Integrated Algebra Regents Review #2. Geometry Relative Error Probability. Geometry. Formulas you need to know!. See reference table. Geometry. Finding Perimeters.
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Formulas you need to know!
See reference table
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 +
AD = BC
Perimeter = 28.853…
= 28.9 units
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.
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
Bags of Wood Chips Needed
Mr. Petri will need 9 bags of wood chips to cover the shaded area.
Surface Area and Volume
V = lwh (not on the reference table)
SA = 2lh + 2hw + 2lw
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.
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?
A = lw
A = (120)(54)
A = 6480
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
ways of ordering popcorn
Letters: 26 (with repeats) Digits:0 – 9 (10 total without repeats)
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!
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 • 6 • 5 • 4 • 3
= 2520 ways
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?
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
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
Sample spaces can also be represented using tree diagrams.
Ex: Using a tree diagram, create the sample space for tossing a coin 3 times.
If A and B are independent events,
then P(A and B) = P(A) • P(B).
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: Denominator decreases!
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:
HT TH 4 outcomes
Based on the information given, the sample space only includes HH and HT.
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!