Integrated AlgebraRegents Review #2

GeometryRelative Error

Probability

GeometryTriangle

1 bhA = bh or A =

2 2

Rectangle

A = bh or A = lw

Paralellogram

A = bh

2

Square

A = s or A = bh

1 2

Trapezoid1

A = h(b + b )2

See reference table

2

Circle

A = πr

Formulas you need to know!

22

Semicircle

1 πrA = πr or A =

2 2

22

Quartercircle

1 πrA = πr or A =

4 4

πdC

GeometryFinding Perimeters

In the diagram, ABCD is an isosceles trapezoid. Its bases are AB and CD. BA is extended to E, and DE and EB are perpendicular. Side BC is a diameter of semicircle O, AB = 4, AE = 3, DE = 4, and DC = 10. Find the perimeter of the figure to the nearest tenth.

Perimeter = distance around the figure

Find the distance around the semicircle

a2 + b2 = c2

32 + 42 = c2

25 = c2

5 = cIf AD = 5 then BC = 5 2

5πC

2πd

C

Perimeter = 10 + 4 + 3 + 4 + 2

5π

Perimeter = 28.853… = 28.9 units

Isosceles TrapezoidAD = BC

GeometryFinding Areas

Find the area of the composite figure pictured below. Represent your answer in terms of pi.

Area of Triangle + Area of SemicircleA = ½ bhA = ½ (12)(14)A = 84

18πA2

36π2

π(6)A

2πr

A

2

2

2cm18π84 Area Remember: An answer left in terms of pi is accurate and exact. Rounding leads to an approximate result. Never round unless otherwise directed.

GeometryShaded Area

Mr. Petri has a rectangular plot of land with a length of 20 feet and a width of 10 feet. He wants to design a flower garden in the shape of a circle with two semicircles at each end of the center circle, as shown in the accompanying diagram. He will fill in the shaded area with wood chips. If one bag of wood chips covers 5 square feet, how many bags must he buy? Area of Rectangle – Area 2 Circles

A = lwA = (10)(20)A = 200

Diameter = 10Radius = 5

2

2

A = πr

A = π(5)A = 25π

50π2 circles2(25π) =

Area of Shaded Region200 – 42.92036…square feet

50π

42.920368.584...

5

Bags of Wood Chips Needed

Mr. Petri will need 9 bags of wood chips to cover the shaded area.

GeometrySurface Area and Volumea. Find the volume of the cylinder to

the nearest hundredth.

3in1809.56V

576πV(16)π(6)V

hπrV2

2

b. Find the surface area to the nearest hundredth of the cylinder if it represents a can which has no lid or bottom.

2in603.19SA

π

2

192SA(6)(16)2SA

SArh2r2rh2SA

V = lwh (not on the reference table)

SA = 2lh + 2hw + 2lw

Relative Error

The relative error expresses the "relative size of the error" of the measurement in relation to the measurement itself.

Any measurement made with a measuring device is approximate.

The error in measurement is a mathematical way to show the

uncertainty in the measurement.  It is the difference between the result of the measurement and the true value of what you were measuring.

ActualDifference

RE Always subtract smaller number from bigger number to create a positive difference.

Relative ErrorA student mistakenly measures the length of a radius to be 24 inches. The actual radius is 25 inches.a.Find the relative error.b.Find the percent of error. 25

125

2425Actual

DifferenceRE

= .04

Percent Error = .04 x 100 = 4%

The groundskeeper is replacing the turf on a football field. His measurements of the field are 130 yards by 60 yards. The actual measurements are 120 yards by 54 yards. What is the relative error, to the nearest ten thousandth, in calculating the area of the football field?

Actual AreaA = lwA = (120)(54)A = 6480

Measured AreaA = lwA = (130)(60)A = 7800

.0.203703..64801320

648064807800

ActualDifference

RE

The relative error is 0.2037

Probability

If there are a ways for one activity to occur, and b ways for a second activity to occur, then there are a • b ways for both to occur. Multiply the number of ways each activity can occur.

The Counting Principle

Examples:

1. Activities: roll a die and flip a coin

There are 6 ways to roll a die and two ways to flip a coin.

There are 6 • 2 = 12 ways to roll a die and flip a coin.

2. Activities: a coin is tossed five times

There are 2 ways to flip a coin when each coin is flipped.

There are 2 • 2 • 2 • 2 •2 = 32 arrangements of heads and tails.

ProbabilityThe Counting Principle

2) Your state issues license plates consisting of letters and numbers. There are 26 letters and the letters may be repeated. There are 10 digits and the digits may be not be repeated. How many possible license plates can be issued with two letters

followed by three numbers?

1) A movie theater sells 4 sizes of popcorn (small, medium, large and extra large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased?

12toppings

3sizes

4ways of ordering popcorn

Letters: 26 (with repeats) Digits: 0 – 9 (10 total without repeats)

486,720digit

8digit

9digit10

letter26

letter26

license plates

ProbabilityPermutations

Consider, four students walking toward their school entrance. How many different ways could they arrange themselves in this side-by-side pattern?

1,2,3,4 2,1,3,4 3,2,1,4 4,2,3,11,2,4,3 2,1,4,3 3,2,4,1 4,2,1,31,3,2,4 2,3,1,4 3,1,2,4 4,3,2,11,3,4,2 2,3,4,1 3,1,4,2 4,3,1,21,4,2,3 2,4,1,3 3,4,2,1 4,1,2,31,4,3,2 2,4,3,1 3,4,1,2 4,1,3,2

The number of different arrangements is 24 or 4! = 4 • 3 • 2 • 1. There are 24 different arrangements, or permutations, of the four students walking side-by-side.

The notation for a permutation: n Pr

n is the total number of objects r is the number of objects chosen 4 P4 = 4!

Consider the example above: There are 4 friends and all 4 friends are being arranged.

A permutation is an arrangement of objects in a specific order. The order of the arrangement is important!!

ProbabilityPermutations

Not all permutations are factorials!

Find the number of ways to arrange 7 books on a shelf.

7 P7 = 7! 7 • 6 • 5 • 4 • 3 • 2 • 1= 5040 ways

Find the number of ways to arrange 5 books on a shelf chosen from a set 7 books.

7 P5 = 7 • 6 • 5 • 4 • 3 = 2520 ways

Calculator Corner:To compute factorials (!)…Example: 7!1)Enter number (7)2)Press Math3)Scroll to the right to PRB4)Press #4 (!)5)Enter

To compute permutations (n Pr)…Example: 7 P5 1)Enter the 1st number (7)2)Press Math3)Scroll to the right to PRB4)Press #2 (n Pr)5)Enter second number (5)6)Enter

ProbabilityTheoretical & Experimental Probability

Theoretical Probability of an event is the number of ways that the event can occur, divided by the total number of outcomes.

outcomesof#totaleventinoutcomesof#

P(event)

Empirical Probability of an event is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). It is based specifically on direct observations or experiences.

trialsof#totaloccurseventtimesof#

P(event)

Ex: What is the probability of landing on an even number if a die is rolled?

Sample Space: 1 2 3 4 5 6Even #’s: 2, 4, 6 6

3)#P(even

Ex: Mary rolled a die 25 times and landed on an even number 9 times. What is the empirical probability that Mary will land on an even number on her next roll?

259

)#P(even

ProbabilityTheoretical & Experimental Probability

Sum of the rolls of two dice

3, 5, 5, 4, 6, 7, 7, 5, 9, 10, 12, 9, 6, 5, 7, 8,  7, 4, 11, 6, 8, 8, 10, 6, 7, 4, 4, 5, 7, 9, 9, 7, 8, 11, 6, 5, 4, 7, 7, 4,3, 6, 7, 7, 7, 8, 6, 7, 8, 9

 

Karen and Jason roll two dice 50 times and record their results in the accompanying chart.1) What is their empirical (experimental) probability of rolling a 7?2) What is the theoretical probability of rolling a 7?

26%5013

P(7) 17%

366

P(7)

Empirical Probability Theoretical Probability

ProbabilitySample Spaces

Ex: Marnie wants to choose an outfit consisting of a blouse (green or red), a pair of pants (jeans or khakis) and a pair of shoes (sandals or sneakers). Create a sample space to show all the different possible outfits she can make.

A sample space is a set of all possible outcomes for an activity or experiment.

Green Blouse, Jeans, SandalsGreen Blouse, Jeans, SneakersGreen Blouse, Khakis, SandalsGreen Blouse, Khakis, Sneakers

Red Blouse, Jeans, SandalsRed Blouse, Jeans, SneakersRed Blouse, Khakis, SandalsRed Blouse, Khakis, Sneakers

8 possible combinations of outfits 2 2 2

blouse pants shoes

1) How many outfits include a pair of jeans?

2) What is the probability that Marnie will choose an outfit with a red blouse or sneakers?

4 outfits GB, Jeans, Sa GB, Jeans, Sn RB, Jeans, Sa RB, Jeans, Sn

RB, K, Sn GB, J, Sn RB, K, Sa RB, J, Sn GB,K, Sn RB, J, Sa 8

6

ProbabilitySample Spaces

Sample spaces can also be represented using tree diagrams.

Ex: Using a tree diagram, create the sample space for tossing a coin 3 times.

outcomes8toss3rd

2toss2nd

2toss1st2

HHHHHTHTHHTT

THHTHTTTHTTT

a) How many outcomes include two heads and a tail?

b) What is the probability of landing on at least two heads out of the three tosses?

HHTHTHTHT

3 outcomes

HHHHHTHTHTHT

4/8

ProbabilityCompound Probability

If A and B are independent events, then P(A and B) = P(A) • P(B). “With Replacement”

Example: A drawer contains 3 red paperclips, 4 green paperclips, 5 blue paperclips, 1 white paperclip and 2 yellow paperclips. One paperclip is taken from the drawer and then replaced. Another paperclip is taken from the drawer. a)What is the probability that the first paperclip is red and the second paperclip is blue?

b)If the first paperclip is not replaced, what is the probability that first paperclip is red and the second is blue?

c)If the first paperclip is not replaced, what is the probability that both paperclips are red?

If A and B are dependent events, and A occurs first,then P(A and B) = P(A) • P(B, once A has occurred) “Without Replacement”

22515

155

153

B)P(R,

21015

145

153

B)P(R,

2106

142

153

R)P(R, Without Replacement: Denominator decreases!

ProbabilityConditional Probability

The conditional probability of an event B, in relation to event A, is the probability that event B will occur given the knowledge that an event A has already occurred.

Example: You toss two pennies. The first penny shows HEADS and the other penny rolls under the table and you cannot see it. What is the probability that they are both HEADS?

Sample Space-Tossing two Coins:HH TTHT TH 4 outcomes

Based on the information given, the sample space only includes HH and HT.

21

coin)secondonP(heads

ProbabilityConditional Probability

Grade Snowboarding Skiing Ice Skating TOTAL6th 68 41 46 1557th 84 56 70 2108th 59 74 47 180

TOTAL 211 171 163 545

Middle school students were surveyed about what their favorite sport is. The results are shown in the following table. If a student is selected at random, what is the probability that the student prefers snowboarding given that he/she is in sixth grade grade?

Conditional Probability is the same as Conditional Relative Frequency

Condition: The student is in 6th grade

What is the probability that the student prefers snowboarding?15568

Now it’s your turn to review on your own!

Use the information presented today to help you practice questions from the Regents Exams in the Green Book.

See halgebra.org for the answer keys.

Integrated Algebra Regents Review #3

Tomorrow (Tuesday), June 17th BE THERE!