This document discusses ratios, proportions, and similar figures. It begins by defining a ratio as a comparison of two quantities using a fraction. It then provides examples of writing and interpreting ratios and unit ratios. The document explains that a proportion is when two ratios are set equal, and proportions can be solved using cross-multiplication. Examples are given for solving various types of proportion problems. The document concludes by defining congruent angles and similar figures, and providing examples of using proportions to solve for missing lengths in similar figures.
Page 1 of 7 Pre‐calculus 12 Final Assignment (22 mark.docxbunyansaturnina
Page 1 of 7
Pre‐calculus 12
Final Assignment (22 marks)
Each question is worth 1 mark. You must show all your work to obtain full marks.
Marks will be deducted for no work shown.
1. What happens to the graph of 1 if the equation is changed to 1?
2. The graph of y = √ undergoes the transformation (x, y) ( 3,2 5)x y . What is the resulting
equation?
3. Determine the equation of the polynomial in factored form of the least degree that is symmetric
to the y‐axis, touches but does not go through the x‐axis at (3, 0), and has P(0) = 27
4. Determine the measure of all angles that satisfy the following conditions. Give exact answers.
csc =2 in the domain 2 2
Page 2 of 7
5. Solve: 3 cos ² 8 cos 4 0, over all real numbers
6. Use factoring to help to prove each identity for all permissible values of x. Must state
restrictions over all real numbers.
2sin sin tan
cos sin cos
x x x
x x x
Page 3 of 7
7. In a population of moths, 78 moths increase to 1000 moths in 40 weeks. What is the
doubling time for this population of moths?
8. Solve the following equation: log 3 log 5 2
9. Solve for x algebraically: 5 2 3 . State your answer to the nearest hundredth.
10. A radioactive substance has a half‐life of 92 hours. If 48g were present initially, how long will it
take for the substance to decay to 3g? Show algebraically.
Page 4 of 7
11. Given the following two functions √ 1 and 1, evaluate
3 .
12. A sample of 5 people is selected from 3 smokers and 12non‐smokers. In how many ways can
the 5 people be selected?
13. Given the functions 7 and √ , determine an explicit equation for
, then state its domain.
14. Determine the 4th term of 3 2 .
15. Solve by algebra √13 1 0
Page 5 of 7
16. Determine the domain, range, and intercepts of 2√4 2 3. Graph the function.
17. For the graph of , determine an non‐permissible values of , write the coordinates of
any hole and write the equation of any vertical asymptote.
Page 6 of 7
18. Sketch the graph of 3 4 5. State the domain, range, and equation of the
horizontal asymptote.
19. Suppose you play a game of cards in which only 5 cards are dealt from a standard 52 deck. How
many ways are there to obtain at least 3 cards of the same suit? An example of a hand that
contains at least 3 cards of the same suit is 4 hearts and 1 club.
20. Given , determine , the inverse of .
Page 7 of 7
21. Consider the digits 0, 2, 4, 5, 6, 8. How many 3‐digit even numbers less than 700 can be
formed if repetition of digits is not allowed? Note: the first digit cannot be zero.
22. If and 2 3, determine the value of 1 .
Lesson 3.4
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 20. .
Page 1 of 7 Pre‐calculus 12 Final Assignment (22 mark.docxbunyansaturnina
Page 1 of 7
Pre‐calculus 12
Final Assignment (22 marks)
Each question is worth 1 mark. You must show all your work to obtain full marks.
Marks will be deducted for no work shown.
1. What happens to the graph of 1 if the equation is changed to 1?
2. The graph of y = √ undergoes the transformation (x, y) ( 3,2 5)x y . What is the resulting
equation?
3. Determine the equation of the polynomial in factored form of the least degree that is symmetric
to the y‐axis, touches but does not go through the x‐axis at (3, 0), and has P(0) = 27
4. Determine the measure of all angles that satisfy the following conditions. Give exact answers.
csc =2 in the domain 2 2
Page 2 of 7
5. Solve: 3 cos ² 8 cos 4 0, over all real numbers
6. Use factoring to help to prove each identity for all permissible values of x. Must state
restrictions over all real numbers.
2sin sin tan
cos sin cos
x x x
x x x
Page 3 of 7
7. In a population of moths, 78 moths increase to 1000 moths in 40 weeks. What is the
doubling time for this population of moths?
8. Solve the following equation: log 3 log 5 2
9. Solve for x algebraically: 5 2 3 . State your answer to the nearest hundredth.
10. A radioactive substance has a half‐life of 92 hours. If 48g were present initially, how long will it
take for the substance to decay to 3g? Show algebraically.
Page 4 of 7
11. Given the following two functions √ 1 and 1, evaluate
3 .
12. A sample of 5 people is selected from 3 smokers and 12non‐smokers. In how many ways can
the 5 people be selected?
13. Given the functions 7 and √ , determine an explicit equation for
, then state its domain.
14. Determine the 4th term of 3 2 .
15. Solve by algebra √13 1 0
Page 5 of 7
16. Determine the domain, range, and intercepts of 2√4 2 3. Graph the function.
17. For the graph of , determine an non‐permissible values of , write the coordinates of
any hole and write the equation of any vertical asymptote.
Page 6 of 7
18. Sketch the graph of 3 4 5. State the domain, range, and equation of the
horizontal asymptote.
19. Suppose you play a game of cards in which only 5 cards are dealt from a standard 52 deck. How
many ways are there to obtain at least 3 cards of the same suit? An example of a hand that
contains at least 3 cards of the same suit is 4 hearts and 1 club.
20. Given , determine , the inverse of .
Page 7 of 7
21. Consider the digits 0, 2, 4, 5, 6, 8. How many 3‐digit even numbers less than 700 can be
formed if repetition of digits is not allowed? Note: the first digit cannot be zero.
22. If and 2 3, determine the value of 1 .
Lesson 3.4
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 20. .
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
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The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Biological screening of herbal drugs: Introduction and Need for
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Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
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How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
1. Ratios and Proportions
3.1
1. Solve problems involving ratios.
2. Solve for a missing number in a proportion.
3. Solve proportion problems.
4. Use proportions to solve for missing lengths in figures
that are similar.
You may use calculators in this chapter!!
2. Ratio: A comparison of two quantities
using a quotient (fraction).
The word to separates the numerator and
denominator quantities.
The ratio of 12 to 17 translates to
12
17
.
Numerator Denominator
Unit ratio: A ratio with a denominator of 1.
3. A bin at a hardware store contains 120 washers
and 85 bolts. Write the ratio of washers to bolts in
simplest form.
Ratios
The ratio of washers to bolts bolts
washers
85
120
17
24
Express the ratio as a unit ratio. Interpret the answer.
17
24
1
41
1.
There are 1.41 washers for every bolt.
4. The price of a 10.5 ounce can of soup is $1.68. Write
the unit ratio that expresses the price to weight.
Ratios
The ratio of price to weight
weight
price
10.5
1.68
1
16
.
Interpret the answer.
The soup costs $.16 per ounce.
5. One molecule of glucose contains 6 carbon atoms, 12
hydrogen atoms, and 6 oxygen atoms. What is the
ratio of hydrogen atoms to the total number of atoms
in the molecule?
Ratios
The ratio of hydrogen atoms to total atoms
atoms
total
atoms
hydrogen
24
12
2
1
6. Proportions
4
3
8
6
Proportion: two ratios set equal.
24
6
4
24
8
3
Cross-products of proportions are always equal!
Only works if there is an equal (=) sign!
4
3
8
6
No!
7. Solving Proportions
1. Calculate the cross products.
2. Set the cross products equal to each other.
3. Solve the equation.
8
5
3 x
24
8
3
x
x 5
5
x
5
24
5
5
5
24
x
8. Solving Proportions
x
18
20
12
x
x 12
12
360
18
20
360
12
x
12
12
30
x
1. Calculate the cross products.
2. Set the cross products equal.
3. Solve the equation.
10. Solving Proportions
3
7
6
5
x
5
3
x 42
42
5
3
x
3
3
9
x
1. Calculate the cross products.
2. Set the cross products equal.
3. Solve the equation.
42
15
3
x
15
15
27
3
x
11. Solving Proportions
Gary notices that his water bill was $24.80 for 600
cubic feet of water. At that rate, what would the
charges be for 940 cubic feet of water?
feet
cubic
dollars
600
80
24.
23312 x
600
x
600
23312
600
600
85
38.
$
x
940
x
12. Solving Proportions
Chevrolet estimates that its 2012 Tahoe will travel
520 miles on one tank of gas. If the tank of the
Tahoe holds 26 gallons, how far can a driver expect
to travel on 20 gallons?
gallon
miles
26
520
10400 x
26
x
26
10400
26
26
miles
400
x
20
x
13. Congruent angles: Angles that have the same measure.
The symbol for congruent is .
Similar figures: Figures with congruent angles and
proportional side lengths.
The two figures are similar. Find the missing length.
10
8
6
5
x
small
large
5
10
x
10 40
40
10
x
10
10
4
x
x
8
14. Similar Figures
The two figures are similar. Find the missing lengths.
Round your answer to the nearest hundredth.
small
large
5
6
8
12
.
.
4
134. x
.5
6
x
.
. 5
6
4
134
5
6
5
6 .
.
km
68
20.
x
5
10.
x
12.8 km
10.5 km
6.5 km 6.5 km
4.6 km
x
y
5
6
8
12
.
.
88
58. y
.5
6
y
.
. 5
6
88
58
5
6
5
6 .
.
km
06
9.
y
6
4.
y