime number.

 

Therefore, the prime factors of 120 are 2, 2, 2, 3, and 5.

 

The other factors of 120 are determined by taking subsets of these prime factors.

 

2 x  2 = 4

 

2 x 2 x 2 = 8

 

2 x 3 = 6

 

2 x  2   3 = 12

 

2 x  2 x  2 x 3 = 24

 

2 x 5 = 10

 

2 x  2   5 = 20

 

2 x  2 x 2 x  5 = 40

 

3 x  5 = 15

 

2 x  3 x  5 = 30

 

2 x  2 x  3   5 = 60

 

Therefore, the factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.

 

Note: Be sure to include 1 and 120 as factors.

 

Find the first differences for each of the following relations.

 

Solution:

 

a) To find the first differences, subtract successive y-values for equal steps of x-values. For example, the first entry in the first difference column is obtained by calculating  . Note: Since the first differences are constant for this relation, it is a linear relation.

 

X	Y
-3	-7
-2	-5	2
-1	-3	2
0	-1	2
1	1	2
2	3	2
3	5	2

 

b) The first entry in the first difference column is obtained by calculating  .
Note: Since the first differences are not constant for this relation, it is a non-linear relation.

 

X	Y
-3	9
-2	4	-5
-1	1	-3
0	0	-1
1	1	1
2	4	3
3	9	5

 

 

 

Draw a top view and a side view for a pyramid with a square base.

 

Solution:

 

The perspective drawing of a pyramid is shown. From the top, you would see the outline of the square base. From the side, you would see the triangle.

 

 

Students at a school were surveyed as to how many litres of soft drinks they consumed, on average, per week. The results are shown in the table.

 

Soft Drinks (L)	Frequency
0	25
1	33
2	47
3	54
4	51
5	42
6	31
7	18
8	9
9	7
10	3

a) How many students were surveyed?

 

b) Find the total number of litres of soft drinks consumed, per week, by those surveyed.
c) Make a bar graph of the results.

 

Solution:

 

a) To find the total number of students polled, add the entries in the frequency column to obtain 320.

 

Soft Drinks (L)	Frequency	Total
0	25	0
1	33	33
2	47	94
3	54	162
4	51	204
5	42	210
6	31	186
7	18	126
8	9	72
9	7	63
10	3	30

 

 

b) To find the total number of litres of soft drinks consumed, add a column to your table that multiplies the first column entries by the second column entries. Then add up the entries in the third column to obtain 1180 L.

 

c) To make a bar graph of the results, place the numbers in the first column along the horizontal axis, and the numbers in the second column along the vertical axis.

 

 

Plot the points (6, -2), (-4, -2) and (1, 5) on a coordinate grid. Join the points with line segments. What is the apparent relation among these points?

 

Solution

 

The points appear to form an isosceles triangle.

 

 

A private pilot kept a log of her flying hours for one year. The results are shown in the graph.

 

 

a) Describe the trend shown in the graph.

 

b) Between which two months did flying time increase the most?

 

c) Between which two months did flying time decrease the most?

 

d) What was the average flying time per month?

 

Solution:

 

a) Flying time increases as the months get warmer, and decreases as the months get colder.

 

b) The increase was greatest from March to April.

 

c) The decrease was greatest from October to November.

d) To find the average, add up the flying times and divide by 12 to obtain 8.25 h.

 

 

 

Graph   using the slope and the y-intercept.

 

Solution:

 

Rearrange the equation to standard form.

 

 

 

The slope is   and the y-intercept is -1. Therefore, a run of 3 results in a rise of 2. Begin your graph at  . Run 3 units to point E, and then rise two units to point B. Draw your line through points A and B.

 

 

 

a) Convert 10 200 cm to kilometres.

 

b) Convert 2.4 L to millilitres.

 

c) Convert 2.52 hg to kilograms.

 

Solution:

 

a) There are 100 cm in a metre, and 1000 m in a kilometre. Therefore, there are  100 x 1000 = 100 000 cm in a kilometre. To change 10 200 cm to kilometres, divide by 100 000, or move the decimal point five places to the left, to obtain 0.102 km.

 

b) There are 1000 mL to a litre. Therefore, multiply by 1000, or move the decimal point three places to the right, to obtain 2400 mL.

 

c) There are 100 g in a hectogram, and 1000 g in a kilogram. Therefore, there are   hg in a kilogram. Therefore, divide by 10, or move the decimal point one place to the left, to obtain 0.252 kg.

 

Evaluate

 

Solution:

 

Follow the order of operations represented by BEDMAS(Brackets, Exponents, Multiplication, Division, Addition, Subtraction). Evaluate the expression in the first set of brackets, then evaluate the exponent. Perform any multiplication or division, working from left to right. Finally, perform addition and subtraction, working from left to right.

 

 

 

Find the next three numbers in this pattern:

 

0, 1, 3, 7, 15,...

 

Solution:

 

Powers of 2, starting with 2 o produce the pattern 1, 2, 4, 8, 16,... . The numbers in the desired pattern are one less than these. Therefore, the next three numbers are

 

31, 63, 127

 

1. a) Convert the fraction   to a decimal, and a percent.

 

b) Convert the decimal 0.12 to a percent, and a fraction.

 

c) Convert 32% to a decimal, and a fraction.

 

Solution:

 

a) To convert to a decimal, divide the numerator by the denominator.

 

 

To convert to a percent, multiply the decimal by 100. This can be done by moving the decimal point two places to the right.

 

0.375 = 37.5%

 

b) To convert to a percent, multiply the decimal by 100. This can be done by moving the decimal point two places to the right.

 

0.12 = 12%

 

To convert to a fraction, express the decimal over 10 for one decimal place, over 100 for two decimal places, over 1000 for three decimal places, and so on. In this case, use 100. Then reduce the resulting fraction, if possible.

 

 

c) To convert to a decimal, divide the percent by 100. This can be done by moving the decimal point two places to the left.

 

32% = 0.32

 

To convert to a fraction, express the decimal over 10 for one decimal place, over 100 for two decimal places, over 1000 for three decimal places, and so on. In this case, use 100. Then reduce the resulting fraction.

 

 

 

Perimeter and Area: Worked Example

 

A pie-shaped building lot for a new house has a radius of 50 m. Find the perimeter and area of the lot.

 

 

Solution:

 

The perimeter of the lot is given by the radius plus the radius plus one-quarter of the circumference.

 

 

The area of the lot is one-quarter of the area of the circle.

 

 

Bread machine mix purchased at a bulk food store must be mixed with water in the proportion 5 parts of mix to 2 parts of water. How much water is necessary for 1.5 L of mix?

 

Solution:

 

Let w be the volume of water needed.

 

 

Therefore, 0.6 L of water is needed.

 

 

 

Use the Pythagorean theorem to find the third side of the triangle shown.

 

 

Solution:

 

The Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides. Apply this to the given triangle to obtain the correct equation.

 

 

Then, solve the equation for x.

 

 

Therefore,

 

 

 

 

Graph the quadratic function   using the vertex and stretch factor. Find the y-intercept, and estimate the x-intercepts.

 

Solution:

 

Inspection of the function shows that the vertex is at (-1, 7). The stretch factor is 2, and the negative sign in front of the stretch factor indicates that the parabola opens down.

 

 

 

 

To find the y-intercept, set x = 0.

 

 

From the graph, the x-intercepts appear to be at  x= -2.9 and x= 0.9 .

 

 

 

Reduce the ratio 255:323.

 

Solution:

 

In order to reduce a ratio, you must find the greatest common factor. Use a calculator to help you. The greatest common factor for the given ratio is 17. Therefore, the reduced ratio is found by dividing each term in the given ratio by 17.

 

 

1. Convert to scientific notation: a) 5 330 000   b) 0.000 000 123.

 

2. Convert to standard form: a) 5.44  x  105   b) 8.21  x   10-3.

 

Solution:

 

1. a) To convert to scientific notation, move the decimal point to the left until only one digit, the 5, remains to the left of the decimal point. This requires six moves to the left. Therefore, the answer is 5.33  x   106.

 

Note: The zeros after the last 3 are usually assumed to be place-holders for the decimal point. If you know that a zero is the result of a measurement, then you must include it with the scientific notation.

 

b) Move the decimal point to the right until you have one digit, the 1, to the left of the decimal point. This requires seven moves to the right. Hence, the answer is 1.23  x   10-7.

 

2. a) The power of ten is positive. Hence, you must move the decimal point five places to the right. This give you the answer of 544 000.

 

b) The power of ten is negative. Hence, you must move the decimal point three places to the left. The answer is 0.008 21.

 

Expand and simplify:

.

 

Solution:

 

Expand the squared binomial first.

 

 

Then, multiply each term in the trinomial by each term in the remaining binomial, and collect like terms.

 

 

Solve for x:

 

Solution:

 

Clear the fractions by multiplying both sides by the lowest common denominator of 15. Alternatively, you can achieve the same result by using the shortcut known as "cross-multiplying". Then, use the distributive property to clear the brackets.

 

 

Subtract 9x from both sides and 5 from both sides to isolate the variable terms. Collect like terms and solve for x.

 

 

1. Determine   correct to four decimal places.

 

2. Determine the measure of an angle whose cosine is 0.3334, correct to one decimal place.

 

3. Evaluate  , correct to two decimal places.

 

Solution:

 

1. Use the sine function (sin) on your calculator.

 

 

2. Use the inverse cosine function (cos -1) on your calculator.

 

 

3. Use the tangent function (tan) on your calculator.