Year End Review.notebook - Grade 12 College Math · ... SOHCAHTOA, Sine Law, Cosine Law • Applications: Word Problems. ... DECIDE WHETHER TO USE SOHCAHTOA, SINE LAW OR COSINE

Year End Review.notebook

1

June 10, 2013

Jan 128:41 AM

Exam Review

Units Covered:

• Trigonometry• Geometry & Measurement• Statistics: Twovariable Data• Graphical Models• Exponential Functions• Financial Math: Annuities & Mortgages

Jan 129:00 AM

Trigonemetry

Key Ideas:

• Pythagorean Theorem• Primary trig ratios: sine, cosine & tangent• Inverses: sin1, cos1, tan1• Obtuse angles • Solving Triangles: SOHCAHTOA, Sine Law, Cosine Law• Applications: Word Problems


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June 10, 2013

Jan 129:13 AM

1. Pythagorean Theorem:c2 = a2 + b2

• always "c"• opposite the right angle• the longest side

WARNING: When solving for one of the other two sides use:

a2 = c2 b2 b2 = c2 a2

They mean the same thing!

hypotenuse

Jan 1212:47 PM

Examples: Solve for the missing side. Round to one decimal place.

108

x

2.3

3.7

x

x2 = 102 82

a2 = c2 b2

x2 = 100 64

x2 = 36

x = √36x = 6

c2 = a2 + b2

x2 = 2.32 + 3.72

x2 = 5.29 + 13.69

x2 = 18.98

x = √19.98x ≈ 4.4


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June 10, 2013

Jan 1211:57 AM

2. Primary Trigonometric Raos: Sine, Cosine & Tangent

Example 1: Evaluate to 3 decimal places.

sin 650 = cos 1240 = tan 3410 =

angles

Example 2: Solve for the angle to the nearest degree.

sin R = 0.25 cos B = 0.92 tan Q = 1.54

Recall: When solving for an angle we must use the inverse functions!

R = sin1(0.25) B = cos1(0.92) R = tan1(1.54)

0.906 0.559 0.344

R = 14.50 B = 23.10 R = 57.00

Jan 125:00 PM

R = 570What about obtuse angle? That is, angles between 900 and 1800.

Sin A Cos A Tan AFor Acute angles (0° <A < 90°) + + +

For Obtuse angles (90° <A < 180°) + - -

Also, given an acute angle A, its supplementary obtuse angle is (1800 A) and ...

sin A = sin (1800 - A)

cos A = - cos (1800 - A)

tan A = - tan (1800 - A)

Let's look at some examples to remember how we apply this knowledge!

Example: Determine the measure of the obtuse angles.

sin B = 0.3cos P = 0.264 tan Q = 1.42

cos1(0.264) ≈ 105.30

∴ P ≈ 105.30

tan1(1.42) ≈ -54.80

∴ Q ≈ 1800 54.80

Q ≈ 125.20

sin1(0.3) ≈ 17.50

∴ B ≈ 1800 17.50

B ≈ 162.50


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June 10, 2013

Jan 1212:12 PM

When do we use these?

When we want to solve for an angle or a side in a right angle triangle.

A

BC

What is the first thing that we must do when solving a right angle triangle?

LABEL THE SIDES: OPPOSITE, ADJACENT, HYPOTENUSE

hypostenuse

adjacent

opposite

Jan 1212:17 PM

A

BC

Recall: SOHCAHTOA

Hence, the primary trig ratios for angle A below are:

hypostenuseadjacent

opposite

sin A =

opposite

hypotenusecos A =

adjacent

hypotenusetan A =

adjacent

opposite


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June 10, 2013

Jan 1212:45 PM

Example 1: Solve for side b.

420

12 cm

b

opposite

adjacenthypotenuse

sin 420 = opposite

hypotenuse

sin 420 = b

12

b = 12sin420 (12 x sin420)

b = 8.0 cm

Jan 124:23 PM

Example 2: Solve for side p.

670

9.7 cm

p

oppositeadjacent

hypotenuse

tan 670 = opposite

adjacent

tan 670 = 9.7

p

p = 9.7

tan 670(9.7 ÷ tan670)

p = 4.1 cm


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June 10, 2013

Jan 124:37 PM

Example 3: Solve for angle Q.

11 m

oppositeadjacent

hypotenuse

cos Q = hypotenuse

adjacent

cos Q = 11

17

Q = (cos1 (11÷17) )

Q = 49.70

Q

RS

17 m

11

17cos1 [ [

Jan 125:11 PM

Can we use the Pythagorean Theorem? NO

Can we use SOHCAHTOA? NO

Then ... what can we use?

• Sine Law • Cosine Law

3. Solving triangle that are NOT right angles


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June 10, 2013

Jan 125:34 PM

Before we begin let's recall a few things about triangles:

#1 Sum of the angle in a triangle:

A

B

C

A + B + C = 1800

#2 Labelling Conventions:

P

Q

R

p

q

r

• Angles are labelled with

• Sides are labelled with

upper case letters

lower case letters

• sides and angles correspond.Opposite

Jan 125:48 PM

Sine Law:

A

B

C

a

b

c

a = b = c sin A sin B sin C

OR

sin A = sin B = sin C a b c

Recall: Look for the OPPOSITE SIDEANGLE PAIR!

If you are given two angles, always start by finding the third angle (i.e. 1800 minus the other two)

i.e. Are you given A & a, B & b or C & c?


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June 10, 2013

Jan 126:02 PM

A

B

C

a

b

12

320

650

Example 1:

A = 1800 320 650 = 830

a = c sin A Sin C

Find the value of side a in the triangle below.

a = 12 sin 83 0 sin 65 0

12sin83 sin65 0

a = 0

a = 13.14

The size of opposite sideangle pairs should correspond (i.e. the biggest side should be across from the biggest angle).

Jan 126:17 PM

A

B

C

2.3

b

3.1

470

Example 2:

sin C = sin A c a

Find the value of angle C in the triangle below.

3.1sin47 2.3

sin C = 0

sin C = sin 47 3.1 2.3

0

3.1sin47 2.3

0C = sin-1 [ [

C = 80.30


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June 10, 2013

Jan 126:26 PM

Cosine Law:

A

B

C

a

b

c

a2 = b2 + c2 - 2bc cosA .OR

cosA = a2 + b2 - c2

2bc

Jan 126:36 PM

Example 1: Find the value of side b in the triangle below.

A

B

C

1411

230

b

b2 = 142 + 112 - 2(14)(11)cos230

b2 = 142 + 112 - 2(14)(11)cos230

b2 = a2 + c2 - 2ac cosB .

b = √(142 + 112 - 2(14)(11)cos230)

b ≈ 5.8


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June 10, 2013

Jan 126:53 PM

Example 2: Find the value of angle A in the triangle below.

A

B

C

26

15

19

cosA = a2 + b2 - c2

2bc

cosA = 192 + 152 - 262

2(19)(15)

A = cos-1 192 + 152 - 262

2(19)(15)[ [

A = cos-1 -90

570[ [

A = 99.10

Jan 127:03 PM

4. Applicaons

Example:

DRAW A PICTURE!

Two planes took off from Pearson International Airport at the same time. The first plane heads due west at an angle of elevation of 25o and a speed of 168 Km/hr. The second plane heads due east at an angle of elevation of 32o and a speed of 156 Km/hr. How far apart are the planes after 3 hours?

Plane 1 Plane 2

Airport

d

504 468

1800 250 320 = 1230

SIMPLIFY YOUR PICTURE!

Plane 1 Plane 2

Airport

250 320

250 320

d

168 km/hr x 3 hr = 504 km 156 km/hr x 3 hr = 468 km

DECIDE WHETHER TO USE SOHCAHTOA, SINE LAW OR COSINE LAW!

SINE LAW d = 468 sin1230 sin25 0

468sin123 sin25 0

d = 0

d = 928.7 km

∴ the planes are 928.7 km apart after 3 hours.


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June 10, 2013

Jan 129:00 AM

Geometry and Measurement

Key Ideas:

• Unit Conversion: Metric and Imperial• Area of Composite Figures • Volume• Surface Area • Optimization (not on exam)

Jan 1311:05 AM

Prefix Valuetera 1012 or 1 000 000 000 000giga 109 or 1 000 000 000mega 106 or 1 000 000kilo 103 or 1 000hecto102 or 100deka101 or 10100 or 1deci10-1 or 0.1centi10-2 or 0.01milli 10-3 or 0.001micro 10-6 or 0.000 001nano 10-9 or 0.000 000 001pico 10-12 or 0.000 000 000 001

The Metric System

When you convert a larger unit to a smaller unit, you MULTIPLY the smaller number by the difference in the 10n exponents

∴ the number get will get BIGGER → move to the RIGHT.

When you convert a smaller unit to a larger unit, you DIVIDE the larger number by the difference in the 10n exponents.

∴ the number get will get SMALLER → move to the LEFT.


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June 10, 2013

Jan 1311:28 AM

4.5 L to mL 12 km to cm

27 000 mL to L 69 500 000 mg to kg

Examples:

big to small > move right

how many units? 3

left or right?

4500 mL

how many units?

left or right?

5

big to small > move right

1,200,000 cm

small to big > move left

how many units? 3

left or right?

27 L

how many units? 6

left or right?

69.5 Kg

small to big > move left

Convert the following to the given units.

Jan 1311:38 AM

Length Mass Volume1 ft = 12 inches 1 lb = 16 ounces 1 gal = 4 qt 1 yd = 3 ft 1 Ton = 2000 lb 1 qt = 2 pt1 mile = 5280 ft1 pt = 16 fluid oz

The Imperial System

When you convert a larger unit to a smaller unit, you MULTIPLY the number by the conversion factor. THE NUMBER SHOULD GET BIGGER!!!

When you convert a smaller unit to a larger unit, you DIVIDE the number by the conversion factor. THE NUMBER SHOULD GET SMALLER!!!


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June 10, 2013

Jan 1311:56 AM

3.2 ft to inches 0.75 lbs to ounces

2456 ft to yd 12.5 qt to gal

Examples: Convert the following to the given units.

conversion factor:

multiply or divide?

3.2 x 12 = 38.4"

12

multiply

conversion factor:

multiply or divide?

0.75 x 16 = 12 oz

16

multiply

conversion factor:

multiply or divide?

2456 ÷ 3 = 818.7 yd

3divide

conversion factor:

multiply or divide?

12.5 ÷ 4 = 3.2 gal

4divide

Jan 1312:16 PM

Converng between Metric and Imperial

LENGTH AREA VOLUME

1 in = 2.54 cm 1 in2 = 6.45 cm2 1 in3 = 16.39 cm3

1 ft = 0.3048 m 1 ft2 = 0.0929 m2 1 ft3 = 0.02832 m3 = 28.32 L

1 mile = 1.609 Km

Note: 1 m3 = 1000 L

CAPACITY MASS SPEED

1 fl oz = 29.6 mL 1 oz = 28.35 g 1 mph = 1.609 Km/h

1 US gal = 3.75 L 1 lb = 0.454 Kg


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June 10, 2013

Jan 1312:39 PM

When converting Imperial to Metric you …

When converting Metric to Imperial you …

MULTIPLY by the conversion factor.

DIVIDE by the conversion factor.

Remember to ask yourself: "should the number get bigger or smaller?"

How do you know? Look to your units!

• If you are converting from a bigger unit to a smaller unit the number should get BIGGER!

• If you are converting from a smaller unit to a bigger unit, the number should get SMALLER!

Jan 1312:45 PM

2.4" to cm 325 mL to ounces

5 Km to miles 2.8 Kg to lbs.

smaller or bigger?

2.4 x 2.54 = 6.096 cm

2.54

bigger smaller or bigger?

325 ÷ 29.6 = 11.0 oz

29.6

smaller

smaller or bigger?

5 ÷ 1.609 = 3.1 miles

1.609smaller

conversion factor:

smaller or bigger?

2.8 ÷ 0.454 = 2.2 lbs.

0.454bigger


conversion factor: conversion factor:

conversion factor:


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June 10, 2013

Jan 132:04 PM

Using conversions to compare priceYou are almost out of gas and desperately need to fill up. You come across two gasstations. The first one is selling their gas at a price of $0.93/L. The second is selling theirgas at $2/gal. Which gas station should you choose?

Conversion factor: 1 US gal = 3.75 L

$2/gal = $2/3.75L = $0.53/L

REMEMER: JUST SUBSTITUTE AND DIVIDE!

∴ The second gas station has the better price.

Only convert one!

Jan 131:04 PM

Recall:

• If you SQUARE your units, you must SQUARE your conversion factor!

• If you CUBE your units, you must CUBE your conversion factor!

i.e. 1 ft = 12" 1 ft2 = 144 in2

i.e. 1 m = 100 cm 1 m3 = 1,000,000 cm3

conversion factor: conversion factor:


71 Km2 to m2 3.7 yd3 to ft3

1 Km = 1000 m

1 Km2 = 1,000,000 m2

Bigger or Smaller? Bigger

71 x 1,000,000 = 71,000,000 m2

1 yd = 3 ft

1 yd3 = 27 ft3

Bigger or Smaller? Bigger

3.7 x 27 = 99.9 ft3


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June 10, 2013

Jan 132:38 PM

Composite Figures

You obtain a summer job mowing lawns. A person with an odd shaped lot shown below hires you to cut his lawn. Given the dimension on the diagram, what area of lawn will you be cutting? State your answer in m2.

AREA OF THE RECTANGLE: ARectangle = l x w = 8' x 5' = 40 ft2

AREA OF THE TRIANGLE: ATriangle = b x h = 4' x 3' = 6 ft2 2 2

AREA OF THE LAWN: ALawn = 40 ft2 6 ft2 = 34 ft2

CONVERSION: 1 ft = 0.3048 m 1 ft2 = 0.0929 m2 34 ft2 = 3.2 m2

Jan 138:01 PM

Volume: The amount of 3dimensional space occupied by an object.

Volume of a prism: Area of Base x Height

See formula page at the back of your practice work. These formula's will be provided on the exam.

Example: Find the volume of the following cylinder with a diameter of 4 cm and a height of 6 cm.

V = area of base x height

V = πr2 x hV = π(2cm)2 x (6 cm)V = 75.4 cm3

Recall: For volume the units should always be cubed.


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June 10, 2013

Jan 138:18 PM

Surface Area: In general, surface area is the sum of all of the areas of the shapes that cover the surface of a 3D object.

Example: Find the surface area for the cylinder in the previous example. Recall: diameter = 4 cm and height = 6 cm.

SA = 2(area of base/circle) + area of the rectangle

SA = 2πr2 + (Circumferance)(height of the prism)

SA = 2πr2 + 2πrh

SA = 2π(2)2 + 2π(2)(6)SA = 100.5 cm2

Recall: The units for surface area should always be squared.

Jun 108:39 AM

Maximizing Area:

Remember: MAKE A SQUARE!

Example: Tony wants to enclose a rock garden in his back yard he has $720 for materials. If the edging he wants costs $22.50/ft, what is the maximum area that he can enclose?


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June 10, 2013

Jan 138:43 PM

Statistics:Two-variable DataKey Ideas:

1. Interpreting Scatter Plots• Independent vs. Dependent variables• Correlation• Line of Best fit• Outliers• Interpolation/Extrapolation

2. Statistical Literacy & Data Analysis• Quartiles & percentiles• Population vs. Sample• Analyzing bias in surveys

Jan 172:35 PM

Part 1: Interpreting Scatter Plots

Scatter plots represent twovariable data as points. Scatter plots may reveal a relationship between the two variables.

In twovariable situations, one variable may be dependent on another; its value changes according to the value of the independent variable.

For example, the value of a car depends on its age.

Typically, we plot the independent variable on the horizontal axis, and the dependent variable on the vertical axis.


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June 10, 2013

Jan 172:40 PM

A correlation is a relationship between two variables.Graphing twovariable data on a scatter plot may showa correlation between the variables.

Types of correlations

A positive correlation describes a situation in which both variables increase together.

A negative correlation describes a situation in which one variable decreases as the other variable increases.

On a scatter plot, the points go up to the right.

On a scatter plot, the points go down to the right.

Jan 172:43 PM

Line of Best FitA line of best fit is a line drawn through data points to best represent a linear relationship between two variables.

Drawing the line of best fit involves more than just finding a pathway through the middle of the data. The line of best fit is the line that is closest to each point. The more varied the position of points, the more difficult it is to draw the line of best fit.

OutliersIn a scatter plot, a point that lies far away from the main cluster ofpoints is an outlier. An outlier may be caused by inaccuratemeasurements, or it may be an unusual, but still valid, result.

Outliers and the Line of Best FitThe line of best fit should reflect ALL valid points from a data set. Thisincludes outliers!

If most of the plotted points are clustered along a linear path, evena single outlier can affect the path of the line of best fit.


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June 10, 2013

Jan 172:49 PM

Example: Exploring the Affect of Outliers and the Line of Best Fit

Which line is the line of best fit? Justify your choice.

SolutionThe line in Graph C seems most likely to be the line of best fit. It passes just below the middle of the main cluster of data. Its path is affected by the outliers, but it is affected more by the main cluster of data. It is the most reasonable choice.

Jan 172:51 PM

Interpolating and Extrapolating from a Line of Best Fit

A line of best fit can be used to estimate or predict values.

• Estimating values that lie among the known values on the graph isinterpolation.

• Predicting values that lie beyond the known values is extrapolation.


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June 10, 2013

Jan 172:55 PM

Example: Using a Line of Best Fit to Make Predictions

These are the preexam term marks and exam marks for some students in a Grade 12 math course.

a) Graph the data and draw the line of best fit.b) Determine the equation of the line of best fit.c) Use the data to predict the exam mark of a student with a preexamterm mark of 98%.d) Use the data to predict the exam mark of a student with a preexamterm mark of 10%.Example 2Term mark (%) 84 76 70 95 92 61 25 55 51 73 62Exam mark (%) 80 72 68 96 90 58 29 60 53 77 67

Jan 172:55 PM

y = mx + b b = 8

m = rise run

= 7043 = 27 = 9 = 0.9 7040 30 10

y = 0.9x + 8

b)

a)

c) y = 0.9(98) + 8 = 96.2%

d) y = 0.9(10) + 8 = 17%


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June 10, 2013

Jan 173:05 PM

Part 2: Statistical Literacy & Data Analysis

Television, radio, newspapers, and Web sites often report statistical data. To understand these reports, you need to be familiar with the statistical language they use.

Percentiles:A percentile tells approximately what percent of the data are less than a particular data value. Percentiles are a good way to rank data when you have a lot of data or want to keep data private.

Percent vs. Percentile:

• If you achieved a 72% on a math test it means that you got 72% of the questions correct.

• If you are in the 72nd percentile on a math test it means that you got a better grade than 71% of the class.

Jan 173:16 PM

QuartilesA quartile is any of three numbers that separate a sorted data set intofour equal parts.

The second quartile is the median. It cuts the data set in half.So, it is the same as the 50th percentile.

The first, or lowest, quartile is the median of the data values less thanthe second quartile. It separates the lowest 25% of the data set.So, it is the same as the 25th percentile.

The third, or upper, quartile is the median of the data values greaterthan the second quartile. It separates the highest 25% of the data set.So, it is the same as the 75th percentile.

Step 1: Line up the data from smallest to biggestStep 2: Count to make sure you haven't missed any.Step 3: Calculate the 2 nd quartile.


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June 10, 2013

Jan 173:19 PM

Example: Calculating Quartiles & PercentilesHere are the hourly pay rates, in dollars, for 17 highschool students with parttime jobs.

a) What are the quartiles for this data set?

7.50 7.75 8.00 8.00 8.25 9.00 9.15 9.25 9.45 9.50 9.75 10.20 10.75 11.25 11.50 12.50 13.00

2nd = 9.451st = (8.00 + 8.25) ÷2 = 8.125 3rd = (10.75 + 11.25) ÷2 = 11.00

b) Damien’s pay is in the 85th percentile for this group. What is Damien’s hourly pay rate?

The 85th percentile means approximately 85% of the students in the group earn less money per hour than Damien.

∴ Damien is the 15th student in the ordered list. He earns $11.50 per hour.

17 × 0.85 = 14.45

Round down to the nearest whole number to determine the number of students who earn less money per hour than Damien: 14

17 = the number of pieces of data

Jan 173:32 PM

Data ReliabilityWhen you read statistical data, you need to think about the reliability of the source. Data from a government agency are usually more reliable than data from someone who is trying to sell a product or promote a point of view.

When assessing the validity of survey consider the following 3 factors:

• Sample size; population size• The method of selecting respondents• The survey questions


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June 10, 2013

Jan 173:44 PM

Population vs. Sample

The ______________ of any set is all the objects in the set.

Collecting data about every individual in a population is called a _______________.

Conducting a census can be costly and time consuming. It may even be physically impossible.

Usually, data are collected for a smaller set of individuals/items selected from the population. This is called a _____________.

population

census

sample

Jan 173:58 PM

There are 3 key points we must consider when analyzing the validity of a survey:

Some sampling techniques are random, which means each member ofthe population has the same chance of being selected. A nonrandomtechnique may not yield a representative sample.

Sample Size

Representative Samples

Sample size can affect survey results. If the sample is too small, thesurvey results may not be reliable. If it is too great, the survey may becostly and difficult to administer.

A sample needs to be typical of the entire population. This is called arepresentative sample. If the sample is not representative, it is __________and the survey results are invalid.

Sampling Technique

biased

Random techniques Nonrandom techniques• Simple random sampling • Convenience sampling• Stratified sampling • Judgement sampling• Cluster sampling • Voluntary sampling• Systematic sampling


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June 10, 2013

Jan 174:00 PM

Other important factors which may affect the validity of a survey ...

Another factor to consider is how the survey is conducted.This is particularly important if any of the questions are about sensitivesubjects. People may be more likely to answer honestly if they can replyanonymously in writing rather than responding to an interviewer inperson or over the phone.

Biased questions restrict people’s choices unnecessarily or use wordsthat could influence people to answer in a certain way.For results to be valid, survey questions must be unbiased.

Biased Questions

Survey Techniques

Jan 174:05 PM

To summarize, to assess the validity of a survey, ask yourself these questions:

• Is the sample size large enough?

• Is the sample representative?

• Are the survey questions unbiased?

• Was the collection method appropriate?


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June 10, 2013

Jan 174:09 PM

Graphical Models

Key Ideas:

1. Comparing linear, quadratic & exponential relations

• first and second differences• ratios

2. Interpreting Graphs

• Linear Equations calculate slope• Quadratic Equations The parabola max/min.

Jan 174:29 PM

Part 1:Linear vs. Quadratic vs. Exponential

Linear Quadratic Exponential

graph line parabola exponential curve

equation y = mx + b y = ax2 + bx + c y = ax

table of values first differences 2nd differences ratios


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June 10, 2013

Jan 174:47 PM

Examples: For each model, determine if the relationship is linear, quadratic, exponential, or none, and give reasons for your answer.

x y 1st 2nd ratio

5 26

4 17 9

3 10 7 2

2 5 5 2

1 2 3 2

x y 1st 2nd ratio

2 16

3 64 48 4

4 256 192 144 4

5 1024 768 576 4

6 4096 3072 2304 4

x y 1st 2nd ratio

5 8

4 6 2 0.75

3 1 5 3 0.17

2 6 5 10 6

1 8 2 3 1.33

x y 1st 2nd ratio

3 23

2 17 6

1 11 6

0 5 6

1 1 6

Jan 175:15 PM

Part 2: Interpreting Graphs

Linear Relations Quadratic Relations Exponential Relations


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June 10, 2013

Jan 175:23 PM

Linear RelationsThe first thing we usually do when we interpret a line is to determine the rate of change a.k.a THE SLOPE. This tells us the rate at which our dependent variable is changing in relation to our independent variable.

• How far a man travels per min.• How much an amusment park's profit increases with each additional 10 customers.• How much fuel is used per 100 km (i.e. a vehicles fuel consumption).

Examples:

If you are looking to extrapolate (predict) information from a line it is helpful to determine the equation of the line; that is, y = mx+ b.

Jan 175:33 PM

Examples:

Step 1: Pick two EASY points.

a.) m = rise = 600 = 60 = 60 run 10 1

b.) m = rise = 01000 = 1000 = 200 run 50 5

The slope represents the cars speed.

∴ the car is travelling at 60 km/h.

The slope represents the value lost on a computer per year.

∴ the computer is losing $250 worth of it's value per year.


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June 10, 2013

Jan 183:20 PM

Example:

The graduation committee is making arrangements for the prom. This graph shows the total cost of the prom based on the number of people who attend it.

a) What is the yintercept? What does it represent?

b) Calculate the rate of change (the slope) in the cost with respect to the number of people. What does this rate of change represent?

c) What will the total cost be if 60 people attend the Prom?

d) The committee receives a donation of $5000 for the prom. How many people can attend for free?

500; the cost if no one attends prom it could be the cost of the deposit for the location, the supplies, ect.

slope = rise = 3500 500 = 3000 = 15 run 2000 200

y = mx + by = 15x + 500

It represents the cost per person; that is, it will cost $15 per person.

y = 15(60) + 500 = 900 + 500 = 1400

∴ it will cost $1400 if 60 people attend.

y = 15x + 500 5000 = 15x + 500

5000 500 = 15x

4500 = 15x 15 15

x = 300

∴ with the $5000 donation, 300 students can attend prom for free.

Jan 175:43 PM

Quadratic Relations

The graph of a quadratic relation is called a parabola. The parabola has some important features:

Everything you ever wanted to know about parabolas…

Ø Parabolas can open up or down

Ø The zero of a parabola is where the graph crosses the x axis

Ø The axis of symmetry divides the parabola into two equal halves

Ø The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value.

Ø The optimal value (max/min) is the value of the y coordinate of the vertex


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June 10, 2013

Jan 175:44 PM

Examples:

a.) Vertex = (150, 22500)

This tells us that a company will make their maximum profit of $22500 if they sell 150 tickets.

b.) Vertex = (2, 22)

This tells us that the ball will reach its maximum height of 22 m, 2 seconds afterthe ball is thrown.

Jan 129:00 AM

Exponential Functions

Key Ideas:

• Exponent Rules• Rational exponents i.e. x1/3 = ∛x• Using rational exponents to solve equations i.e. x3 = 64• Solving exponential equations i.e. 2(4y2) = 128• Rearranging formulas: i.e. Solve for r in A=πr2


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June 10, 2013

Jan 176:16 PM

Exponent RulesPlease refer to the handout "Exponent Rules" for a summary of these rules as well as some examples.

NOTE: These will not be provided on the exam!

Jan 176:24 PM

Question

x4(y3)2(z0)3

(x2)4(y2)3(z2)5x12y0z10= =

x4y6z0

x8y6z10= z10

x12


32

June 10, 2013

Jan 187:21 PM

Rational Exponents

Examples:

1) 49½ = √49 = 7

2) (64)1/3 = ∛(-64) = 4

3) 324/5 = (√32)4 = 165

4) 0.043/2 = (√0.04) 3 = 0.008

Jan 187:24 PM

Using Rational Exponents to Solve Equations

Rational exponents are useful for solving equations involving powers.For example, take both sides of the equation x3 = 125 to the power 1/3 tofind the solution x = 5.

Examples:

1) x4 = 16

(x4)1/4 = 161/4

x = ∜16 = 2

2) x3/2 = 27

(x3/2)2/3 = 272/3

x = (∛27)2 = 9


33

June 10, 2013

Jan 187:43 PM

Solving Exponential Equations

An exponential equation is an equation that contains a variable in the exponent. Some examples of exponential equations are:

2x = 32 9x+1 = 27x (0.8)x = 0.18

Some exponential equations can be solved by writing both sides of theequation as powers of the same base. This allows us to use the following property.

For example, since 4x and 43 are both powers of 4, the solution to 4x = 43 is x = 3.

Jan 187:50 PM

Examples:

1) 5x = 56

x = 6

2) 2x = 32

x = 52x = 25

3) 2(73x4) = 98

73x4 = 72

3x4 = 2

3x = 2 + 4

3x = 6 3 3

x = 2

2(73x4 ) = 982 2

73x4 = 49

4) 35x+8 = 273x

35x+8 = 39x

5x+8 = 9x

8 = 9x 5x

8 = 4x4 4

x = 2

5) 8x5 = 43x1

23(x5) = 22(3x1)

3(x5) = 2(3x1)

3x15 = 6x2

3x 6x = 2 +15

3x = 13

x = 13/3


34

June 10, 2013

Jan 188:17 PM

Rearranging Formulas

Using the formula for the volume of a cylinder, V = πr2h, substute the values V=16964.6 m3 and r=15 m, and solve for h.

Example:

Use inverse operations and balance strategies!

Solution:

16964.6 = π(15)2h

16964.6 = π(225)h 225π 225π

h = 16964.6 ÷ 225 ÷ π ≈24 m

Jan 188:36 PM

Financial Mathematics:Annuities & Mortgages

Key Ideas:

• Ordinary Simple Annuites (A)• Solving for the payment amount (R)• Present Value (PV)• Mortgages: Amortization Tables


35

June 10, 2013

Jan 188:40 PM

Ordinary Simple Annuities

An annuity is a series of equal payments made at regular intervals.In an ordinary simple annuity , payments are made at the end of each compounding period. The amount of an annuity is the sum of the regular deposits plus interest.

The amount formula can only be used when:• The payment interval is the same as the compounding period.• A payment is made at the end of each compounding period.• The first payment is made at the end of the first compounding period.

Jan 188:58 PM

# of compounding periods per yeari = rate per annum ÷

n = # of compounding periods per year x # of years

NOTE:


36

June 10, 2013

Jan 188:53 PM

Example 1: Using the Amount Formula$450 is deposited at the end of each quarter for 1.5 years at 10% per year compounded quarterly. What is the amount of the annuity? How much interest does the annuity earn?

R = 450

i = 0.1 ÷ 4 = 0.025

Solution:

n = 1.5 x 4 = 6

A = R[(1+i)n1]

iA = 450[(1+0.025)61]

0.025

A = 2874.48

∴ the amount of the annuity is $2874.48.

Total Deposits: 450 x 6 = 2700

Interest = 2874.48 2700 = 174.48 ∴ the annuity earned $174.48 in interest.

Jan 188:52 PM

Example 2: Solving for the Payment Amount

Tyler would like to have $5000 saved in 2 years so she can afford a trip to Ireland. He will make equal deposits every month, into an account that pays 6% annual interest compounded monthly. What would be the value of Tyler’s monthly deposits?

Solution:A = 5000

i = 0.06÷12 = 0.005n = 2 x 12 = 24

R = ? We must rearrange the formula!

A = R[(1+i)n1]

iR = Ai

[(1+i)n1]

NOT PROVIDED ON THE EXAM!

R = 5000(0.005)

[(1+0.005)241]= $196.60∴

∴ Tyler will have to make monthly deposits of $196.60 in order to have $5000 saved in two years.


37

June 10, 2013

Jan 209:00 PM

Present Value of an AnnuityThe present value of an annuity is the principal that must be invested today to provide the regular payments of an annuity.

The present value formula can only be used when:• The payment interval is the same as the compounding period.• A payment is made at the end of each compounding period.• The first payment is made at the end of the first compounding period.

Jan 209:32 PM

Example 1: Using the Present Value FormulaVictor wants to withdraw $700 at the end of each month for 3 years, starting 1 month from now. His bank account earns 5.4% per year compounded monthly. How much must Victor deposit in his bank account today to pay for the withdrawals?

Solution:

R = 700

i = 0.054 ÷ 12 = 0.0045

n = 3 x 12 = 36

PV = R[1 (1+i)n]

i

PV = 700[1(1+0.0045)36]

0.0045

PV = 23216.62

∴ Victor will have to deposit $23,216.62 now in order to withdraw $700 at the end of every month for 3 years.


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June 10, 2013

Jan 209:46 PM

Example 1: Solving for the withdrawal/payment amount.

Sarah just bought a new car for $23,900. She has saved up a $2500 down payment but must finance the rest. The dealership is offering a 2.4% financing rate, compounded monthly for 5 years. Calculate Sarah's monthly payments?

Solution:

First we must decide how much money Sarah needs to finance/borrow?

$23,900 $2,500 = $21,400 The total cost less the down payment

PV = 21400

i = 0.024 ÷ 12 = 0.02

n = 5 x 12 = 60

R = PV(i)

[1 (1+i)n]

R = 615.63

∴ Sarah's monthly paymentswill be $615.63 for 5 years.

This is our PRESENT VALUE.

R = 21400(0.02)

[1 (1+0.02)60]

Jan 2010:05 PM

How do we decide whether to use the Simple Annuity formula or the Present Value formula?

Simple Annuity: THE BALANCE IS GETTING BIGGER!

Present Value: THE BALANCE IS GETTING SMALLER!

i.e. • Investments• Putting money into RRSPs

i.e. • Loans• Taking money out of RRSPs


39

June 10, 2013

Jan 2010:11 PM

Amortizing a Mortgage

A loan to finance the purchase of real estate, usually with specified payment periods and interest rates.

What is a Mortgage?

We can use an amortization table to analyse how a mortgage is repaid. The amortization table gives a detailed breakdown of the interest and principal paid by each payment and the loan balance after the payment.

What is an Amortization Table?

What is the amortization period?

The amortization period is the length of time it takes to pay off your mortgage. Note: the shorter the amortization period the less interest we will pay!

Jan 2010:21 PM

Let's take a look at an amortization table.

What was the principal amount borrowed on this mortgage? $210,000

How much are the monthly payments? $1221.62

What is the amortization period? 300 ÷ 12 = 25 years

Explain what is meant by the interest paid vs. the principal paid with each payment. What happens to this as me moves on in your mortgage?

With every payment you make, part of the money goes to pay back the Principal (i.e. the original amount that you borrowed) and the rest goes to pay the interest. Unfortunately, at the beginning of your mortgage the majority of your payment amount is going to interest; however, as time moves on this ratio changes and more and more of your payment will go towards actually paying back the principal.

Documents

Year End Review.notebook - Grade 12 College Math · ... SOHCAHTOA, Sine Law, Cosine Law • Applications: Word Problems. ... DECIDE WHETHER TO USE SOHCAHTOA, SINE LAW OR COSINE