MAT 080-Algebra II Applications of Quadratic Equations
|
|
|
- Myra Scott
- 7 years ago
- Views:
Transcription
1 MAT 080-Algebra II Applications of Quadratic Equations Objectives a Applications involving rectangles b Applications involving right triangles a Applications involving rectangles One of the common applications of quadratic equations is to find the unknown length and width of a rectangle. To solve these types of problems, we need to use the formula for the area of a rectangle: length width = area or l w A This is a formula that is frequently used, and you will need to memorize it. Eample 1 A rectangle has a length that is 4 meters more than the width. The area of the rectangle is 117 square meters. Find the dimensions of the rectangle. Step 1 Draw a diagram of the rectangle. Label the length and the width. Since we know nothing at all about the width, we will call it. In the problem we are told that the length is 4 meters more than the width, so we will let 4 represent the length. length 4 Area = 117 sq. meters width =
2 MAT 080: Applications of Quadratic Equations Step Write the equation using the formula for the area of a rectangle and the information from the diagram. Formula: length width area or l w A From diagram: width, length 4, and area 117 sq. meters length width area Formula (4 ) 117 Substitute (4 ) for length, for width, and 117 for area Distribute through (4 ) Write the equation in standard form by subtracting 117 from both sides. ( 13)( 9) 0 Factor Take each factor, set it equal to 0, and solve the resulting equations: Since length must be positive, the solution that fits the problem is discard the We Step 3 Substitute 9 in for in the diagram in order to determine the dimensions of the rectangle. length 4 length width = width = 9 So, the width is 9 meters, and the length is 13 meters.
3 Applications Involving Rectangles 3 Practice Problem 1 A rectangle whose area is 11 square feet has a length that is 6 feet greater than the width. Find the dimensions of the rectangle. Solution to this Practice Problem may be found on page 1. Eample A rectangle has a width that is 5 feet less than the length. The area of the rectangle is 16 square feet. Find the dimensions of the rectangle. Step 1 Draw a diagram of the rectangle. Label the length and the width. Since we know nothing at all about the length, we will call it. In the problem we are told that the width is 5 feet less than the length, so we will let 5 represent the width. (Note that 5 is not correct!) length Area = 16 sq. feet width = 5 Step Write the equation using the formula for the area of a rectangle and the information from the diagram. Formula: length width area or l w A From diagram: length, width 5, and area 16 sq. feet length width area Formula ( 5) 16 Substitute for length, 5 for width, and 16 for area. Eample continues on net page
4 4 MAT 080: Applications of Quadratic Equations Eample -continued ( 5) 16 Equation from last step on previous page 5 16 Distribute through ( 5) Write the equation in standard form by subtracting 16 from both sides. ( 14)( 9) 0 Factor Take each factor, set it equal to 0, and solve the resulting equations: Since length must be positive, the solution that fits the problem is discard the We Step 3 Substitute 14 in for in the diagram in order to determine the dimensions of the rectangle. length length 14 width = 5 width = 14 5 = 9 So, the length is 14 feet, and the width is 9 feet. Practice Problem A rectangle whose area is 19 square meters has a width that is 4 meters less than the length. Find the dimensions of the rectangle. Solution to this Practice Problem may be found on page 13.
5 Applications Involving Right Triangles 5 b Applications involving right triangles In your tetbook in the section on Applications of Quadratic Equations you were introduced to the Pythagorean Theorem: a b c. This formula is used when working with the lengths of the sides of a right triangle. A right triangle is a triangle that has a right angle (90 ). a (leg) c (hypotenuse) b (leg) Note that the hypotenuse must be c. The hypotenuse is the longest side, and is always opposite the right (90 ) angle. It makes no difference which leg is a and which leg is b. This formula is used often, so you will need to memorize it. Eample 3 The length of one leg of a right triangle is feet longer than the other leg. The length of the hypotenuse is 10 feet. Find the lengths of the two legs. Step 1 Draw a diagram of the triangle. Label each of the sides. Since we know nothing at all about the shorter leg, we will call it. In the problem we are told that the other leg is feet longer. So, we will let represent the longer leg. The hypotenuse is 10. a c 10 b
6 6 MAT 080: Applications of Quadratic Equations Step Write the equation using the Pythagorean Theorem and the information from the diagram. Formula: a b c From diagram: side a, side b, and hypotenuse 10 a b c Formula ( ) 10 Substitute for a, for b, and 10 for c. ( )( ) 100 Important: ( ) means ( )( ), which must be multiplied using FOIL Multiply ( )( ) 4. to make Combine like terms. make and make Write the equation in standard form by subtracting 100 from both sides Equation in standard form. ( 48) 0 Factor the common factor of. ( 48) Divide both sides by to cancel the common factor of. Note that 0 will still equal 0. ( 8)( 6) 0 Factor.
7 Applications Involving Right Triangles 7 Take each factor, set it equal to 0, and solve the resulting equations: Since length must be positive, the solution that fits the problem is discard the We Step 3 Substitute 6 in for in the diagram in order to determine the lengths of the legs of the right triangle. a c 10 a 6 c 10 b b 6 8 So, one leg is 6 feet and the other leg is 8 feet. Practice Problem 3 The length of one leg of a right triangle is 3 feet longer than the other leg. The length of the hypotenuse is 15 feet. Find the lengths of the two legs. Solution to this Practice Problem may be found on page 15.
8 8 MAT 080: Applications of Quadratic Equations Eample 4 The length of one leg of a right triangle is 3 feet shorter than the other leg. The length of the hypotenuse is 15 feet. Find the lengths of the two legs. Step 1 Draw a diagram of the triangle. Label each of the sides. Since we know nothing at all about the longer leg, we will call it. In the problem we are told that the other leg is 3 feet shorter. So, we will let 3 represent the shorter leg (Note that 3 is not correct!). The hypotenuse is 15. a 3 c 15 Step Write the equation using the Pythagorean Theorem and the information from the diagram. Formula: a b c From diagram: side a 3, side b, and hypotenuse 15 a b c Formula b ( 3) 15 Substitute 3 for a, for b, and 15 for c. ( 3)( 3) 5 Important: ( 3) means ( 3)( 3), which must be multiplied using FOIL Multiply ( 3)( 3) to make Combine like terms. make and 3 3 make Write the equation in standard form by subtracting 5 from both sides.
9 Applications Involving Right Triangles Equation in standard form. ( 3 108) 0 Factor the common factor of. ( 3 108) Divide both sides by to cancel the common factor of. Note that 0 will still equal 0. ( 1)( 9) 0 Factor. Take each factor, set it equal to 0, and solve the resulting equations: Since length must be positive, the solution that fits the problem is discard the We Step 3 Substitute 1 in for in the diagram in order to determine the lengths of the legs of the right triangle. a 3 c 15 a 1 3 = 9 c 15 b b 1 So, one leg is 1 feet and the other leg is 9 feet. Practice Problem 4 The length of one leg of a right triangle is 4 feet shorter than the other leg. The length of the hypotenuse is 0 feet. Find the lengths of the two legs. Solution to this Practice Problem may be found on page 17.
10 10 MAT 080: Applications of Quadratic Equations Homework Problems Answers to Homework Problems are on page 19 a Applications involving rectangles 1. A rectangle whose area is 180 square feet has a width that is 3 feet less than the length. Find the dimensions of the rectangle.. A rectangle has a length that is meters more than the width. The area of the rectangle is 88 square meters. Find the dimensions of the rectangle. 3. A rectangle has a width that is 6 meters less than the length. The area of the rectangle is 80 square meters. Find the dimensions of the rectangle. 4. A rectangle whose area is 336 square feet has a length that is 4 meters less than the twice the width. Find the dimensions of the rectangle. 5. A rectangle has a length that is 3 meters more than twice the width. The area of the rectangle is 44 square meters. Find the dimensions of the rectangle. b Applications involving right triangles 6. The hypotenuse of a right triangle is 5 meters in length. One leg is 17 meters longer than the other. Find the length of each leg. 7. The hypotenuse of a right triangle is 15 yards in length. One leg is 3 yards shorter than the other. Find the length of each leg. 8. The hypotenuse of a right triangle is 40 yards in length. One leg is 8 yards longer than the other. Find the length of each leg.
11 Homework Problems The sporting goods store at the mall is a perfect rectangle. Its diagonal is 13 meters and the width of the rectangle is 7 meters shorter than its length. Find the length and the width of the store. 10. Manual placed a ladder 0 feet long against the side of his house. The distance from the top of the ladder to the bottom of the house is 4 feet greater than the distance from the bottom of the house to the foot of the ladder. How far is the foot of the ladder from the house? How far up on the house does the top of the ladder touch the building?
12 1 MAT 080: Applications of Quadratic Equations Solutions to Practice Problems Practice Problem 1 A rectangle whose area is 11 square feet has a length that is 6 feet greater than the width. Find the dimensions of the rectangle. Step 1 Draw a diagram of the rectangle. Label the length and the width. Since we know nothing at all about the width, we will call it. In the problem we are told that the length is 6 meters more than the width, so we will let 6 represent the length. length 6 Area = 11 sq. meters width = Step Write the equation using the formula for the area of a rectangle and the information from the diagram. Formula: length width area or l w A From diagram: width, length 6, and area 11 sq. meters length width area Formula (6 ) 11 Substitute (6 ) for length, for width, and 11 for area Distribute through (6 ) Write the equation in standard form by subtracting 11 from both sides. ( 14)( 8) 0 Factor
13 Solutions to Practice Problems 13 Take each factor, set it equal to 0, and solve the resulting equations: Since length must be positive, the solution that fits the problem is discard the We Step 3 Substitute 8 in for in the diagram in order to determine the dimensions of the rectangle. length 6 length width = width = 8 So, the width is 8 meters, and the length is 14 meters. Practice Problem A rectangle whose area is 19 square feet has a width that is 4 feet less than the length. Find the dimensions of the rectangle. Step 1 Draw a diagram of the rectangle. Label the length and the width. Since we know nothing at all about the length, we will call it. In the problem we are told that the width is 4 feet less than the length, so we will let 4 represent the width. (Note that 4 is not correct!) length Area = 19 sq. feet width = 4
14 14 MAT 080: Applications of Quadratic Equations Step Write the equation using the formula for the area of a rectangle and the information from the diagram. Formula: length width area or l w A From diagram: length, width 4, and area 19 sq. feet length width area Formula ( 4) 19 Substitute for length, 4 for width, and 19 for area Distribute through ( 4) Write the equation in standard form by subtracting 19 from both sides. ( 16)( 1) 0 Factor Take each factor, set it equal to 0, and solve the resulting equations: Since length must be positive, the solution that fits the problem is discard the We Step 3 Substitute 16 in for in the diagram in order to determine the dimensions of the rectangle. length length 16 width = 4 width = 16 4 = 1 So, the length is 16 feet, and the width is 1 feet.
15 Solutions to Practice Problems 15 Practice Problem 3 The length of one leg of a right triangle is 3 feet longer than the other leg. The length of the hypotenuse is 15 feet. Find the lengths of the two legs. Step 1 Draw a diagram of the triangle. Label each of the sides. Since we know nothing at all about the shorter leg, we will call it. In the problem we are told that the other leg is 3 feet longer than the shorter leg. So, we will let 3 represent the longer leg. The hypotenuse is 15. a c 15 Step Write the equation using the Pythagorean theorem and the information from the diagram. Formula: a b c From diagram: side a, side b 3, and hypotenuse 15 b 3 a b c Formula ( 3) 15 Substitute for a, 3 for b, and 15 for c. ( 3)( 3) 5 Important: ( 3) means ( 3)( 3), which must be multiplied using FOIL Multiply ( 3)( 3) to make Combine like terms. make and 3 3 make Write equation in standard form by subtracting 5 from both sides.
16 16 MAT 080: Applications of Quadratic Equations Equation in standard form. ( 3 108) 0 Factor the common factor of. ( 3 108) Divide both sides by to cancel the common factor of. Note that 0 will still equal 0. ( 1)( 9) 0 Factor. Take each factor, set it equal to 0, and solve the resulting equations: Since length must be positive, the solution that fits the problem is discard the We Step 3 Substitute 9 in for in the diagram in order to determine the lengths of the legs of the right triangle. a c 10 a 9 c 10 b 3 b So, one leg is 9 feet and the other leg is 1 feet.
17 Solutions to Practice Problems 17 Practice Problem 4 The length of one leg of a right triangle is 4 feet shorter than the other leg. The length of the hypotenuse is 0 feet. Find the lengths of the two legs. Step 1 Draw a diagram of the triangle. Label each of the sides. Since we know nothing at all about the longer leg, we will call it. In the problem we are told that the other leg is 4 feet shorter. So, we will let 4 represent the shorter leg (Note that 4 is not correct!). The hypotenuse is 0. a 4 c 0 Step Write the equation using the Pythagorean theorem and the information from the diagram. Formula: a b c From diagram: side a 4, side b, and hypotenuse 0 a b c Formula b ( 4) 0 Substitute 4 for a, for b, and 0 for c. ( 4)( 4) 400 Important: ( 4) means ( 4)( 4), which must be multiplied using FOIL Multiply ( 4)( 4) to make Combine like terms. make and 4 4 make Write the equation in standard form by subtracting 400 from both sides.
18 18 MAT 080: Applications of Quadratic Equations Equation in standard form. ( 4 19) 0 Factor the common factor of. ( 4 19) Divide both sides by to cancel the common factor of. Note that 0 will still equal 0. ( 16)( 1) 0 Factor. Take each factor, set it equal to 0, and solve the resulting equations: Since length must be positive, the solution that fits the problem is discard the We Step 3 Substitute 16 in for in the diagram in order to determine the lengths of the legs of the right triangle. a 4 c 0 a c 0 b b 16 So, one leg is 16 feet and the other leg is 1 feet.
19 Answers to Homework Problems 19 Answers to Homework Problems 1. length 15 feet and width 1 feet. length 18 meters and width 16 meters 3. length 0 meters and width 14 meters 4. length 4 feet and width 14 feet 5. length 11 meters and width 4 meters 6. one leg is 7 meters; the other leg is 4 meters 7. one leg is 1 yards; the other leg is 9 yards 8. one leg is 3 yards; the other leg is 4 yards 9. length is 1 meters; width is 5 meters 10. the foot of the ladder is 1 feet from the bottom of the house; the top of the ladder touches the building at a point 16 feet above the ground
Sect 6.7 - Solving Equations Using the Zero Product Rule
Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
Square Roots and the Pythagorean Theorem
4.8 Square Roots and the Pythagorean Theorem 4.8 OBJECTIVES 1. Find the square root of a perfect square 2. Use the Pythagorean theorem to find the length of a missing side of a right triangle 3. Approximate
Applications of the Pythagorean Theorem
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
Pythagorean Theorem: 9. x 2 2
Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
Geometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
Lesson 18 Pythagorean Triples & Special Right Triangles
Student Name: Date: Contact Person Name: Phone Number: Teas Assessment of Knowledge and Skills Eit Level Math Review Lesson 18 Pythagorean Triples & Special Right Triangles TAKS Objective 6 Demonstrate
MATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
Pythagoras Theorem. Page I can... 1... identify and label right-angled triangles. 2... explain Pythagoras Theorem. 4... calculate the hypotenuse
Pythagoras Theorem Page I can... 1... identify and label right-angled triangles 2... eplain Pythagoras Theorem 4... calculate the hypotenuse 5... calculate a shorter side 6... determine whether a triangle
Geometry: Classifying, Identifying, and Constructing Triangles
Geometry: Classifying, Identifying, and Constructing Triangles Lesson Objectives Teacher's Notes Lesson Notes 1) Identify acute, right, and obtuse triangles. 2) Identify scalene, isosceles, equilateral
Law of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.
Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where
12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2
DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal
Geometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
Free Pre-Algebra Lesson 55! page 1
Free Pre-Algebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
Factoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
Solving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
Characteristics of the Four Main Geometrical Figures
Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.
Veterans Upward Bound Algebra I Concepts - Honors
Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
MATH 100 PRACTICE FINAL EXAM
MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number
Perimeter is the length of the boundary of a two dimensional figure.
Section 2.2: Perimeter and Area Perimeter is the length of the boundary of a two dimensional figure. The perimeter of a circle is called the circumference. The perimeter of any two dimensional figure whose
How To Solve Factoring Problems
05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
POLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
Hiker. A hiker sets off at 10am and walks at a steady speed for 2 hours due north, then turns and walks for a further 5 hours due west.
Hiker A hiker sets off at 10am and walks at a steady speed for hours due north, then turns and walks for a further 5 hours due west. If he continues at the same speed, what s the earliest time he could
Sample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
MAIN IDEA The rectangle at the right has an area of 20 square units. The distance around the rectangle is 5 + 4 + 5 + 4, or 18 units.
1-9 Algebra: Area Formulas MAIN IDEA The rectangle at the right has an area of 20 square units. The distance around the rectangle is 5 + 4 + 5 + 4, or 1. Find the areas of rectangles and squares. New Vocabulary
7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
Right Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
Nonlinear Systems and the Conic Sections
C H A P T E R 11 Nonlinear Systems and the Conic Sections x y 0 40 Width of boom carpet Most intense sonic boom is between these lines t a cruising speed of 1,40 miles per hour, the Concorde can fly from
MATHCOUNTS TOOLBOX Facts, Formulas and Tricks
MATHCOUNTS TOOLBOX Facts, Formulas and Tricks MATHCOUNTS Coaching Kit 40 I. PRIME NUMBERS from 1 through 100 (1 is not prime!) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 II.
Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.
Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 9-1.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles
Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
COMPETENCY TEST SAMPLE TEST. A scientific, non-graphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh
BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, non-graphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
Algebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
Pre-Algebra Lesson 6-1 to 6-3 Quiz
Pre-lgebra Lesson 6-1 to 6-3 Quiz Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the area of the triangle. 17 ft 74 ft Not drawn to scale a. 629 ft
Basic Lesson: Pythagorean Theorem
Basic Lesson: Pythagorean Theorem Basic skill One leg of a triangle is 10 cm and other leg is of 24 cm. Find out the hypotenuse? Here we have AB = 10 and BC = 24 Using the Pythagorean Theorem AC 2 = AB
Basic Math for the Small Public Water Systems Operator
Basic Math for the Small Public Water Systems Operator Small Public Water Systems Technology Assistance Center Penn State Harrisburg Introduction Area In this module we will learn how to calculate the
How To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
Inv 1 5. Draw 2 different shapes, each with an area of 15 square units and perimeter of 16 units.
Covering and Surrounding: Homework Examples from ACE Investigation 1: Questions 5, 8, 21 Investigation 2: Questions 6, 7, 11, 27 Investigation 3: Questions 6, 8, 11 Investigation 5: Questions 15, 26 ACE
How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
Geometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
Area, Perimeter, Volume and Pythagorean Theorem Assessment
Area, Perimeter, Volume and Pythagorean Theorem Assessment Name: 1. Find the perimeter of a right triangle with legs measuring 10 inches and 24 inches a. 34 inches b. 60 inches c. 120 inches d. 240 inches
Review of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
Quadratics - Rectangles
9.7 Quadratics - Rectangles Objective: Solve applications of quadratic equations using rectangles. An application of solving quadratic equations comes from the formula for the area of a rectangle. The
SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property
498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1
TEST A CHAPTER 6, EQUATIONS, INEQUALITIES, PROBLEM SOLVING. 1. Factor x 2-5x + 6. 2. Factor x 2-4x - 5.
TEST A CHAPTER 6, EQUATIONS, INEQUALITIES, PROBLEM SOLVING. Factor x 2-5x + 6. 2. Factor x 2-4x - 5. 3. Solve: (x + 2)(x - 3) = 0 x(x - 3)(x + 4) = 0 4. Solve by factoring: x 2 + x + 2 = 0. 5. Solve by
Polynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
MATD 0390 - Intermediate Algebra Review for Pretest
MATD 090 - Intermediate Algebra Review for Pretest. Evaluate: a) - b) - c) (-) d) 0. Evaluate: [ - ( - )]. Evaluate: - -(-7) + (-8). Evaluate: - - + [6 - ( - 9)]. Simplify: [x - (x - )] 6. Solve: -(x +
SECTION 1-6 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
1. The volume of the object below is 186 cm 3. Calculate the Length of x. (a) 3.1 cm (b) 2.5 cm (c) 1.75 cm (d) 1.25 cm
Volume and Surface Area On the provincial exam students will need to use the formulas for volume and surface area of geometric solids to solve problems. These problems will not simply ask, Find the volume
Pythagorean Theorem: Proof and Applications
Pythagorean Theorem: Proof and Applications Kamel Al-Khaled & Ameen Alawneh Department of Mathematics and Statistics, Jordan University of Science and Technology IRBID 22110, JORDAN E-mail: kamel@just.edu.jo,
Solutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
FACTORING QUADRATICS 8.1.1 through 8.1.4
Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten
The GED math test gives you a page of math formulas that
Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding
Area and Perimeter. Name: Class: Date: Short Answer
Name: Class: Date: ID: A Area and Perimeter Short Answer 1. The squares on this grid are 1 centimeter long and 1 centimeter wide. Outline two different figures with an area of 12 square centimeters and
Time needed: each worksheet will take approximately 1 hour to complete
Pythagoras Theorem Teacher s Notes Subject: Mathematics Topic: Pythagoras theorem Level: Pre-intermediate, intermediate Time needed: each worksheet will take approximately 1 hour to complete Learning objectives:
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
8-2 The Pythagorean Theorem and Its Converse. Find x.
1 8- The Pythagorean Theorem and Its Converse Find x. 1. hypotenuse is 13 and the lengths of the legs are 5 and x.. equaltothesquareofthelengthofthehypotenuse. The length of the hypotenuse is x and the
Perimeter, Area, and Volume
Perimeter, Area, and Volume Perimeter of Common Geometric Figures The perimeter of a geometric figure is defined as the distance around the outside of the figure. Perimeter is calculated by adding all
TSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
Answers to Basic Algebra Review
Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
Equations Involving Fractions
. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation
MATH 110 Landscape Horticulture Worksheet #4
MATH 110 Landscape Horticulture Worksheet #4 Ratios The math name for a fraction is ratio. It is just a comparison of one quantity with another quantity that is similar. As a Landscape Horticulturist,
MATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
TRIGONOMETRY Compound & Double angle formulae
TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae
6. LECTURE 6. Objectives
6. LECTURE 6 Objectives I understand how to use vectors to understand displacement. I can find the magnitude of a vector. I can sketch a vector. I can add and subtract vector. I can multiply a vector by
Example 1: Suppose the demand function is p = 50 2q, and the supply function is p = 10 + 3q. a) Find the equilibrium point b) Sketch a graph
The Effect of Taxes on Equilibrium Example 1: Suppose the demand function is p = 50 2q, and the supply function is p = 10 + 3q. a) Find the equilibrium point b) Sketch a graph Solution to part a: Set the
Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
Lesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
Tallahassee Community College PERIMETER
Tallahassee Community College 47 PERIMETER The perimeter of a plane figure is the distance around it. Perimeter is measured in linear units because we are finding the total of the lengths of the sides
A Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
Quick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its
(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:
Lesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
PowerScore Test Preparation (800) 545-1750
Question 1 Test 1, Second QR Section (version 2) Two triangles QA: x QB: y Geometry: Triangles Answer: Quantity A is greater 1. The astute student might recognize the 0:60:90 and 45:45:90 triangle right
RIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
Perimeter. 14ft. 5ft. 11ft.
Perimeter The perimeter of a geometric figure is the distance around the figure. The perimeter could be thought of as walking around the figure while keeping track of the distance traveled. To determine
Sequences. A sequence is a list of numbers, or a pattern, which obeys a rule.
Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the
FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST. Mathematics Reference Sheets. Copyright Statement for this Assessment and Evaluation Services Publication
FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST Mathematics Reference Sheets Copyright Statement for this Assessment and Evaluation Services Publication Authorization for reproduction of this document is hereby
Section 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
Factoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
DigitalCommons@University of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University
SQUARE-SQUARE ROOT AND CUBE-CUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
43 Perimeter and Area
43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study
Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)
Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.
GAP CLOSING. 2D Measurement GAP CLOSING. Intermeditate / Senior Facilitator s Guide. 2D Measurement
GAP CLOSING 2D Measurement GAP CLOSING 2D Measurement Intermeditate / Senior Facilitator s Guide 2-D Measurement Diagnostic...4 Administer the diagnostic...4 Using diagnostic results to personalize interventions...4
STRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
a. You can t do the simple trick of finding two integers that multiply to give 6 and add to give 5 because the a (a = 4) is not equal to one.
FACTORING TRINOMIALS USING THE AC METHOD. Factoring trinomial epressions in one unknown is an important skill necessary to eventually solve quadratic equations. Trinomial epressions are of the form a 2
Algebra Word Problems
WORKPLACE LINK: Nancy works at a clothing store. A customer wants to know the original price of a pair of slacks that are now on sale for 40% off. The sale price is $6.50. Nancy knows that 40% of the original
( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:
Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (
Algebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials