Measurement: Length, Area, and Volume 10.1 The Measurement Process 10.2Area and Perimeter 10.3The...
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Transcript of Measurement: Length, Area, and Volume 10.1 The Measurement Process 10.2Area and Perimeter 10.3The...
Measurement: Length, Area, and Volume
10.1 The Measurement Process10.2 Area and Perimeter10.3 The Pythagorean Theorem10.4 Volume10.5 Surface Area
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
10.1
The Measurement Process
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-2
THE MEASUREMENT PROCESS
1. Choose the property (length, area, volume, etc.) to be measured.
2. Select a unit of measurement.
3. Compare the size of the object with the size of the unit.
4. Express the measurement as the number of units used.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-3
THE MEASUREMENT PROCESS
A measurement is most often only an approximation, and decisions must be made to choose appropriate measurement tools and units to provide accuracy and precision.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-4
EARLY MEASUREMENT
Units of measurement originally were defined for convenience rather than accuracy.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-5
THE U.S. CUSTOMARY SYSTEM
The U.S. customary system of measurement is also known as the “English” system.
Learning the customary system requires extensive memorization.
Using the system involves computations with cumbersome numerical factors.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-6
CUSTOMARY UNITS OF LENGTH
12 inches = 1 foot
3 feet = 1 yard
16.5 feet = 1 rod
660 feet = 1 furlong
5280 feet = 1 mile
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-7
CUSTOMARY UNITS OF AREA
21 ft
2
2 2
1 ft
12 in 144 in
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-8
CUSTOMARY UNITS OF AREA
21 yd
2
2 2
1 yd
3 ft 9 ft
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-9
CUSTOMARY UNITS OF VOLUME
31 ft
3
3 3
1 ft
12 in 1728 in
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-10
CUSTOMARY UNITS OF VOLUME
31 yd
3
3 3
1 yd
3 ft 27 ft
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-11
CUSTOMARY UNITS OF CAPACITY
3 teaspoons = 1 tablespoon
1 tablespoon = ½ fluid ounce
8 fluid ounces = 1 cup
4 cups = 1 quart
4 quarts = 1 gallon
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-12
THE METRIC SYSTEM
The metric system of measurement is also known as the SI system, after its French name, Systeme Internationale.
The principal advantage of the metric system – other than its universality – is the ease of comparison of units. The ratio of one unit to another is always a power of 10.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-13
FUNDAMENTAL UNITS IN THE METRIC SYSTEM
Length meter (m)
Area square meter (m2)square kilometer (km2)
Volume cubic centimeter (cm3)cubic meter (m3)liter (L)
Weight kilogram (kg)
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-14
THE SI DECIMAL PREFIXES
PREFIX FACTOR SYMBOL
kilo 1000 = 103 k
hecto 100 = 102 h
deka (or deca) 10 = 101 da
(none for basic unit) 1 = 100 (none)
deci 0.1= 10-1 d
centi 0.01 = 10-2 c
milli 0.001= 10-3 m
micro 0.000001= 10-6 (mu)
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-15
Example 10.3: Changing Units in the Metric System
Convert each of these measurements to the unit shown:
a. 1495 mm = ________ m
b. 29.5 cm = _________ mm
c. 38.741 m = ________ km
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-16
1.495
294
38.741
Example 10.4: Estimating Weights in the Metric System
Match each term to the approximate weight of the item taken from the list that follows:
a. Nickelb. Compact automobilec. Two-liter bottle of sodad. Recommended daily allowance of vitamin B-6e. Size D batteryf. Large watermelon
List: 2 mg, 2 kg, 100g, 120kg, 9 kg, 5g
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-17
5 g1200 kg
2 kg2 mg
100 g9 kg
TEMPERATURE
FAHRENHEIT SCALE
CELSIUS SCALE
freezing pt of water 32 0boiling pt of water 212 100
10032
180C F 180
32100
F C
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-18
Example 10.5: Computing Speeds and Capacity with Units
A cheetah can run 60 miles per hour. What is the speed in feet per second?
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-19
mi60
hr
mi 5280 ft 1 hr 1 min= 60
hr 1 mi 60 min 60 sec
60 5280 ft=
60 60 sec
ft= 88
sec
10.2
Area and Perimeter
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-20
AREA OF A REGION IN THE PLANE
Let R be a region and assume that a unit of area is chosen. The number of units required to cover a region in the plane without overlap is area of the region R.
Usually, squares are chosen to define a unit of area, but any shape that tiles the plane can serve equally well.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-21
AREA OF A RECTANGLE
A rectangle of length l and width w has area A given by the formula A = lw.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-22
The are of a rectangle is the product of its length and width.
AREA OF A PARALLELOGRAM
A parallelogram of base b and altitude h has area A given by A = bh.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-23
Example 10.9: Using the Parallelogram Area Formula
Find the area of the parallelogram.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-24
The base is 10 cm and the height is 4 cm, so the area is A = (10 cm)(4 cm) = 40 cm2..
Example 10.9: Using the Parallelogram Area Formula
Find the area of the parallelogram.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-25
The area is A = (30 cm)(12 cm) = 36 cm2..
AREA OF A TRIANGLE
A triangle of base b and altitude h has area A given by A = ½ bh.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-26
Example 10.10: Using the Triangle Area Formula
Find the area of the triangle.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-27
Using the formula 2110 cm 7 cm 35 cm .
2A
AREA OF A TRAPEZOID
A trapezoid with bases of length a and b and altitude h has area A = ½(a + b)h.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-28
PERIMETER OF A REGION
If a region is bounded by a simple closed curve, then the perimeter of the region is the length of the curve. More generally, the perimeter of a region is the length of its boundary.
Perimeter is a length measurement.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-29
DEFINITION:
The ratio of the circumference C to the diameter d of a circle is .
Therefore,
and .C
C dd
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-30
Example 10.14: Calculating the Equatorial Circumference of the Earth
The equatorial diameter of the earth is 7926 miles. Calculate the distance around the earth at the equator, using the following for pi.
a. 3.14 b. 3.1416
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-31
a. (3.14)(7926 miles) = 24,887.64 miles
b. (3.1416)(7926 miles) = 24,900.322 miles
The two different approximations for pi account for the distance of about 12.7 miles in the answers.
DEFINITION:AREA OF A CIRCLE
The area A enclosed by a circle of radius r is 2.A r
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-32
Example 10.15: Determining the Size of a Pizza
A 14-inch pizza has the same thickness as a 10-inch pizza. How many times more ingredients are there on the larger pizza?
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-33
Pizzas are measured by their diameters. So the radii of the two pizzas are 7-inch and 5-inch, respectively. Since the thicknesses are the same, the amount of ingredients used is proportional to the areas of the pizza.
p
Example 10.15: continued
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-34
Larger pizza =
Smaller pizza =
2 27 in 49 in
2 25 in 25 in
The ratio of areas is 2 249 in / 25 in 1.96.
The 14-inch pizza has about twice the ingredients of the 10-inch pizza.
10.3
The Pythagorean Theorem
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-35
THE PYTHAGOREAN THEOREM
If a right triangle has legs of length a and b and its hypotenuse has length c, then
2 2 2.a b c
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-36
PROVING THE PYTHAGOREAN THEOREM
The sum of the areas of the squares on the leg of a right triangle is equal to the area of the square on the hypotenuse.
2 2 2.a b c
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-37
THE CONVERSE OFTHE PYTHAGOREAN THEOREM
Let a triangle have sides of length a, b, and c.
If , then the triangle is a right triangle and the angle opposite the side of length c is its right angle.
2 2 2a b c
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-38
Example 10.16: Using the Pythagorean Theorem
Find the lengths x and y in the figure.
2 2 213 37
169 1369
1538
x
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-39
1538 39.2.x Therefore,
Example 10.18: Checking for Right Triangles
Determine whether the three lengths given are the lengths of the sides of a right triangle.a. 15, 17, 18 b. 10, 5, 5 3
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-40
a. 82 + 152 = 64 + 225 = 289 = 172
The lengths are the sides of a right triangle.
b. The lengths are the sides of a right triangle.
22 25 + 5 3 25 25 3 25 75 100 10
10.4
Volume
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-41
SURFACE AREA AND VOLUME
The surface area of a figure in space measures the boundary of the space figure.
The volume measures the amount of space enclosed within the boundary.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-42
Example 10.19: Computing the Volume of a Right Prism and A Right Cylinder
Find the volume of the gift box.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-43
The base area, B, consists of a square of length 20 cm and four 5-cm-by-20-cm rectangles, so B = 800 cm2. The height is h = 10 cm, so the volume is V = Bh = 8000 cm3, which can also be expressed as 8 liters.
Example 10.19: continued
Find the volume of the juice can.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-44
The height is h = 9.5 cm, so the volume V = Bh =
2 22.75 cm 7.5625 cm .
3 371.84375 cm , or about 226 cm .
The area of the circular base of the juice can is
VOLUME OF A GENERAL PRISM OR CYLINDER
A prism or cylinder of height h and base area of B has volume
.V Bh
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-45
VOLUME OF A PYRAMID
V Bh
A cube can be dissected into three congruent pyramids. 1
3V Bh
Volume of a cube.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-46
VOLUME OF A PYRAMID OR CONE
The volume of a pyramid or cone of height h and base area of B is given by
1.
3V Bh
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-47
Example 10.20: Determining the Volume of an Egyptian Pyramid
The pyramid of Khufu is 147 m high, and its square base is 231 m on each side. What is the volume of the pyramid?
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-48
The area of the base is (231 m)2. Therefore, the volume is
2 3153.361 m 147 m 2,614.689 m .
3
VOLUME OF A SPHERE
The volume of a sphere of radius r is given by
34.
3V r
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-49
10.5
Surface Area
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-50
DISSECTING A RIGHT PRISM
Two congruent bases
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-51
Example 10.22: Finding the Surface Area of a Prism-Shaped Gift Box
The height of the gift box is 10 cm, the longer edges are 20 cm long, and the short edges of the square corner cutouts are each 5 cm long. What is the surface area of the box?
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-52
Example 10.22: continued
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-53
Each base has the area B = 800 cm2. The lateral area is that of a rectangle 10 cm high and 120 cm long. That is, the lateral surface area is 1200 cm2. Altogether, the surface area of the box is SA= 2 × 800 cm2 + 1200 cm2 = 2800 cm2
SURFACE AREA OF A RIGHT CIRCULAR CONE
Let a right circular cone have slant height s and a base of radius r .
Then the surface area SA of the cone is given by
2 .SA r rs
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-54
SURFACE AREA OF A SPHERE
The surface area of a sphere of radius r is given by
24 .S r
Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-55