4-1 Study Guide and Intervention 8 Chapter 4... · Study Guide and Intervention Ratios and Rates 4-1 ... Express this ratio in simplest form. 2. ... For each exercise below, rates
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Chapter 4 10 Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Rates are useful and meaningful when expressed as a unit rate.For example, which is the better buy—one orange for $0.29 or12 oranges for $3.00?
To find the unit rate for 12 oranges, divide $3.00 by 12. Theresult is $0.25 per orange. If a shopper needs to buy at least12 oranges, then 12 oranges for $3.00 is the better buy.
For each exercise below, rates are given in Column A and Column B. In the blank next to each exercise number, write the letter of the column that contains the better buy.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
OR
AN
GE
S
Column A Column B
1 apple for $0.19 3 apples for $0.59
20 pounds of pet food for $14.99 50 pounds of pet food for $37.99
A car that travels 308 miles on11 gallons of gasoline
A car that travels 406 miles on14 gallons of gasoline
10 floppy discs for $8.99 25 floppy discs for $19.75
1-gallon can of paint for $13.995-gallon bucket of paint for$67.45
84 ounces of liquid detergentfor $10.64
48 ounces of liquid detergentfor $6.19
5,000 square feet of lawn foodfor $11.99
12,500 square feet of lawn foodfor $29.99
2 compact discs for $26.50 3 compact discs for $40.00
8 pencils for $0.99 12 pencils for $1.49
1,000 sheets of computer paperfor $8.95
5,000 sheets of computer paperfor $41.99
Exercises
Chapter 4 15 Course 3
TI-83/84 Plus Activity
Ratios and Rates
NAME ________________________________________ DATE ______________ PERIOD _____
You can simplify many ratios and rates mentally. A graphingcalculator can help you simplify more difficult ratios and rates byusing the FRAC mode under the Math menu.
Express the ratio 72 to 216 in simplest form.
Enter: 72 216 1 1/3
The ratio 72 to 216 in simplest form is }1
3}.
If the ratio or rate cannot be simplified, the input numbers will appear as theanswer in fraction form, or the decimal equivalent may be given as theanswer.
Express the ratio 25 inches to 4 yards in simplest form.(Remember: Both measurements must have the same unit.)
Enter: 25 144 1 25/144
The ratio of 25 inches to 4 yards does not simplify.
Express the ratio 50,000,000 people to 3,237 magazine subscribers in simplest form.
Enter: 50000000 3237 1 15446.40099
The ratio 50,000,000 people to 3,237 subscribers cannot
be simplified.
Express each ratio or rate in simplest form.
1. 88 to 104 2. 16 out of 228 3. 84:56
4. 35 inches to 2 yards 5. 115 miles per 5 gallons 6. 76.5 meters in 8.5 seconds
7. 3,000,000 seconds of television advertisements to 14,897 new customers
ENTERMATH
ENTERMATH
ENTERMATH
Example 1
Example 3
4-1
Get Ready for the Lesson
Read the introduction at the top of page 194 in your textbook.
Write your answers below.
1. Copy and complete the table to determine the cost for different numbers
of pizzas ordered.
2. For each number of pizzas, write the relationship of the cost and number
of pizzas as a ratio in simplest form. What do you notice?
Read the Lesson
3. At Better Shirts, an order of 10 printed T-shirts is $45 and an order of
250 printed T-shirts is $875. What must you do before you can compare
the ratios to see if they are proportional?
4. What must be true in order for the ratios to be proportional?
5. What are the simplified ratios for the T-shirt orders? Are the ratios
proportional or nonproportional?
Remember What You Learned
6. A delivery service charges $7 per package delivered locally. There is also a $2 service
charge for registering an order of packages for any number of packages. Create a table
to show what the costs of sending 1, 2, 3, and 4 packages are, using the service. Is the
relationship between total cost and number of packages proportional or
nonproportional? Explain your reasoning.
Chapter 4 16 Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
The cost of one CD at a record store is $12. Create a table to showthe total cost for different numbers of CDs. Is the total cost proportional to thenumber of CDs purchased?
Total Cost 5
125
245
365
485 $12 per CD Divide the total cost for each by the number
Number of CDs 1 2 3 4 of CDs to find a ratio. Compare the ratios.
Since the ratios are the same, the total cost is proportional to the number of CDs purchased.
The cost to rent a lane at a bowling alley is $9 per hour plus $4 forshoe rental. Create a table to show the total cost for each hour a bowling lane isrented if one person rents shoes. Is the total cost proportional to the number ofhours rented?
Total Cost →
13 or 13 22 or 11 31 or 10.34 40 or 10 Divide each cost by the
Number of Hours 1 2 3 4 number of hours.
Since the ratios are not the same, the total cost is nonproportional to the number of hoursrented with shoes.
Use a table of values to explain your reasoning.
1. PICTURES A photo developer charges $0.25 per photo developed. Is the total cost
proportional to the number of photos developed?
2. SOCCER A soccer club has 15 players for every team, with the exception of two teams
that have 16 players each. Is the number of players proportional to the number of
teams?
NAME ________________________________________ DATE ______________ PERIOD _____
Stores are able to make a profit by buying in bulk. They are charged less per item becausethey agree to buy a large number of items. When the unit price of bulk items is less thanthe unit price for a small order, the rates are nonproportional.
For each exercise below, the number of units and the total cost of the units isgiven in Column A and Column B. In the blank next to each exercise number,write whether the columns are proportional or nonproportional.
Column A Column B
1. ________ 100 T-shirts printed for $428.00 3,000 T-shirts printed for $12,180.00
2. ________ 3 compact discs for $23.34 10,000 compact discs for $32,900.00
3. ________ 8 books for $43.20 250 books for $1,350.00
4. ________ $546.00 earned in 42 hours $31,200.00 earned in 2,400 hours
5. ________ 36 photos printed for $7.20 800 photos printed for $96.00
6. ________ 1 bus token for $1.75 20 bus tokens for $34.00
7. ________ $3,840.00 earned in 160 hours $52,000 earned in 2,000 hours
8. ________ $730.00 earned in 40 hours $9,125.00 earned in 500 hours
9. ________ 1 bagel for $0.65 13 bagels for $7.80
10. _______ a series of 5 concert tickets for a series of 20 concert tickets for $187.50 $685.00
NAME ________________________________________ DATE ______________ PERIOD _____
Study Guide and Intervention
Rate of Change
4-3
Example INCOME The graph shows Mr. Jackson’s annual income between 1998 and 2006. Find the rate ofchange in Mr. Jackson’s income between 1998 and 2001.
Use the formula for the rate of change.Let (x1, y1) 5 (1998, 48,500) and (x2, y2) 5 (2001, 53,000).
}y
x2
2
–
–
y
x1
1} 5 }
532,000001
2
2
4189,95800
} Write the formula for rate of change.
5 }4,5
300} Simplify.
5 }1,5
100} Express this rate as a unit rate.
Between 1998 and 2001, Mr. Jackson’s income increased an average of $1,500 per year.
SURF For Exercises 1–3, use the graph that shows the average daily wave height as measured by an ocean buoy over a nine-day period.
1. Find the rate of change in the average daily
wave height between day 1 and day 3.
2. Find the rate of change in the average daily
wave height between day 3 and day 7.
3. Find the rate of change in the average daily
wave height between day 7 and day 9.
y
x
Wav
e H
eigh
t
7
9
11
13
15
1 3 5 7 9
Day
Wave Height
(7, 14)
(9, 11)
(3, 12)
(1, 8)
0
'98 '00 '02 '04 '06
50,000
45,000
55,000
60,000
65,000
Annual
Inco
me
($)
Year
Mr. Jackson's Income
y
x
1998, 48,500
2006, 57,000
2001, 53,000
0
To find the rate of change between two data points, divide the difference of the y-coordinates by the
difference of the x-coordinates. The rate of change between (x1, y1) and (x2, y2) is }x
NAME ________________________________________ DATE ______________ PERIOD _____
Enrichment4-3
Analyzing Graphs
A graph can be used to represent many real-life situations. Graphs such asthese often have time as the dimension on the horizontal axis. By analyzingthe rate of change of different parts of a graph, you can draw conclusionsabout what was happening in the real-life situation at that time.
TRAVEL The graph at the right represents the speedof a car as it travels along the road. Describe whatis happening in the graph.
• At the origin, the car is stopped.
• Where the line shows a fast, positive rate of
change, the car is speeding up.
• Then the car is going at a constant speed,
shown by the horizontal part of the graph.
• The car is slowing down where the graph
shows a negative rate of change.
• The car stops and stays still for a short time. Then it speeds up again.
The starting and stopping process repeats continually.
Analyze each graph.
1. Ashley is riding her bicycle along a scenic trail.
Describe what is happening in the graph.
2. The graph shows the depth of water in a pond as you
NAME ________________________________________ DATE ______________ PERIOD _____
Study Guide and Intervention
Constant Rate of Change
4-4
Exercises
The table shows the relationship between feet and seconds. Is therelationship between feet and seconds linear? If so, find theconstant rate of change. If not, explain your reasoning.
rate of change 5 }se
fceoentds
} }cchhaannggee
iinn
xy
}
5 }2
34} or 2}
34
}
The rate of change is 2}34
} feet per second.
Find the constant rate of change for the number of feet per second.Interpret its meaning.
Choose two points on the line. The vertical change from pointA to point B is 4 units while the horizontal change is 2 units.
rate of change 5 }se
f
c
e
o
e
n
t
ds} Definition of rate of change
5 }42
} Difference in feet between two points
divided by the difference in seconds
for those two points
5 2 Simplify.
The rate of change is 2 feet per second.
Find the rate of change for each line.
1. 2. 3.
The points given in each table lie on a line. Find the rate of change for the line.
4. 5.
y
xO
y
xO
y
xO
The slope of a line is the ratio of the rise, or vertical change, to the run, or horizontal change.
NAME ________________________________________ DATE ______________ PERIOD _____
4-4
1. $10 per hour
3. No, the change in number of
magazines sold is not constant
5. 3 hours per volunteer
9. 228 F per hour; the temperature if
decreasing by 28 F per hour.
2. 48 per hour
4. No, the change in number of apples
per tree is not constant
6. decreasing 2 gallons per 50 miles; or 1
gallon per 25 miles
10. 3 lbs per person; Three pounds of meat
are needed per person
Time Temperature
9 60
10 64
11 68
12 72
Number of Students Number ofMagazines Sold
10 100
15 110
20 200
25 240
Number of Trees Number of Apples
5 100
10 120
15 150
20 130
Gas Left in Tank Miles Driven
12 0
10 50
8 100
6 150
Number of Number ofVolunteers Hours Logged
5 15
10 30
15 45
20 60
O
10
2 4 6 8 10
20
30
40
50
60
70
80
90
Tem
per
ature
Hours After Storm
O
3
21 3 54 6 8 10
6
9
12
15
18
21
Poun
d o
f M
eal
Number of People
7 9
Determine whether the relationship between the two quantities described in eachtable is linear. If so, find the constant rate of change. If not, explain your reasoning.
Find the constant rate of change for each graph and interpret its meaning.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice
Constant Rate of Change
4-4
Determine whether the relationship between the two quantitiesdescribed in each table is linear. If so, find the constant rate ofchange. If not, explain your reasoning.
1. Fabric Needed for Costumes 2. Distance Traveled on Bike Trip
For Exercises 3 and 4, refer to the graphs below.
3. 4.
Day 1 2 3 4
Distance (mi) 21.8 43.6 68.8 90.6
Number of Costumes
Fabric (yd)
2 4 6 8
7 14 21 28
a. Find the constant rate of change andinterpret its meaning.
b. Determine whether a proportionallinear relationship exists between thetwo quantities shown in the graph.Explain your reasoning.
Hawk Diving Toward Prey
Alt
itude
(ft.
)
Time (s)
100
80
60
40
20
02 4 6 8 10
x
y
Book Sales Sa
les
($)
Day
5,000
4,000
3,000
2,000
1,000
02 4 6 8 10
x
y
a. Find the constant rate of change andinterpret its meaning.
b. Determine whether a proportionallinear relationship exists between thetwo quantities shown in the graph.Explain your reasoning.
NAME ________________________________________ DATE ______________ PERIOD _____
4-4
FLOWERS For Exercises 1 and 2, use LONG DISTANCE For Exercises 3–6, use the graph that shows the depth of the the graph that compares the costs of water in a vase of flowers over 8 days. long distance phone calls with three
different companies.
y
x
0
1
1 2 3 4 5 6 7 8 9 10
2
3
4
5
6
7
8
9
10
Depth of Water in Vase
Dep
th (
in.)
Day
1. Find the rate of change for the line. 2. Interpret the difference between depth
NAME ________________________________________ DATE ______________ PERIOD _____
Enrichment4-4
Chien-Shiung Wu
American physicist Chien-Shiung Wu (1912–1997) was born in Shanghai, China. In 1936, she came to the United States to further her studies in science. She received her doctorate in physics in 1940 from the University of California, and became known as one of the world’s leading physicists. In 1975, she was awarded the National Medal of Science.
Wu is most famous for an experiment that she conducted in 1957. The outcome of the experiment was considered the most significant discovery in physics in more than seventy years. The exercise below will help you learn some facts about it.
The points given in each table lie on a line. Find the rate of change ofthe line. The word or phrase following the solution will complete thestatement correctly.
1. 2.
At the time of the experiment, Wu The site of the experiment was the
was a professor at . in Washington, D.C.
rate of change 5 3: Columbia University rate of change 5 0: National Bureau of
rate of change 5 1: Stanford University Standards
rate of change 5 3: Smithsonian
Institution
3. 4.
The experiment involved a substance In the experiment, the substance was
called . cooled to .
rate of change 5 2: carbon 14 rate of change 5 }43
}: 22738C
rate of change 5 1: cobalt 60 rate of change 5 }34
A national census is taken every ten years. The 1990 census revealed thatthere were about 250,000,000 people in the United States, and that about 8 out of 100 of these people were 5–13 years old. To find the number ofpeople in the United States that were 5–13 years old, use the ratio ofpeople 5–13 years old and create a proportion.
}1800} 5 }
250,0n00,000}
To solve the proportion, find cross products.
8 3 250,000,000 5 2,000,000,000 and n 3 100 5 100n
Then divide: 2,000,000,000 4 100 5 20,000,000.
In 1990, about 20,000,000 people in the United States were 5–13 years old.
Use the approximate ratios in each exercise to create a proportion,given that there were about 250,000,000 people in the United States.Then solve and choose the correct answer from the choices at the right.
1. The United States is a diverse collection of different ________ A. 200,000,000
races and ethnic origins. Asians or Pacific Islanders
accounted for about }1
300} of the population of the
United States. About how many people of Asian or
Pacific-Island origin lived in the United States?
2. African-Americans accounted for about }235} of the ________ B. 22,500,000
population of the United States. About how many
African-American people lived in the United States?
3. People of Hispanic origin accounted for about }1
900} ________ C. 7,500,000
of the population of the United States. About how
many people of Hispanic origin lived in the
United States?
4. Caucasian people accounted for about }45
} of the ________ D. 2,000,000
population of the United States. About how many
people of white or Caucasian origin lived in the
United States?
5. People of American-Indian, Eskimo, or Aluet origin ______ E. 30,000,000
accounted for about }1,0
800} of the population of the
United States. About how many people of
American-Indian, Eskimo, or Aluet origin lived in
the United States?
4-5
Exercises
Example
Chapter 4 40 Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
It takes a worker 4 minutes to stack 2 rows of 8 boxes in awarehouse. How long will it take to stack 8 rows of 8 boxes? Use the draw adiagram strategy to solve the problem.
Understand After 4 minutes, a worker has stacked a 2 rows of 8 boxes. At this rate, howlong would it take to stack 8 rows of boxes?
Plan Draw a diagram showing the level of boxes after 4 minutes.
Solve 2 rows of 8 boxes 5 4 minutes8 rows 5 4 3 2 rows, so multiply the time by 4.4 3 4 minutes 5 16 minutes
Check
8 boxes 3 2 rows of boxes 5 16 boxes Multiply to find the total number of boxes in the stack.
4 minutes 4 16 boxes 5 0.25 min. per box Divide the number of minutes by the number of boxes.8 boxes 3 8 rows of boxes 5 64 boxes Multiply to find the number of boxes in the new stack.
64 boxes 3 0.25 min. 5 16 minutes Multiply the number of boxes by the time per box.
It will take 16 minutes to stack an 8 3 8 wall of boxes.
For Exercises 1–4, use the draw a diagram strategy to solve the problem.
1. GAS A car’s gas tank holds 16 gallons. After filling it for 20 seconds, the
tank contains 2.5 gallons. How many more seconds will it take to fill the
tank?
2. TILING It takes 96 tiles to fill a 2-foot by 3-foot rectangle. How many tiles
would it take to fill a 4-foot by 6-foot rectangle?
3. BEVERAGES Four juice cartons can fill 36 glasses of juice equally. How many
juice cartons are needed to fill 126 glasses equally?
4. PACKAGING It takes 5 large shipping boxes to hold 120 boxes of an action
figure. How many action figures would 8 large shipping boxes hold?
Exercises
Study Guide and Intervention
Problem-Solving Investigation: Draw a Diagram
Chapter 4 42 Course 3
For Exercises 1–5, use the draw a diagram strategy to solve the problem.
1. AQUARIUM An aquarium holds 60 gallons of water. After 6 minutes, the
tank has 15 gallons of water in it. How many more minutes will it take to
fill the tank?
2. TILING Meredith has a set of ninety 1-inch tiles. If she starts with one tile,
then surrounds it with a ring of tiles to create a larger square, how many
surrounding rings can she make before she runs out of tiles?
3. SEWING Judith has a 30-yard by 1-yard roll of fabric. She needs to use 1.5
square yards to create one costume. How many costumes can she create?
4. DRIVING It takes 3 gallons of gas to drive 102 miles. How many miles can
be driven on 16 gallons of gas?
5. PACKING Hector can fit 75 compact discs into 5 boxes. How many compact
discs can he fit into 14 boxes?
NAME ________________________________________ DATE ______________ PERIOD _____
Use the draw a diagram strategy tosolve Exercises 1 and 2.
1. SWIMMING Jon is separating the widthof the swimming pool into equal-sizedlanes with rope. It took him 30 minutesto create 6 equal-sized lanes. How longwould it take him to create 4 equal-sized lanes in a similar swimming pool?
2. TRAVEL Two planes are flying from SanFrancisco to Chicago, a distance of 1,800miles. They leave San Francisco at thesame time. After 30 minutes, one planehas traveled 25 more miles than theother plane. How much longer will ittake the slower plane to get to Chicagothan the faster plane if the faster planeis traveling at 500 miles per hour?
Use any strategy to solve Exercises 3–6.Some strategies are shown below.
3. TALENT SHOW In a solo singing andpiano playing show, 18 people sang and14 played piano. Six people both sangand played piano. How many peoplewere in the singing and piano playingshow?
4. LETTERS Suppose you have three stripsof paper as shown. How many capitalletters of the alphabet could you formusing one or more of these three stripsfor each letter? List them according tothe number of strips.
5. CLOTHING A store has 255 wool ponchosto sell. There are 112 adult-sizedponchos that sell for $45 each. The restare kid-sized and sell for $32 each. Ifthe store sells all the ponchos, howmuch money will the store receive?
6. DINOSAURS Brad mad a model of aStegosaurus. If you multiply the model'slength by 8 and subtract 4, you will findthe length of an average Stegosaurus. Ifthe actual Stegosaurus is 30 ft long, howlong is Brad’s model.
NAME ________________________________________ DATE ______________ PERIOD _____
If a 4-inch by 5-inch photograph is enlarged to an 8-inch by 10-inchphotograph, the photographs are said to be similar. They have the sameshape, but they do not have the same size. If a 4-inch by 5-inch photographis duplicated and a new 4-inch by 5-inch photograph is made, thephotographs are said to be congruent. They have the same size and shape.
In each exercise, identify which of the triangles are congruent toeach other and identify which of the shapes are similar to each other.Use the symbol for > congruence and the symbol , for similarity.
1. 2.
3. 4.
5.
a
b
c
d
f
a
b
c
d
AB
C
D
J L
H NK M
A B C D
H K LM N
P
Q
R T
AB
T
C
D
Q
G
H
J
LKM
NP
Example
Chapter 4 51 Course 3
TI-83/84 Plus Activity
Similar Polygons
NAME ________________________________________ DATE ______________ PERIOD _____
Segment M9N9 is a dilation of segment MN. Find the scale factor of the dilation, and classify it as an enlargement or a reduction.
Write the ratio of the x- or y-coordinate of one vertex of thedilated figure to the x- or y-coordinate of the correspondingvertex of the original figure. Use the x-coordinates of N(1, 22)and N9(2, 24).
5 }21
} or 2
The scale factor is 2. Since the image is larger than the originalfigure, the dilation is an enlargement.
1. Polygon ABCD has vertices A(2, 4), B(21, 5), C(23, 25), and
D(3, 24). Find the coordinates of its image after a dilation
with a scale factor of }12
}. Then graph polygon ABCD and its
dilation.
2. Segment P9Q9 is a dilation of segment PQ. Find the scale
factor of the dilation, and classify it as an enlargement or a
reduction.
x-coordinate of point N9}}}x-coordinate of point N
y
xO
C
B
B9
A9 C9A
y
xO
M9M
N
N9
y
xO
y
xO
PP9
Q9 Q
Example 1
Example 2
Exercises
The image produced by enlarging or reducing a figure is called a dilation.
Chapter 4 54 Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Find the coordinates of the vertices of polygon F9G9H9J9 afterpolygon FGHJ is dilated using the given scale factor. Then graphpolygon FGHJ and polygon F9G9H9J9.
In the exercises below, figure R9S9T9 is a dilation of figure RST andfigure A9B9C9D9 is a dilation of figure ABCD. Find the scale factor ofeach dilation and classify it as an enlargement or as a reduction.
5. 6.
7. GLASS BLOWING The diameter of a vase is now 4 centimeters. If the diameter increases
by a factor of }7
3}, what will be the diameter then?
3}4
C
B
A
x
y
R
R
S
S
TT x
y
‘
‘
‘
O
A
B
D
D
C
x
y
‘ C‘
B‘A‘
O
x
y
O 4
4
8
8
-4
-4
-8
-8
N
L
P
O
M
Chapter 4 56 Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
A dilation of a shape creates a new shape that is similar to theoriginal. The ratio of the new image to the original is called thescale factor.
Plot and draw each shape. Then perform the dilation ofeach shape using a scale factor of two. After the new imagehas been drawn, determine the area of both the originalshape and its dilation.
Use Lists and Plots to draw figures and their dilations. Enter thecoordinates of one polygon. Then multiply those coordinates by ascale factor. Plot both sets of coordinates.
Graph nABC with vertices at A(1, 1), B(3, 5),and C(2, 8). Graph an image of nABC
with the scale factor of 3.
Step 1 Clear all lists and open the List feature.
[MEM] 6
Step 2 In L1, enter the x-coordinates of the vertices.In L2, enter the y-coordinates. Repeat the first ordered pair at the end of the list. Pressafter each value.
Step 3 In L3, enter a formula to multiply the x-coordinates in L1 by 3.
[TEXT] “Done [STAT] 1
3 [TEXT] “Done
Step 4 In L4, enter a formula to multiply the y-coordinates in L2 by 3.
[TEXT] “Done [STAT] 2 3 [TEXT]
“Done
Step 5 Create two plots.
[PLOT] 1 Turn on Plot 1. Choose line graph. Choose L1 and L2.
[PLOT] 2 Turn on Plot 2. Choose line graph. Choose L3 and L4.
Step 6 Set the graph window and display the plots.
7
Use your calculator to draw the polygons and their dilations. Sketch thedrawings. Label the vertices and coordinates.
LIGHTING George is standing next to alightpole in the middle of the day. George’s shadow is 1.5 feet long, and the lightpole’s shadow is 4.5 feet long. If George is 6 feet tall, how tall is the lightpole?
Write a proportion and solve.
George’s shadow → 1.5 6 George’s height
lightpole’s shadow → 4.5 h lightpole’s height
1.5 ? h 5 4.5 ? 6 Find the cross products.
1.5h 5 27 Multiply.
}11.5.5h
} 5 }12.75} Divide each side by 1.5.
h 5 18 Simplify.
The lightpole is 18 feet tall.
1. MONUMENTS A statue casts a shadow 30 feet
long. At the same time, a person who is 5 feet
tall casts a shadow that is 6 feet long. How
tall is the statue?
2. BUILDINGS A building casts a shadow 72 meters
long. At the same time, a parking meter that is
1.2 meters tall casts a shadow that is 0.8 meter
long. How tall is the building?
3. SURVEYING The two
triangles shown in the
figure are similar. Find
the distance d across
Red River.
0.9 km
1 km
1.8 km
h ft
30 ft
5 ft
6 ft
Indirect measurement allows you to find distances or lengths that are difficult to measure directly
using the properties of similar polygons.
h ft
6 ft
4.5 ft 1.5 ft
Example
Exercises
Chapter 4 61 Course 3
Skills Practice
Indirect Measurement
NAME ________________________________________ DATE ______________ PERIOD _____
A proportion can be used to determine the heightof tall structures if three variables of the proportionare known. The three known variables are usuallythe height a of the observer, the length b of theobserver’s shadow, and the length d of thestructure’s shadow. However, a proportion can besolved given any three of the four variables.
This chart contains information about various observers and tallbuildings. Use proportions and your calculator to complete the chartof tall buildings of the world.
a
b
c
d
ab
cd
5
Height ofObserver
Length ofShadow
BuildingLocation
Height ofBuilding
Length ofShadow
1. 5 ft 8 in.Natwest,London
80 ft
2. 4 ft 9 in.Columbia SeafirstCenter, Seattle
954 ft 318 ft
3. 5 ft 10 in. 14 in.Wachovia Building,Winston-Salem
NAME ________________________________________ DATE ______________ PERIOD _____
4-10
Exercises
Example 1 INTERIOR DESIGN A designer has made
a scale drawing of a living room for one of her
clients. The scale of the drawing is 1 inch 5 1}13
}
feet. On the drawing, the sofa is 6 inches long.
Find the actual length of the sofa.
Let x represent the actual length of the sofa. Write and solve a proportion.
The actual length of the sofa is 8 feet.
Find the scale factor for the drawing in Example 1.
Write the ratio of 1 inch to 1}13
} feet in simplest form.
5 }116
iinn.@.@}
Convert 1}13
} feet to inches.
The scale factor is }116} or 1:16. This means that each distance on the drawing is }
116} the
actual distance.
LANDSCAPING Yutaka has made a scale drawingof his yard. The scale of the drawing is1 centimeter 5 0.5 meter.
1. The length of the patio is 4.5 centimeters in the
drawing. Find the actual length.
2. The actual distance between the water faucet and
the pear tree is 11.2 meters. Find the corresponding
distance on the drawing.
3. Find the scale factor for the drawing.
1 in.}
1}13
} ft
To find the scale factor for scale drawings and models, write the ratio given by the scale in simplest
form.
1 in. 6 in.
1 ft1
3
x ft5
1 · x 5 1 · 61
3
x 5 8
Find the cross products.
Simplify.
Drawing Scale Actual Length
1 in. = 1 ft
Sofa
13
1 cm = 0.5 m
Patio
PondPath
Water Faucet
PearTree
Garden
Distances on a scale drawing or model are proportional to real-life distances. The scale is determined
by the ratio of a given length on a drawing or model to its corresponding actual length.
Example 2
Chapter 4 67 Course 3
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on
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NAME ________________________________________ DATE ______________ PERIOD _____
Skills Practice
Scale Drawings and Models
4-10
ARCHITECTURE The scale on a set of architectural drawings for a houseis 1.5 inches 5 2 feet. Find the length of each part of the house.
7. What is the scale factor of these drawings?
TOWN PLANNING For Exercises 8–11, use the following information.
As part of a downtown renewal project, businesses have constructeda scale model of the town square to present to the city commissionfor its approval. The scale of the model is 1 inch 5 7 feet.
8. The courthouse is the tallest building in the town square. If it is
5}12
} inches tall in the model, how tall is the actual building?
9. The business owners would like to install new lampposts that are
each 12 feet tall. How tall are the lampposts in the model?
10. In the model, the lampposts are 3}37
} inches apart. How far apart
will they be when they are installed?
11. What is the scale factor?
12. MAPS On a map, two cities are 6}12
} inches apart. The actual distance
between the cities is 104 miles. What is the scale of the map?
NAME ________________________________________ DATE ______________ PERIOD _____
4-10
LANDSCAPE PLANS For Exercises 1–4, use the drawing and an inch rulerto find the actual length and width of each section of the park.Measure to the nearest eighth of an inch.
1. Playground
2. Restrooms
3. Picnic Area
4. What is the scale factor of the
park plan? Explain its meaning.
5. SPIDERS The smallest spider, the Patu marples of Samoa, is 0.43 millimeter long.
A scale model of this spider is 8 centimeters long. What is the scale of the model?
What is the scale factor of the model?
6. ANIMALS An average adult giraffe is 18 feet tall. A newborn giraffe is about 6 feet tall.
Kayla is building a model of a mother giraffe and her newborn. She wants the model to
be no more than 17 inches high. Choose an appropriate scale for a model of the giraffes.
Then use it to find the height of the mother and the height of the newborn giraffe.
7. TRAVEL On a map, the distance between Charleston and Columbia, South Carolina, is 5
inches. If the scale of the map is }7
8} inch 5 20 miles, about how long would it take the
Garcia family to drive from Charleston to Columbia if they drove 60 miles per hour?
Lawn
Picnic
Area
Dog
Run
Playground
Key1 in. 5 68 ft
Restrooms
Chapter 4 69 Course 3
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on
4–10
NAME ________________________________________ DATE ______________ PERIOD _____
Word Problem Practice
Scale Drawings and Models
4-10
CAMPUS PLANNING For Exercises 1–3, use thefollowing information.
The local school district has made a scale modelof the campus of Engels Middle School includinga proposed new building. The scale of the modelis 1 inch 5 3 feet.
1. An existing gymnasium is 8 inches tall
in the model. How tall is the actual
gymnasium?
2. The new building is 22.5 inches from
the gymnasium in the model. What
will be the actual distance from the
gymnasium to the new building if it
is built?
3. What is the scale factor of the model? 4. MAPS On a map, two cities are 5}34
NAME ________________________________________ DATE ______________ PERIOD _____
4-10
Scale Drawings
The figure at the right has an area of 6 square units. If the figurerepresented a map, and was drawn to a scale of 1 unit 5 3 feet, thelengths of the sides would be 6 ft and 9 ft. So, the figure wouldrepresent an area of 54 square feet.
The ratio of the actual area of the figure to the scale area of thefigure can be expressed as a ratio.
}ascctaulaelaarreeaa
} 5 }564} 5 }
19
} or 1 to 9
Find the actual area and the scale area of these figures. Thendetermine the ratio of actual area to scale area.
1. Scale: 1 unit 5 4 ft 2. Scale: 1 unit 5 50 cm
actual area actual area
scale area scale area
ratio ratio
3. Scale: 1 unit 5 8 mi 4. Scale: 1 unit 5 12 m
actual area actual area
scale area scale area
ratio ratio
5. Scale: 1 unit 5 18 in. 6. Scale: 1 unit 5 6 km
actual area actual area
scale area scale area
ratio ratio
Chapter 4 71 Course 3
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on
4–10
NAME ________________________________________ DATE ______________ PERIOD _____
Spreadsheet Activity
Scale Drawings
4-10
Example
You can use a spreadsheet to calculate the actual measurements froma scale drawing.
The table shows the measurements on anarchitect’s drawing for various rooms in a house. If 2centimeters represents 1.5 feet, find the actualmeasurements for each room.
Step 1 Use the first column for all of the scale measurements.Use the formula bar to enter the numbers. Then press ENTER to move to the next cell.
Step 2 In cell B1, enter an equals sign followed by A1*1.5/2.Then press ENTER to return the actual size of the living room.
Step 3 Click on the bottom right side of cell B1 and drag the formula down through cell B6 to return the actual measurements for the other rooms.
The actual measurements for all of the rooms are 22.5 feet, 18.75 feet,30 feet, 33.75 feet, 18.75 feet,and 31.5 feet.
Find the actual measurements for each scale measurement if 1 inch 5 2 feet.