Square Roots and Irrational Numbers

1
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
A blue cube is 3 times as tall as a red cube. How
many red cubes can fit into the blue cube?
27
11-1
2
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
(For help, go to Lesson 4-2.)
Write the numbers in each list without
exponents. 1. 12, 22, 32, . . ., 122 2. 102,
202, 302, . . ., 1202
Check Skills Youll Need
11-1
3
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Solutions 1. 12, 22, 32, . . . , 122 1, 4,
9, 16, 25, 36, 49, 64, 81, 100, 121,
144 2. 102, 202, 302, . . . , 1202 100,
400, 900, 1,600, 2,500, 3,600, 4,900, 6,400,
8,100, 10,000, 12,100, 14,400
11-1
4
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Simplify each square root.
Quick Check
11-1
5
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
You can use the formula d 1.5h to estimate
the distance d, in miles, to a horizon line when
your eyes are h feet above the ground. Estimate
the distance to the horizon seen by a lifeguard
whose eyes are 20 feet above the ground.
Quick Check
The lifeguard can see about 5 miles to the
horizon.
11-1
6
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Identify each number as rational or irrational.
Explain.
rational, because 49 is a perfect square
b. 0.16
rational, because it is a terminating decimal
irrational, because 3 is not a perfect square
d. 0.3333 . . .
rational, because it is a repeating decimal
irrational, because 15 is not a perfect square
f. 12.69
rational, because it is a terminating decimal
g. 0.1234567 . . .
Quick Check
irrational, because it neither terminates nor
repeats
11-1
7
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Simplify each square root or estimate to the
nearest integer. 1. 100 2.
57 Identify each number as rational or
irrational. 3. 48 4. 0.0125 5. The
formula d 1.5h , where h equals the height,
in feet, of the viewers eyes, estimates the
distance d, in miles, to the horizon from the
viewer. Find the distance to the horizon for a
person whose eyes are 6 ft above the ground.
10
8
irrational
rational
3 mi
11-1
8
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
The Jones Organic Farm has 18 tomato plants and
30 string bean plants. Farmer Jones wants every
row to contain at least two tomato plants and two
bean plants. There should be as many rows as
possible, and all the rows must be the same. How
should Farmer Jones plant the rows?
6 rows, with each row containing 5 bean plants
and 3 tomato plants
11-2
9
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
(For help, go to Lesson 4-2.)
Simplify. 1. 42 62 2. 52 82 3. 72
92 4. 92 32
Check Skills Youll Need
11-2
10
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Solutions 1. 42 62 2. 52 82 16 36
52 25 64 89 3. 72 92 4. 92 32
49 81 130 81 9 90
11-2
11
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find c, the length of the hypotenuse.
The length of the hypotenuse is 35 cm.
Quick Check
11-2
12
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find the value of x in the triangle. Round to
the nearest tenth.
11-2
13
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
(continued)
Quick Check
The value of x is about 12.1 in.
11-2
14
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
The carpentry terms span, rise, and rafter
length are illustrated in the diagram. A
carpenter wants to make a roof that has a span of
20 ft and a rise of 10 ft. What should the rafter
length be?
Quick Check
The rafter length should be about 14.1 ft.
11-2
15
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Is a triangle with sides 10 cm, 24 cm, and 26 cm
a right triangle?
The triangle is a right triangle.
Quick Check
11-2
16
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find the missing length. Round to the nearest
tenth. 1. a 7, b 8, c 2. a 9, c
17, b 3. Is a triangle with sides 6.9
ft, 9.2 ft, and 11.5 ft a right triangle?
Explain. 4. What is the rise of a roof if
the span is 30 ft and the rafter length is 16
ft? Refer to the diagram on page 586.
10.6
14.4
yes 6.92 9.22 11.52
about 5.6 ft
11-2
17
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the number halfway between 0.784 and 0.76.
0.772
11-3
18
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
(For help, go to Lesson 1-10.)
Write the coordinates of each point. 1. A 2. D 3.
G 4. J
Check Skills Youll Need
11-3
19
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Solutions 1. A (3, 4) 2. D (0, 3) 3. G (4,
2) 4. J (3, 1)
11-3
20
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the distance between T(3, 2) and V(8, 3).
The distance between T and V is about 7.1 units.
Quick Check
11-3
21
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the perimeter of WXYZ.
The points are W (3, 2), X (2, 1), Y (4, 0),
Z (1, 5). Use the Distance Formula to find the
side lengths.
11-3
22
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
(continued)
11-3
23
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
(continued)
The perimeter is about 20.1 units.
Quick Check
11-3
24
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the midpoint of TV.
Quick Check
11-3
25
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the length (to the nearest tenth) and
midpoint of each segment with the given
endpoints. 1. A(2, 5) and B(3, 4) 2. D(4,
6) and E(7, 2) 3. Find the perimeter of
ABC, with coordinates A(3, 0), B(0, 4),
and C(3, 0).
16
11-3
26
Problem Solving Strategy Write a Proportion
PRE-ALGEBRA LESSON 11-4
Use these numbers to write as many proportions as
you can 5, 8, 15, 24
11-4
27
Problem Solving Strategy Write a Proportion
PRE-ALGEBRA LESSON 11-4
(For help, go to Lesson 6-2.)
Solve each proportion. 1. 2. 3.
4.
Check Skills Youll Need
11-4
28
Problem Solving Strategy Write a Proportion
PRE-ALGEBRA LESSON 11-4
Solutions 1. 2.
3 a 1 12 25 h 5
20 3a 12 25h 100
a 4 h 4 3. 4.
1 x 4 8 7
c 2 35 x 32 7c
70 c 10
a 12
20 25
c 35
11-4
29
Problem Solving Strategy Write a Proportion
PRE-ALGEBRA LESSON 11-4
At a given time of day, a building of unknown
height casts a shadow that is 24 feet long. At
the same time of day, a post that is 8 feet tall
casts a shadow that is 4 feet long. What is the
height x of the building?
11-4
30
Problem Solving Strategy Write a Proportion
PRE-ALGEBRA LESSON 11-4
(continued)
Write a proportion using the legs of the similar
right triangles.
4x 24(8) Write cross products.
4x 192 Simplify.
x 48 Divide each side by 4.
Quick Check
The height of the building is 48 ft.
11-4
31
Problem Solving Strategy Write a Proportion
PRE-ALGEBRA LESSON 11-4
Write a proportion and solve. 1. On the
blueprints for a rectangular floor, the width of
the floor is 6 in. The diagonal distance
across the floor is 10 in. If the width of the
actual floor is 32 ft, what is the actual
diagonal distance across the floor? 2. A
right triangle with side lengths 3 cm, 4 cm, and
5 cm is similar to a right triangle with a
20-cm hypotenuse. Find the perimeter of the
larger triangle. 3. A 6-ft-tall man
standing near a geyser has a shadow 4.5 ft long.
The geyser has a shadow 15 ft long. What is
the height of the geyser?
about 53 ft
48 cm
20 ft
11-4
32
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
One angle measure of a right triangle is 75
degrees. What is the measurement, in degrees, of
the other acute angle of the triangle?
15 degrees
11-5
33
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
(For help, go to Lesson 11-2.)
Find the missing side of each right
triangle. 1. legs 6 m and 8 m 2. leg 9 m
hypotenuse 15 m 3. legs 27 m and 36
m 4. leg 48 m hypotenuse 60 m
Check Skills Youll Need
11-5
34
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Solutions 1. c2 a2 b2 2. a2 b2
c2 c2 62 82 92 b2 152 c2
100 81 b2 225 c 100 10 m
b2 144 b 144
12 m 3. c2 a2 b2 4. a2 b2 c2
c2 272 362 482 b2 602
c2 2025 2304 b2 3600 c 2025
45 m b2 1296
b 1296 36 m
11-5
35
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Find the length of the hypotenuse in the
triangle.
The length of the hypotenuse is about 14.1 cm.
Quick Check
11-5
36
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Patrice folds square napkins diagonally to put
on a table. The side length of each napkin is 20
in. How long is the diagonal?
The diagonal length is about 28.3 in.
Quick Check
11-5
37
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Find the missing lengths in the triangle.
The length of the shorter leg is 7 ft. The length
of the longer leg is about 12.1 ft.
Quick Check
11-5
38
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Find each missing length. 1. Find the length
of the legs of a 45-45-90 triangle with a
hypotenuse of 4 2 cm. 2. Find the
length of the longer leg of a 30-60-90
triangle with a hypotenuse of 6 in.
3. Kit folds a bandana diagonally before
tying it around her head. The side length of
the bandana is 16 in. About how long is the
diagonal?
4 cm
about 22.6 in.
11-5
39
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
A piece of rope 68 in. long is to be cut into two
pieces. How long will each piece be if one piece
is cut three times longer than the other piece?
17 in. and 51 in.
11-6
40
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
(For help, go to Lesson 6-3.)
Solve each problem. 1. A 6-ft man casts an
8-ft shadow while a nearby flagpole casts a
20-ft shadow. How tall is the flagpole?
2. When a 12-ft tall building casts a 22-ft
shadow, how long is the shadow of a nearby
14-ft tree?
Check Skills Youll Need
11-6
41
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Solutions 1. 2.
6 20 8 x 22 14 12 x
120 8x 308 12x
x 15 ft x
25 ft
11-6
42
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Find the sine, cosine, and tangent of A.
Quick Check
11-6
43
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Find the trigonometric ratios of 18 using a
scientific calculator or the table on page 779.
Round to four decimal places.
Quick Check
11-6
44
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
The diagram shows a doorstop in the shape of a
wedge. What is the length of the hypotenuse of
the doorstop?
You know the angle and the side opposite the
angle. You want to find w, the length of the
hypotenuse.
w(sin 40) 10 Multiply each side by w.
Quick Check
The hypotenuse is about 15.6 cm long.
11-6
45
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Solve. 1. In ABC, AB 5, AC 12, and BC
13. If A is a right angle, find the sine,
cosine, and tangent of B. 2. One angle of
a right triangle is 35, and the adjacent leg is
15. a. What is the length of the opposite leg?
b. What is the length of the
hypotenuse? 3. Find the sine, cosine, and
tangent of 72 using a calculator or a table.
about 10.5
about 18.3
11-6
46
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
An airplane flies at an average speed of 275
miles per hour. How far does the airplane fly in
150 minutes?
687.5 miles
11-7
47
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
(For help, go to Lesson 2-3.)
Find each trigonometric ratio. 1. sin
45 2. cos 32 3. tan 18 4. sin
68 5. cos 88 6. tan 84
Check Skills Youll Need
11-7
48
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
Solutions 1. sin 45 0.7071 2. cos 32
0.8480 3. tan 18 0.3249 4. sin 68
0.9272 5. cos 88 0.0349 6. tan 84
9.5144
11-7
49
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
Janine is flying a kite. She lets out 30 yd of
string and anchors it to the ground. She
determines that the angle of elevation of the
kite is 52. What is the height h of the kite
from the ground?
30(sin 52) h Multiply each side by 30.
Quick Check
The kite is about 24 yd from the ground.
11-7
50
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
Quick Check
Greg wants to find the height of a tree. From
his position 30 ft from the base of the tree, he
sees the top of the tree at an angle of elevation
of 61. Gregs eyes are 6 ft from the ground. How
tall is the tree, to the nearest foot?
30(tan 61) h Multiply each side by 30.
54 6 60 Add 6 to account for the height of
Gregs eyes from the ground.
The tree is about 60 ft tall.
11-7
51
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
An airplane is flying 1.5 mi above the ground.
If the pilot must begin a 3 descent to an
airport runway at that altitude, how far is the
airplane from the beginning of the runway (in
ground distance)?
Draw a picture (not to scale).
d tan 3 1.5 Multiply each side by d.
11-7
52
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
(continued)
The airplane is about 28.6 mi from the airport.
Quick Check
11-7
53
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
Solve. Round answers to the nearest unit. 1. The
angle of elevation from a boat to the top of a
lighthouse is 35. The lighthouse is 96 ft
tall. How far from the base of the lighthouse
is the boat? 2. Ming launched a model
rocket from 20 m away. The rocket traveled
straight up. Ming saw it peak at an angle of 70.
If she is 1.5 m tall, how high did the rocket
fly? 3. An airplane is flying 2.5 mi above
the ground. If the pilot must begin a 3
descent to an airport runway at that altitude,
how far is the airplane from the beginning of
the runway (in ground distance)?
137 ft
57 m
48 mi
11-7