c Kendra Kilmer June 12, 2009 Section 11.1 - Linear Measure The English System Originally, a yard was the distance from the tip of the nose to the end of an outstetched arm of an adult person and a foot was the length of a human foot. Since then it has gone through many definitions until now the definitions are based on the meter. Unit Equivalent in Other Units yard (yd) 3 feet foot (ft) 12 inches mile (mi) 1760 yards or 5280 feet Dimensional Analysis (Unit Analysis) A process used to convert from one unit of measure to another using unit ratios (ratios equivalent to 1). Example 1: Convert each of the following: a) 200 feet = yards b) 3.75 yards = inches c) 8690 feet = miles d) 940 inches = yards The Metric System Unit Symbol Relationship to Base Unit kilometer km 1000 m hectometer km 100 m dekameter dam 10 m meter m base unit decimeter dm 0.1 m centimeter cm 0.01 m millimeter mm 0.001 m Approximate Conversions Between English and Metric Systems • 1 kilometer ≈ 0.62 miles • 1 meter ≈ 1.09 yards • 2.54 centimeters ≈ 1 inch 1
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Originally, a yard was the distance from the tip of the nose tothe end of an outstetched arm ofan adult person and a foot was the length of a human foot. Sincethen it has gone through manydefinitions until now the definitions are based on the meter.
Unit Equivalent in Other Unitsyard (yd) 3 feetfoot (ft) 12 inchesmile (mi) 1760 yards or 5280 feet
Dimensional Analysis (Unit Analysis)
A process used to convert from one unit of measure to another usingunit ratios (ratios equivalentto 1).
Example 1: Convert each of the following:
a) 200 feet= yards
b) 3.75 yards= inches
c) 8690 feet= miles
d) 940 inches= yards
The Metric System
Unit Symbol Relationship to Base Unitkilometer km 1000 mhectometer km 100 mdekameter dam 10 mmeter m base unitdecimeter dm 0.1 mcentimeter cm 0.01 mmillimeter mm 0.001 m
Approximate Conversions Between English and Metric Systems
Thegreatest possible error (GPE)of a measurement is one-half the smallest unit used.
Example 4: Determine the GPE for each of the following measurements andinterpret.
a) 25 inches
b) 10.8 cm
c) 5.64 m
Distance Properties
1. The distance between any two pointsA andB is greater than or equal to 0, writtenAB ≥ 0.
2. The distance between any two pointsA andB is the same as the distance betweenB andA, writtenAB = BA.
3. For any three pointsA, B, andC, the distance betweenA andB plus the distance betweenB andC isgreater than or equal to the distance betweenA andC, writtenAB+BC ≥ AC.
Distance Around a Plane Figure
The perimeter of a simple closed curve is the length of the curve. If a figure is a polygon, itsperimeter is the sum of the lengths of its sides. A perimeter has linear measure.
Example 5: Find the perimeter of each of the shapes below:
Example 6: Given a square of any size, stretch a rope tightly around it. Now take the rope off, add 100 inchesto it, and put the extended rope back around the square so thatthe new rope makes a square around theoriginal square. Findd, the distance between the squares.
Circumference of a Circle
A circle is defined as the set of all points in a plane that are the same distance from a given point,thecenter. The perimeter of a circle is itscircumference.
C = 2πr = πd
wherer is the radius,d is the diameter, andπ = 3.14159....
Example 7: Find the circumference of the circle pictured below:
Area is measured using square units and the area of a region isthe number of non-overlappingsquare units that covers the region. For instance, a square measuring 1 cm on a side has an area of1 square centimeter denoted 1 cm2.
Areas on a Geoboard
In teaching the concept of area, intuitive activities should preced the development of formulas.
Example 1: Find the area of each of the figures below:
Converting Units of Area
The most commonly used units of area in the English system arethe square inch (in.2), the squarefoot (ft2), the square yard (yd2), and the square mile (mi2). In the metric system, the most com-monly used units are the square millimeter (mm2), the square centimeter (cm2), the square meter(m2), and the square kilometer (km2). We must be careful when coverting between these units.
One application of area today is in land measures. The commonunit of land measure in the Englishsystem is the acre. Historically, an acre was the amount of land a man with one horse could plowin one day.
Unit of Area Equivalent in Other Units1 acre 4840 yd2
1 mi2 640 acres
1 a (are) 100 m2
1 ha (hectare) 100a or 10,000m2 or 1hm2
1 km2 1,000,000 m2
Example 3:
a) A square field has a side of 400 yards. Find the area of the field in acres.
b) A square field has a side of 400 meters. Find the area of the field in hectares.
Example 3: The size of a rectangular television screen is given as the length of the diagonal of the screen. Ifthe width of the screen is 24 inches, and the height of the screen is 18 inches, what is the length of thediagonal?
Example 4: A pole BD, 28 feet high, is perpendicular to the ground. Two wiresBC andBA, each 35 feetlong, are attahced to the top of the pole and to stakesA andC on the ground. If pointsA, D, andC arecollinear, how far are the stakesA andC from each other?
Special Right Triangles
Property of 45◦−45◦−90◦ triangle: In an isosceles right triangle, if the length of each leg isa,then the hypotenuse has lengtha
√2.
Property of 30◦−60◦−90◦ triangle: In a 30◦−60◦−90◦triangle, the length of the hypotenuse istwo times as long as the leg opposite the 30◦ angle and the leg opposite the 60◦ angle is
Theorem 11-2 (Converse of the Pythagorean Theorem)If △ABC is a triangle with sides of lengthsa, b,andc such thata2 + b2 = c2, then△ABC is a right triangle with the right angle opposite the side oflengthc.
Example 6: Determine if the following can be the lengths of the sides of aright triangle:
a) 51,68,85
b) 3,4,7
Distance Formula
The distance between the pointsA(x1,y1) andB(x2,y2) is given by
AB =√
(x2− x1)2+(y2− y1)2
Example 7: Let’s convince ourselves of the distance formula.
Example 8: Find the distance between the points(−5,−3) and(2,−1)
The surface areais the sum of the areas of the faces (lateral and bases) of a three-dimensionalobject. Thelateral surface areais the sum of the areas of the lateral faces.
Surface Area of a Cube
ss
s
The surface area of a cube is , wheres is the length of a side.
Example 1: Find the surface area of a cube with length of a side 8 inches.
Surface Area of a Right Prism with Regularn-gon Bases
h
The surface area of a Right Prism with regularn-gon bases is , wherel is the lengthof a side of the base,h is the height of a lateral face, andB is the area of the base.
Example 2: Find the surface area of a right regular-hexagonal prism with height 7 feet and length of eachside of the hexagon 4 feet.
The surface area of a right circular cylinder is , wherer is the radius of the circle,andh is the height of the cylinder.
Example 3: Find the surface area of a right circular cylinder in which the radius of the circular base is 5 cmand the height of the cylinder is 25 cm.
Surface Area of a Right Regular Pyramid
l
b
The surface area of a right regular pyramid is , wheren is the number of sides ofthe regular polygon,b is the length of a side of the base,B is the area of the base, andl is the slantheight.
Example 4: Find the surface area of a right regular triangular pyramid with slant height 5 inches and lengthof side of the base 4 inches.
Example 7: The napkin ring pictured in the following figure is to be resilvered. How many square millime-ters must be covered?
Example 8: The base of a right pyramid is a regular hexagon with sides of length 12 meters. The altitude ofthe pyramid is 9 meters. Find the total surface area of the pyramid.
Surface Areais the number of square units covering a three dimensional figure;Volume describeshow much space a three-dimensional figure contains.
The unit of measure for volume must be a shape that tessellates space (can be stacked so that theyleave no gaps and fill space). Standard units of volume are based on cubes and arecubic units. Acubic unit is the amount of space enclosed within a cube that measures 1 unit on a side.
Example 1: Determine the surface area and volume of the following figure:
Volume of Right Rectangular Prisms
The volume of a right rectangular prism can be measured by determining how many cubes areneeded to build it as a solid.
Thus, the volume of aright rectangular prism is wherel is the length of thebase,w is the width of the base, andh is the height of the prism.
The most commonly used metric units of volume are thecubic centimeterand thecubic meter.
Example 2: Convert each of the following:
a) 5m3 = cm3
b) 12,300cm3 = m3
In the metric system, cubic units may be used for either dry orliquid measure, although units suchas liters and milliliters are usually used for liquid measures. By definition, aliter , symbolized L,equals the capacity of a cubic decimeter (1L= 1dm3)
Basic units of volume in the English system are the cubic foot(1ft3), the cubic yard (1yd3), andthe cubic inch (1in3). For liquid measures we use the gallon and the quart.
The volume of a sphere is ‘ , wherer is the radius of the sphere.
Example 9: Find the volume of a sphere with diameter 16 inches.
Mass
Mass is a quantity of matter.Weight is a force exerted by gravitational pull. On Earth, the termsare commonly interchanged.
In the metric system, the fundamental unit for mass is thegram, denoted g. A paper clip and athumbtack each have a mass of about 1 gram. One mL of water weighs about 1 gram.
Example 10: How many liters of water can a 90 cm by 160 cm by 65 cm rectangular prism hold? What isthe mass in kilograms?
To measure temperature in the metric system thedegree Celsiusis used. To measure temperature inthe English System, theFahrenheit scaleis used. These two scales have the following relationship:
Example 11: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms ofCelsius temperature. Use it to convert 65◦ C to ◦F.
Example 12: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms ofFahrenheit. Use it to convert 100◦F to ◦C.