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Integrated Algebra Regents Review #2

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Integrated AlgebraRegents Review #2

Geometry

Relative Error

Probability

Formulas you need to know!

See reference table

Finding 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 nearesttenth.

Perimeter = distance around the figure

Find the distance around the semicircle

a2 + b2 = c2

32 + 42 = c2

25 = c2

5 = c

IfAD = 5 thenBC = 5

Perimeter = 10 + 4 + 3 + 4 +

Isosceles Trapezoid

AD = BC

Perimeter = 28.853…

= 28.9 units

Finding Areas

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

Area of Triangle + Area of Semicircle

A = ½ bh

A = ½ (12)(14)

A = 84

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

Shaded 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 = lw

A = (10)(20)

A = 200

Diameter = 10

Radius = 5

Area of Shaded Region

200 –

42.92036…square feet

Bags of Wood Chips Needed

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

Surface Area and Volume

- Find the volume of the cylinder to the nearest hundredth.

V = lwh (not on the reference table)

SA = 2lh + 2hw + 2lw

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

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.

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

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

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

= .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 Area

A = lw

A = (120)(54)

A = 6480

Measured Area

A = lw

A = (130)(60)

A = 7800

The relative error is 0.2037

The Counting Principle

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.

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.

The Counting Principle

- 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?

ways of ordering popcorn

- 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?

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

license plates

Permutations

A permutationis an arrangement of objects in a specific order. The order of the arrangement isimportant!!

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.

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

The notation for a permutation: n Prn is the total number of objects r is the number of objects chosen

4 P4 = 4!

Permutations

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

= 5040 ways

7 • 6 • 5 • 4 • 3 • 2 • 1

7 P7 = 7!

Not all permutations are factorials!

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!
- Enter number (7)
- Press Math
- Scroll to the right to PRB
- Press #4 (!)
- Enter

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

Theoretical & Experimental Probability

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

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

Sample Space: 1 2 3 4 5 6

Even #’s: 2, 4, 6

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.

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?

Theoretical & Experimental Probability

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?

Theoretical Probability

Empirical Probability

Sample Spaces

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

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.

Red Blouse, Jeans, Sandals

Red Blouse, Jeans, Sneakers

Red Blouse, Khakis, Sandals

Red Blouse, Khakis, Sneakers

Green Blouse, Jeans, Sandals

Green Blouse, Jeans, Sneakers

Green Blouse, Khakis, Sandals

Green Blouse, Khakis, Sneakers

8 possible combinations of outfits

- How many outfits include a pair of jeans?
- 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, SnGB, J, SnRB, K, Sa

RB, J, SnGB,K, SnRB, J, Sa

Sample 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.

- How many outcomes include two heads and a tail?
- What is the probability of landing on at least two heads out of the three tosses?

HHH

HHT

HTH

HTT

HHT

HTH

THT

3 outcomes

THH

THT

TTH

TTT

HHH

HHT

HTH

THT

4/8

Compound Probability

If A and B are independent events,

then P(A and B) = P(A) • P(B).

“With Replacement”

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”

- 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.
- What is the probability that the first paperclip is red and the second paperclip is blue?
- If the first paperclip is not replaced, what is the probability that first paperclip is red and the second is blue?
- If the first paperclip is not replaced, what is the probability that both paperclips are red?

Without Replacement: Denominator decreases!

Conditional 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:

HHTT

HTTH 4 outcomes

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

Conditional Probability

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?

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 17thBE THERE!