# Solving Geometric Applications

Size: px
Start display at page:

Transcription

1 1.8 Solving Geometric Applications 1.8 OBJECTIVES 1. Find a perimeter 2. Solve applications that involve perimeter 3. Find the area of a rectangular figure 4. Apply area formulas 5. Apply volume formulas One application of addition is in finding the perimeter of a figure. Definitions: Perimeter The perimeter is the distance around a closed figure. If the figure has straight sides, the perimeter is the sum of the lengths of its sides. Example 1 Finding the Perimeter We wish to fence in the field shown in Figure 1. How much fencing, in feet (ft), will be needed? 30 ft 20 ft 45 ft 18 ft 25 ft Figure 1 NOTE Make sure to include the unit with each number. The fencing needed is the perimeter of (or the distance around) the field. We must add the lengths of the five sides. 20 ft 30 ft 45 ft 25 ft 18 ft 138 ft So the perimeter is 138 ft. CHECK YOURSELF 1 What is the perimeter of the region shown? 2 50 in. 28 in. 15 in. A rectangle is a figure, like a sheet of paper, with four equal corners. The perimeter of a rectangle is found by adding the lengths of the four sides. 103

2 104 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS Example 2 Finding the Perimeter of a Rectangle Find the perimeter in inches (in.) of the rectangle pictured below. 8 in. 5 in. 5 in. 8 in. The perimeter is the sum of the lengths 8 in., 5 in., 8 in., and 5 in. 8 in. 5 in. 8 in. 5 in. 26 in. The perimeter of the rectangle is 26 in. CHECK YOURSELF 2 Find the perimeter of the rectangle pictured below. 1 7 in. 7 in. 1 In general, we can find the perimeter of a rectangle by using a formula. A formula is a set of symbols that describe a general solution to a problem. Let s look at a picture of a rectangle. Length Width Width Length The perimeter can be found by adding the distances, so Perimeter length width length width To make this formula a little more readable, we abbreviate each of the words, using just the first letter.

3 SOLVING GEOMETRIC APPLICATIONS SECTION Rules and Properties: Formula for the Perimeter of a Rectangle P L W L W (1) There is one other version of this formula that we can use. Because we re adding the length (L) twice, we could write that as 2 L. Because we re adding the width (W) twice, we could write that as 2 W. This gives us another version of the formula. Rules and Properties: Formula for the Perimeter of a Rectangle P 2 L 2 W (2) In words, we say that the perimeter of a rectangle is twice its length plus twice its width. Example 3 uses formula (1). Example 3 Finding the Perimeter of a Rectangle A rectangle has length 1 and width 8 in. What is its perimeter? Start by drawing a picture of the problem. 1 NOTE We say the rectangle is 8 in. by 1 8 in. 8 in. 1 Now use formula (1) P 1 8 in. 1 8 in. 38 in. The perimeter is 38 in. CHECK YOURSELF 3 A bedroom is 9 ft by 12 ft. What is its perimeter?

4 106 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS Units Analysis What happens when we multiply two denominate numbers? The units of the result turn out to be the product of the units. This makes sense when we look at an example from geometry. The area of a square is the square of one side. As a formula, we write that as A s 2 1 ft 1 ft 1 ft 1 ft This tile is 1 foot by 1 foot. A s 2 (1 ft) 2 1 ft 1 ft 1 (ft) (ft) 1 ft 2 In other words, its area is one square foot. If we want to find the area of a room we are actually finding how many of these square feet can be placed in the room. Let s look now at the idea of area. Area is a measure that we give to a surface. It is measured in terms of square units. The area is the number of square units that are needed to cover the surface. One standard unit of area measure is the square inch, written in. 2. This is the measure of the surface contained in a square with sides of See Figure 2. NOTE The unit inch (in.) can be treated as though it were a number. So in. in. can be written in. 2. It is read square inches. NOTE The length and width must be in terms of the same unit. Other units of area measure are the square foot (ft 2 ), the square yard (yd 2 ), the square centimeter (cm 2 ), and the square meter (m 2 ). Finding the area of a figure means finding the number of square units it contains. One simple case is a rectangle. Figure 3 shows a rectangle. The length of the rectangle is 4 inches (in.), and the width is The area of the rectangle is measured in terms of square inches. We can simply count to find the area, 12 square inches (in. 2 ). However, because each of the four vertical strips contains 2, we can multiply: Area 1 2 Width One square inch Figure 2 1in. 2 Length Figure 3

5 SOLVING GEOMETRIC APPLICATIONS SECTION Rules and Properties: Formula for the Area of a Rectangle In general, we can write the formula for the area of a rectangle: If the length of a rectangle is L units and the width is W units, then the formula for the area, A, of the rectangle can be written as A L W (square units) (3) Example 4 Find the Area of a Rectangle A room has dimensions 12 feet (ft) by 15 feet (ft). Find its area. 12 ft 15 ft Use formula (3), with L 15 ft and W 12 ft. A L W 15 ft 12 ft 180 ft 2 The area of the room is 180 ft 2. CHECK YOURSELF 4 A desktop has dimensions 50 in. by 25 in. What is the area of its surface? We can also write a convenient formula for the area of a square. If the sides of the square have length S, we can write Rules and Properties: Formula for the Area of a Square NOTE S 2 is read S squared. A S S S 2 (4) Example 5 Finding Area 3' 3' You wish to cover a square table with a plastic laminate that costs 60 a square foot. If each side of the table measures 3 ft, what will it cost to cover the table?

6 108 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS We first must find the area of the table. Use formula (4), with S 3 ft. A S 2 (3 ft) 2 3 ft 3 ft 9 ft 2 Now, multiply by the cost per square foot. Cost 9 60 \$5.40 CHECK YOURSELF 5 You wish to carpet a room that is a square, 4 yd by 4 yd, with carpet that costs \$12 per square yard. What will be the total cost of the carpeting? Sometimes the total area of an oddly shaped figure is found by adding the smaller areas. The next example shows how this is done. Example 6 Finding the Area of an Oddly Shaped Figure Find the area of Figure in. Region 1 Region 2 Figure 4 The area of the figure is found by adding the areas of regions 1 and 2. Region 1 is a by rectangle; the area of region Region 2 is a by rectangle; the area of region 2 2 The total area is the sum of the two areas: Total area CHECK YOURSELF 6 Find the area of Figure 5. Figure 5 Hint: You can find the area by adding the areas of three rectangles, or by subtracting the area of the missing rectangle from the area of the completed larger rectangle.

7 SOLVING GEOMETRIC APPLICATIONS SECTION Our next measurement deals with finding volumes. The volume of a solid is the measure of the space contained in the solid. Definitions: Definition of a Solid A solid is a three-dimensional figure. It has length, width, and height. 1 cubic inch Figure 6 Volume is measured in cubic units. Examples include cubic inches (in. 3 ), cubic feet (ft 3 ), and cubic centimeters (cm 3 ). A cubic inch, for instance, is the measure of the space contained in a cube that is on each edge. See Figure 6. In finding the volume of a figure, we want to know how many cubic units are contained in that figure. Let s start with a simple example, a rectangular solid. A rectangular solid is a very familiar figure. A box, a crate, and most rooms are rectangular solids. Say that the dimensions of the solid are 5 in. by by as pictured in Figure 7. If we divide the solid into units of, we have two layers, each containing 3 units by 5 units, or 15 in. 3 Because there are two layers, the volume is 30 in. 3 5 in. Figure 7 In general, we can see that the volume of a rectangular solid is the product of its length, width, and height. Rules and Properties: Formula for the Volume of a Rectangular Solid V L W H (5) Example 7 Finding Volume A crate has dimensions 4 ft by 2 ft by 3 ft. Find its volume. 3' 4' 2'

8 110 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS NOTE We are not particularly worried about which is the length, which is the width, and which is the height, because the order in which we multiply won t change the result. Use formula (5), with L 4 ft, W 2 ft, and H 3 ft. V L W H 4 ft 2 ft 3 ft 24 ft 3 CHECK YOURSELF 7 A room is 15 ft long, 10 ft wide, and 8 ft high. What is its volume? Overcoming Math Anxiety Taking a Test Earlier in this chapter, we discussed test preparation. Now that you are thoroughly prepared for the test, you must learn how to take it. There is much to the psychology of anxiety that we can t readily address. There is, however, a physical aspect to anxiety that can be addressed rather easily. When people are in a stressful situation, they frequently start to panic. One symptom of the panic is shallow breathing. In a test situation, this starts a vicious cycle. If you breathe too shallowly, then not enough oxygen reaches the brain. When that happens, you are unable to think clearly. In a test situation, being unable to think clearly can cause you to panic. Hence we have a vicious cycle. How do you break that cycle? It s pretty simple. Take a few deep breaths. We have seen students whose performance on math tests improved markedly after they got in the habit of writing remember to breathe! at the bottom of every test page. Try breathing, it will almost certainly improve your math test scores! CHECK YOURSELF ANSWERS in in ft in \$ ft 3

9 Name 1.8 Exercises Section Date Find the perimeter of each figure ft 2. ANSWERS 4 ft 7 ft yd 8 yd 5 ft 5 ft yd 10 ft in in. 8 yd 5 yd 10 yd yd 7 yd yd Multiply the following. Be sure to use the proper units in your answer ft 2 ft mi 13 mi in yd 26 yd Label the following statements true or false. 13. (10 ft) ft 14. (5 mi) 2 25 mi (8 yd) yd (9 in.) 2 9 in

10 ANSWERS Find the area of each figure yd yd 9 in ft in. 4 ft ft ft 10 ft 8 in. 25 ft 40 ft 10 in ft 5 in. 3 ft 2 ft 5 ft 7 ft in in. 18 ft 15 ft 3 ft 112

11 ANSWERS Find the volume of each solid shown yd yd yd in. 8 in. 8 in yd 3 yd 3 yd Solve the following applications. 35. Window size. A rectangular picture window is 4 feet (ft) by 5 ft. Meg wants to put a trim molding around the window. How many feet of molding should she buy? 36. Fencing material. You are fencing in a backyard that measures 30 ft by 20 ft. How much fencing should you buy? 113

12 ANSWERS Tile costs. You wish to cover a bathroom floor with 1-square-foot (1 ft 2 ) tiles that cost \$2 each. If the bathroom is rectangular, 5 ft by 8 ft, how much will the tile cost? Roofing. A rectangular shed roof is 30 ft long and 20 ft wide. Roofing is sold in squares of 100 ft 2. How many squares will be needed to roof the shed? 39. House repairs. A plate glass window measures 5 ft by 7 ft. If glass costs \$8 per square foot, how much will it cost to replace the window? 40. Paint costs. In a hallway, Bill is painting two walls that are 10 ft high by 22 ft long. The instructions on the paint can say that it will cover 400 ft 2 per gallon (gal). Will one gal be enough for the job? 41. Tile costs. Tile for a kitchen counter will cost \$7 per square foot to install. If the counter measures 12 ft by 3 ft, what will the tile cost? 42. Carpet costs. You wish to cover a floor 4 yards (yd) by 5 yd with a carpet costing \$13 per square yard (yd 2 ). What will the carpeting cost? 43. Frame costs. A mountain cabin has a rectangular front that measures 30 ft long and 20 ft high. If the front is to be glass that costs \$12 per square foot, what will the glass cost? 44. Posters. You are making posters 3 ft by 4 ft. How many square feet of material will you need for six posters? 114

13 ANSWERS 45. Shipping. A shipping container is 5 ft by 3 ft by 2 ft. What is its volume? 46. Size of a cord. A cord of wood is 4 ft by 4 ft by 8 ft. What is its volume? Storage. The inside dimensions of a meat market s cooler are 9 ft by 9 ft by. What is the capacity of the cooler in cubic feet? 48. Storage. A storage bin is 18 ft long, wide, and 3 ft high. What is its volume in cubic feet? A rancher wants to build cattle pens as pictured below. Each pen will have a gate 8 ft wide on one end. What is the total cost of the pens if the fencing is \$6 per linear foot and each gate is \$25? 10 ft. 8 ft. 50. Approximate the total area of the sides and ends of the building shown. 30 ft 83 ft 60 ft 51. Suppose you wish to build a small, rectangular pen, and you have enough fencing for the pen s perimeter to be 3. Assuming that the length and width are to be whole numbers, answer the following. (a) List the possible dimensions that the pen could have. (Note: a square is a type of rectangle.) (b) For each set of dimensions (length and width), find the area that the pen would enclose. (c) Which dimensions give the greatest area? (d) What is the greatest area? 115

14 ANSWERS Suppose you wish to build a rectangular kennel that encloses 100 square feet. Assuming that the length and width are to be whole numbers, answer the following. (a) List the possible dimensions that the kennel could have. (Note: a square is a type of rectangle.) (b) For each set of dimensions (length and width), find the perimeter that would surround the kennel. (c) Which dimensions give the least perimeter? (d) What is the least perimeter? Answers ft yd in yd in False 15. True yd in ft in in ft 37. \$ \$ \$ \$ ft \$

### VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region.

Math 6 NOTES 7.5 Name VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. **The formula for the volume of a rectangular prism is:** l = length w = width h = height Study Tip:

### Calculating Area, Perimeter and Volume

Calculating Area, Perimeter and Volume You will be given a formula table to complete your math assessment; however, we strongly recommend that you memorize the following formulae which will be used regularly

### Area of Parallelograms, Triangles, and Trapezoids (pages 314 318)

Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base

### 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

### Area and Circumference

4.4 Area and Circumference 4.4 OBJECTIVES 1. Use p to find the circumference of a circle 2. Use p to find the area of a circle 3. Find the area of a parallelogram 4. Find the area of a triangle 5. Convert

### 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

### 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

### 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

### Area of Parallelograms (pages 546 549)

A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

### Area of a triangle: The area of a triangle can be found with the following formula: 1. 2. 3. 12in

Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 Solve: Find the area of each triangle. 1. 2. 3. 5in4in 11in 12in 9in 21in 14in 19in 13in

### Area is a measure of how much space is occupied by a figure. 1cm 1cm

Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number

### MATH STUDENT BOOK. 6th Grade Unit 8

MATH STUDENT BOOK 6th Grade Unit 8 Unit 8 Geometry and Measurement MATH 608 Geometry and Measurement INTRODUCTION 3 1. PLANE FIGURES 5 PERIMETER 5 AREA OF PARALLELOGRAMS 11 AREA OF TRIANGLES 17 AREA OF

### 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.

### 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

### MD5-26 Stacking Blocks Pages 115 116

MD5-26 Stacking Blocks Pages 115 116 STANDARDS 5.MD.C.4 Goals Students will find the number of cubes in a rectangular stack and develop the formula length width height for the number of cubes in a stack.

### Area & Volume. 1. Surface Area to Volume Ratio

1 1. Surface Area to Volume Ratio Area & Volume For most cells, passage of all materials gases, food molecules, water, waste products, etc. in and out of the cell must occur through the plasma membrane.

### DIMENSIONAL ANALYSIS #2

DIMENSIONAL ANALYSIS #2 Area is measured in square units, such as square feet or square centimeters. These units can be abbreviated as ft 2 (square feet) and cm 2 (square centimeters). For example, we

### 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

### Mathematics as Reasoning Students will use reasoning skills to determine the best method for maximizing area.

Title: A Pen for Penny Brief Overview: This unit is a reinforcement of the concepts of area and perimeter of rectangles. Methods for maximizing area while perimeter remains the same are also included.

### Show that when a circle is inscribed inside a square the diameter of the circle is the same length as the side of the square.

Week & Day Week 6 Day 1 Concept/Skill Perimeter of a square when given the radius of an inscribed circle Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional

### Section 7.2 Area. The Area of Rectangles and Triangles

Section 7. Area The Area of Rectangles and Triangles We encounter two dimensional objects all the time. We see objects that take on the shapes similar to squares, rectangle, trapezoids, triangles, and

### 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

### Geometry Notes VOLUME AND SURFACE AREA

Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate

### Grade 5 Work Sta on Perimeter, Area, Volume

Grade 5 Work Sta on Perimeter, Area, Volume #ThankATeacher #TeacherDay #TeacherApprecia onweek 6. 12. Folder tab label: RC 3 TEKS 5(4)(H) Perimeter, Area, and Volume Cover: Reporting Category 3 Geometry

### 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

### Solids. Objective A: Volume of a Solids

Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular

### 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

### YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

DETAILED SOLUTIONS AND CONCEPTS - SIMPLE GEOMETRIC FIGURES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST

### SURFACE AREA AND VOLUME

SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has

### Area of a triangle: The area of a triangle can be found with the following formula: You can see why this works with the following diagrams:

Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 You can see why this works with the following diagrams: h h b b Solve: Find the area of

### Finding Volume of Rectangular Prisms

MA.FL.7.G.2.1 Justify and apply formulas for surface area and volume of pyramids, prisms, cylinders, and cones. MA.7.G.2.2 Use formulas to find surface areas and volume of three-dimensional composite shapes.

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

### PART 3 MODULE 8 PROBLEMS INVOLVING AREA

PART 3 MODULE 8 PROBLEMS INVOLVING AREA We will be examining a variety of real-world problems that can be solved by referring to familiar facts from elementary geometry. These problems will usually require

### TEKS TAKS 2010 STAAR RELEASED ITEM STAAR MODIFIED RELEASED ITEM

7 th Grade Math TAKS-STAAR-STAAR-M Comparison Spacing has been deleted and graphics minimized to fit table. (1) Number, operation, and quantitative reasoning. The student represents and uses numbers in

### Geometry - Calculating Area and Perimeter

Geometry - Calculating Area and Perimeter In order to complete any of mechanical trades assessments, you will need to memorize certain formulas. These are listed below: (The formulas for circle geometry

### Lesson 21. Circles. Objectives

Student Name: Date: Contact Person Name: Phone Number: Lesson 1 Circles Objectives Understand the concepts of radius and diameter Determine the circumference of a circle, given the diameter or radius Determine

### Imperial Length Measurements

Unit I Measuring Length 1 Section 2.1 Imperial Length Measurements Goals Reading Fractions Reading Halves on a Measuring Tape Reading Quarters on a Measuring Tape Reading Eights on a Measuring Tape Reading

### STATE GOAL 7: Estimate, make and use measurements of objects, quantities and relationships and determine acceptable

C 1 Measurement H OW MUCH SPACE DO YOU N EED? STATE GOAL 7: Estimate, make and use measurements of objects, quantities and relationships and determine acceptable levels of accuracy Statement of Purpose:

### Linear, Square and Cubic Units Grade Five

Ohio Standards Connection Measurement Benchmark F Analyze and explain what happens to area and perimeter or surface area and volume when the dimensions of an object are changed. Indicator 4 Demonstrate

### 10-3 Area of Parallelograms

0-3 Area of Parallelograms MAIN IDEA Find the areas of parallelograms. NYS Core Curriculum 6.A.6 Evaluate formulas for given input values (circumference, area, volume, distance, temperature, interest,

### 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

### Calculating Perimeter

Calculating Perimeter and Area Formulas are equations used to make specific calculations. Common formulas (equations) include: P = 2l + 2w perimeter of a rectangle A = l + w area of a square or rectangle

### Math. Finding Perimeter and Area. Answers. Name: Solve the problems.

1) The woods behind Adam's house were 2 miles wide and 5 miles long. What is the perimeter of the woods? 2) Janet was cutting out some fabric for a friend. She cut a piece that was 7 centimeters wide and

### MEASUREMENTS. U.S. CUSTOMARY SYSTEM OF MEASUREMENT LENGTH The standard U.S. Customary System units of length are inch, foot, yard, and mile.

MEASUREMENTS A measurement includes a number and a unit. 3 feet 7 minutes 12 gallons Standard units of measurement have been established to simplify trade and commerce. TIME Equivalences between units

### Drawing a Bedroom Floorplan

Appendix A Drawing a Bedroom Floorplan In this chapter, you will learn the following to World Class standards: Draw a Bedroom Floorplan Draw the Bedroom Walls Draw and Dimension the Bedroom Door Draw and

### Unit Conversions. Ben Logan <ben.logan@gmail.com> Feb 10, 2005

Unit Conversions Ben Logan Feb 0, 2005 Abstract Conversion between different units of measurement is one of the first concepts covered at the start of a course in chemistry or physics.

### 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

### ACTIVITY: Finding a Formula Experimentally. Work with a partner. Use a paper cup that is shaped like a cone.

8. Volumes of Cones How can you find the volume of a cone? You already know how the volume of a pyramid relates to the volume of a prism. In this activity, you will discover how the volume of a cone relates

### Georgia Department of Education Georgia Standards of Excellence Framework GSE Grade 6 Mathematics Unit 5

**Volume and Cubes Back to Task Table In this problem-based task, students will examine the mathematical relationship between the volume of a rectangular prism in cubic units and the number of unit cubes

### Lesson 3.1 Factors and Multiples of Whole Numbers Exercises (pages 140 141)

Lesson 3.1 Factors and Multiples of Whole Numbers Exercises (pages 140 141) A 3. Multiply each number by 1, 2, 3, 4, 5, and 6. a) 6 1 = 6 6 2 = 12 6 3 = 18 6 4 = 24 6 5 = 30 6 6 = 36 So, the first 6 multiples

### Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms.

Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game

### 1. Kyle stacks 30 sheets of paper as shown to the right. Each sheet weighs about 5 g. How can you find the weight of the whole stack?

Prisms and Cylinders Answer Key Vocabulary: cylinder, height (of a cylinder or prism), prism, volume Prior Knowledge Questions (Do these BEFORE using the Gizmo.) [Note: The purpose of these questions is

### 1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) =

Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is

### Calculating Area and Volume of Ponds and Tanks

SRAC Publication No. 103 Southern Regional Aquaculture Center August 1991 Calculating Area and Volume of Ponds and Tanks Michael P. Masser and John W. Jensen* Good fish farm managers must know the area

### To Multiply Decimals

4.3 Multiplying Decimals 4.3 OBJECTIVES 1. Multiply two or more decimals 2. Use multiplication of decimals to solve application problems 3. Multiply a decimal by a power of ten 4. Use multiplication by

### 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

### Math 10 - Unit 3 Final Review - Numbers

Class: Date: Math 10 - Unit Final Review - Numbers Multiple Choice Identify the choice that best answers the question. 1. Write the prime factorization of 60. a. 2 7 9 b. 2 6 c. 2 2 7 d. 2 7 2. Write the

### Scale Factors and Volume. Discovering the effect on the volume of a prism when its dimensions are multiplied by a scale factor

Scale Factors and Discovering the effect on the volume of a prism when its dimensions are multiplied by a scale factor Find the volume of each prism 1. 2. 15cm 14m 11m 24m 38cm 9cm V = 1,848m 3 V = 5,130cm

### 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

### Metric Units of Length

7.2 Metric Units of Length 7.2 OBJECTIVES. Know the meaning of metric prefixes 2. Estimate metric units of length 3. Convert metric units of length NOTE Even in the United States, the metric system is

### Pizza! Pizza! Assessment

Pizza! Pizza! Assessment 1. A local pizza restaurant sends pizzas to the high school twelve to a carton. If the pizzas are one inch thick, what is the volume of the cylindrical shipping carton for the

### EDEXCEL FUNCTIONAL SKILLS PILOT TEACHER S NOTES. Maths Level 2. Chapter 5. Shape and space

Shape and space 5 EDEXCEL FUNCTIONAL SKILLS PILOT TEACHER S NOTES Maths Level 2 Chapter 5 Shape and space SECTION H 1 Perimeter 2 Area 3 Volume 4 2-D Representations of 3-D Objects 5 Remember what you

### Mathematical Modeling and Optimization Problems Answers

MATH& 141 Mathematical Modeling and Optimization Problems Answers 1. You are designing a rectangular poster which is to have 150 square inches of tet with -inch margins at the top and bottom of the poster

### 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

### 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

### Lesson 11: Volume with Fractional Edge Lengths and Unit Cubes

Lesson : Volume with Fractional Edge Lengths and Unit Cubes Student Outcomes Students extend their understanding of the volume of a right rectangular prism with integer side lengths to right rectangular

### CBA Volume: Student Sheet 1

CBA Volume: Student Sheet 1 For each problem, decide which cube building has more room inside, or if they have the same amount of room. Then find two ways to use cubes to check your answers, one way that

### Build your skills: Perimeter and area Part 1. Working out the perimeter and area of different shapes

Working out the perimeter and area of different shapes This task has two parts. Part 1 In this part, you can brush up your skills and find out about perimeter and area. Part 2 In the second part, you can

### Measurement: Converting Distances

Measurement: Converting Distances Measuring Distances Measuring distances is done by measuring length. You may use a different system to measure length differently than other places in the world. This

### Chapter 19. Mensuration of Sphere

8 Chapter 19 19.1 Sphere: A sphere is a solid bounded by a closed surface every point of which is equidistant from a fixed point called the centre. Most familiar examples of a sphere are baseball, tennis

### Lesson 13: The Formulas for Volume

Student Outcomes Students develop, understand, and apply formulas for finding the volume of right rectangular prisms and cubes. Lesson Notes This lesson is a continuation of Lessons 11, 12, and Module

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

### Mathematics Common Core Sample Questions

New York State Testing Program Mathematics Common Core Sample Questions Grade6 The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and

### SOMETIMES the cost of installing a landscape plan

Pricing the Landscape Plan SOMETIMES the cost of installing a landscape plan can be quite shocking for those not familiar with all the elements of a landscape. When one breaks down the various expenses

### 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

### Understanding Division of Fractions

Understanding Division of Fractions Reteaching - Reteaching - Divide a fraction by a whole number. Find _. Use a model to show _. Divide each eighth into equal parts. Each section shows _ ( ). _. Divide

### Chapter 4: The Concept of Area

Chapter 4: The Concept of Area Defining Area The area of a shape or object can be defined in everyday words as the amount of stuff needed to cover the shape. Common uses of the concept of area are finding

### 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

### 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

### Lateral and Surface Area of Right Prisms

CHAPTER A Lateral and Surface Area of Right Prisms c GOAL Calculate lateral area and surface area of right prisms. You will need a ruler a calculator Learn about the Math A prism is a polyhedron (solid

### A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

### Volume of Pyramids and Cones

Volume of Pyramids and Cones Objective To provide experiences with investigating the relationships between the volumes of geometric solids. www.everydaymathonline.com epresentations etoolkit Algorithms

### Interpreting Graphs. Interpreting a Bar Graph

1.1 Interpreting Graphs Before You compared quantities. Now You ll use graphs to analyze data. Why? So you can make conclusions about data, as in Example 1. KEY VOCABULARY bar graph, p. 3 data, p. 3 frequency

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Santa Monica College COMPASS Geometry Sample Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the area of the shaded region. 1) 5 yd 6 yd

### Vocabulary Cards and Word Walls Revised: June 29, 2011

Vocabulary Cards and Word Walls Revised: June 29, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,

### 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

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### CHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder.

TEST A CHAPTER 8, GEOMETRY 1. A rectangular plot of ground is to be enclosed with 180 yd of fencing. If the plot is twice as long as it is wide, what are its dimensions? 2. A 4 cm by 6 cm rectangle has

### 6.4 Factoring Polynomials

Name Class Date 6.4 Factoring Polynomials Essential Question: What are some ways to factor a polynomial, and how is factoring useful? Resource Locker Explore Analyzing a Visual Model for Polynomial Factorization

### Lesson 4: Surface Area

Lesson 4: Surface Area Selected Content Standards Benchmark Assessed M.3 Estimating, computing, and applying physical measurement using suitable units (e.g., calculate perimeter and area of plane figures,

### 12 Surface Area and Volume

12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

### Exercise 11.1. Q.1. A square and a rectangular field with measurements as given in the figure have the same perimeter. Which field has a larger area?

11 MENSURATION Exercise 11.1 Q.1. A square and a rectangular field with measurements as given in the figure have the same perimeter. Which field has a larger area? (a) Side = 60 m (Given) Perimeter of

### 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

### Filling and Wrapping: Homework Examples from ACE

Filling and Wrapping: Homework Examples from ACE Investigation 1: Building Smart Boxes: Rectangular Prisms, ACE #3 Investigation 2: Polygonal Prisms, ACE #12 Investigation 3: Area and Circumference of

### What You ll Learn. Why It s Important

These students are setting up a tent. How do the students know how to set up the tent? How is the shape of the tent created? How could students find the amount of material needed to make the tent? Why

### Geometry Unit 6 Areas and Perimeters

Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose

### Applications for Triangles

Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given