Transcript Chapter 11 - Crestwood Local Schools

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
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 You’ll Need
11-1
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
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Simplify each square root.
a.
144
144 = 12
b. –
–
81
81 = – 9
Quick Check
11-1
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.
d=
1.5h
Use the formula.
d=
1.5(20)
Replace h with 20.
d=
30
Multiply.
25 <
25 = 5
30 <
36
Find perfect squares close to 30.
Find the square root of the
closest perfect square.
The lifeguard can see about 5 miles to the horizon.
11-1
Quick Check
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Identify each number as rational or irrational. Explain.
a.
49
rational, because 49 is a perfect square
b. 0.16
rational, because it is a terminating decimal
c. 3
irrational, because 3 is not a perfect square
d. 0.3333 . . .
rational, because it is a repeating decimal
e. – 15
irrational, because 15 is not a perfect square
f. 12.69
rational, because it is a terminating decimal
g. 0.1234567 . . .
irrational, because it neither terminates nor repeats
11-1
Quick Check
Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Simplify each square root or estimate to the nearest integer.
1. –
100
2.
–10
57
8
Identify each number as rational or irrational.
3.
48
irrational
4. 0.0125
rational
5. The formula d = 1.5h , where h equals the height, in feet, of the
viewer’s 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.
3 mi
11-1
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
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 You’ll Need
11-2
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Solutions
1. 42 + 62
16 + 36 = 52
2.
52 + 82
25 + 64 = 89
3. 72 + 92
49 + 81 = 130
4.
92 + 32
81 + 9 = 90
11-2
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find c, the length of the hypotenuse.
c2 = a2 + b2
Use the Pythagorean Theorem.
c2 = 282 + 212
Replace a with 28, and b with 21.
c2 = 1,225
Simplify.
c=
1,225 = 35
Find the positive square root of each side.
The length of the hypotenuse is 35 cm.
Quick Check
11-2
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find the value of x in the triangle.
Round to the nearest tenth.
a2 + b2 = c2
Use the Pythagorean Theorem.
72 + x2 = 142
Replace a with 7, b with x, and c with 14.
49 + x2 = 196
Simplify.
x2 = 147
x=
147
Subtract 49 from each side.
Find the positive square root of each side.
11-2
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
(continued)
Then use one of the two methods below to
approximate 147 .
Method 1 Use a calculator.
A calculator value for
x
12.1
147 is 12.124356.
Round to the nearest tenth.
Method 2 Use a table of square roots.
Use the table on page 778. Find the number closest
to 147 in the N2 column. Then find the corresponding
value in the N column. It is a little over 12.
x
12.1
Estimate the nearest tenth.
Quick Check
The value of x is about 12.1 in.
11-2
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?
c2 = a2 + b2
Use the Pythagorean Theorem.
c2 = 102 + 102
Replace a with 10 (half the span), and b with 10.
c2 = 100 + 100
Square 10.
c2 = 200
Add.
c=
c
200
14.1
Find the positive square root.
Round to the nearest tenth.
The rafter length should be about 14.1 ft.
11-2
Quick Check
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Is a triangle with sides 10 cm, 24 cm, and 26 cm
a right triangle?
a2 + b2 = c2
Write the equation to check.
102 + 242
262
Replace a and b with the shorter lengths
and c with the longest length.
100 + 576
676
Simplify.
676 = 676
The triangle is a right triangle.
Quick Check
11-2
The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find the missing length. Round to the
nearest tenth.
1. a = 7, b = 8, c =
10.6
2. a = 9, c = 17, b =
14.4
3. Is a triangle with sides 6.9 ft, 9.2 ft, and 11.5 ft a right triangle?
Explain.
yes; 6.92 + 9.22 = 11.52
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.
about 5.6 ft
11-2
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the number halfway between 0.784 and 0.76.
0.772
11-3
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 You’ll Need
11-3
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Solutions
1. A (–3, 4)
2. D (0, 3)
11-3
3. G (–4, –2)
4. J (3, –1)
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the distance between T(3, –2) and V(8, 3).
d=
(x2 – x1)2 + (y2 – y1)2
Use the Distance Formula.
d=
(8 – 3)2 + (3 – (–2 ))2
Replace (x2, y2) with (8, 3)
and (x1, y1) with (3, –2).
d=
52 + 52
Simplify.
d=
50
Find the exact distance.
d
7.1
Round to the nearest tenth.
The distance between T and V is about 7.1 units.
Quick Check
11-3
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.
WX =
=
XY =
=
(–2 – (–3))2 + (–1 – 2)2
Replace (x2, y2) with (–2, –1)
and (x1, y1) with (–3, 2).
1+9=
Simplify.
10
(4 – (–2))2 + (0 – (–1)2
Replace (x2, y2) with (4, 0)
and (x1, y1) with (–2, –1).
36 + 1 =
Simplify.
37
11-3
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
(continued)
YZ =
=
ZW =
=
(1 – 4)2 + (5 – 0)2
Replace (x2, y2) with (1, 5)
and (x1, y1) with (4, 0).
9 + 25 =
Simplify.
34
(–3 – 1)2 + (2 – 5)2
Replace (x2, y2) with (–3, 2)
and (x1, y1) with (1, 5).
16 + 9 =
Simplify.
25 = 5
11-3
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
(continued)
perimeter =
10 +
37 +
34 + 5
20.1
The perimeter is about 20.1 units.
Quick Check
11-3
Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the midpoint of TV.
x1 + x2 y1 + y2
,
2
2
Use the Midpoint Formula.
=
4 + 9 –3 + 2
,
2
2
Replace (x1, y1) with (4, –3) and
(x2, y2) with (9, 2).
=
13 –1
,
2 2
Simplify the numerators.
1
2
= 6 ,–
1
2
Write the fractions in simplest form.
1
2
The coordinates of the midpoint of TV are 6 , –
11-3
1
.
2
Quick Check
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)
1 1
9.1; (–2 2, – 2 )
3. Find the perimeter of
and C(3, 0).
1
13.6; (12 , 2)
ABC, with coordinates A(–3, 0), B(0, 4),
16
11-3
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
8 , 15
5
=
24 5
15
=
15 8
24 , 5
= 24 , 5
8
8
= 24
11-4
15
Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
(For help, go to Lesson 6-2.)
Solve each proportion.
1.
a
1
= 12
3
2. 5 = 25
h
20
3.
1
8
=
4
x
4.
2
c
=
7
35
Check Skills You’ll Need
11-4
Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
Solutions
1.
1
= a
3
12
2.
1
8
=
4
x
4.
3 • a = 1 • 12
3a = 12
a=4
3.
h
= 20
5
25
25 • h = 5 • 20
25h = 100
h=4
1•x=4•8
x = 32
2
c
=
7
35
7 • c = 2 • 35
7c = 70
c = 10
11-4
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?
Since the triangles are similar,
and you know three lengths,
writing and solving a proportion
is a good strategy to use.
It is helpful to draw the triangles
as separate figures.
11-4
Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
(continued)
Write a proportion using the legs
of the similar right triangles.
8
4
=
x
24
Write a proportion.
4x = 24(8)
Write cross products.
4x = 192
Simplify.
x = 48
Divide each side by 4.
The height of the building is 48 ft.
11-4
Quick Check
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?
about 53 ft
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.
48 cm
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?
20 ft
11-4
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
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 You’ll Need
11-5
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Solutions
1. c2 = a2 + b2
c2 = 62 + 82
c2 = 100
c = 100 = 10 m
2. a2 + b2 = c2
92 + b2 = 152
81 + b2 = 225
b2 = 144
b = 144 = 12 m
3. c2 = a2 + b2
c2 = 272 + 362
c2 = 2025
c = 2025 = 45 m
4.
11-5
a2 + b2 = c2
482 + b2 = 602
2304 + b2 = 3600
b2 = 1296
b = 1296 = 36 m
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Find the length of the hypotenuse in the triangle.
hypotenuse = leg •
2
Use the 45°-45°-90° relationship.
y = 10 •
2
The length of the leg is 10.
14.1
Use a calculator.
The length of the hypotenuse is about 14.1 cm.
11-5
Quick Check
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?
hypotenuse = leg •
2
Use the 45°-45°-90° relationship.
y = 20 •
2
The length of the leg is 20.
28.3
Use a calculator.
The diagonal length is about 28.3 in.
Quick Check
11-5
Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Find the missing lengths in the triangle.
hypotenuse = 2 • shorter leg
14 = 2 • b
The length of the hypotenuse is 14.
14 = 2b
2
2
Divide each side by 2.
7=b
longer leg = shorter leg •
a=7• 3
a 12.1
Simplify.
3
The length of the shorter leg is 7.
Use a calculator.
The length of the shorter leg is 7 ft. The length of the
longer leg is about 12.1 ft.
11-5
Quick Check
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.
4 cm
2. Find the length of the longer leg of a 30°-60°-90° triangle with a
hypotenuse of 6 in.
3
3 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?
about 22.6 in.
11-5
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
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 You’ll Need
11-6
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Solutions
1.
6
x
=
8
20
2.
12
14
=
22
x
6 • 20 = 8 • x
22 • 14 = 12 • x
120 = 8x
308 = 12x
120
= 8x
8
8
308 12x
=
12
12
x = 25 2 ft
3
x = 15 ft
11-6
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Find the sine, cosine, and tangent of
opposite
12
3
adjacent
16
4
12
3
sin
A = hypotenuse = 20 = 5
cos
A = hypotenuse =
=
20
5
tan
A = adjacent
opposite
A.
= 16 = 4
Quick Check
11-6
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.
sin 18°
0.3090
Scientific calculator: Enter 18 and press
the key labeled SIN, COS, or TAN.
cos 18°
0.9511
tan 18°
Table: Find 18° in the first column. Look
0.3249 across to find the appropriate ratio.
Quick Check
11-6
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.
opposite
sin A = hypotenuse
10
sin 40° = w
w(sin 40°) = 10
10
w = sin 40°
w 15.6
Use the sine ratio.
Substitute 40° for the angle, 10 for
the height, and w for the hypotenuse.
Multiply each side by w.
Divide each side by sin 40°.
Use a calculator.
The hypotenuse is about 15.6 cm long.
11-6
Quick Check
Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Solve.
1. In ABC, AB = 5, AC = 12, and BC = 13. If
find the sine, cosine, and tangent of B.
A is a right angle,
12 5 12
, ,
13 13 5
2. One angle of a right triangle is 35°, and the adjacent leg is 15.
a. What is the length of the opposite leg?
about 10.5
b. What is the length of the hypotenuse?
about 18.3
3. Find the sine, cosine, and tangent of 72° using a calculator or
a table.
sin 72° 0.9511; cos 72° 0.3090; tan 72° 3.0777
11-6
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
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 You’ll Need
11-7
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
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?
Draw a picture.
sin A =
opposite
hypotenuse
h
sin 52° = 30
30(sin 52°) = h
24 h
The kite is about 24 yd from the ground.
11-7
Choose an appropriate
trigonometric ratio.
Substitute.
Multiply each side by 30.
Simplify.
Quick Check
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°. Greg’s eyes are 6 ft from the ground. How tall is the
tree, to the nearest foot?
Draw a picture.
opposite
Choose an appropriate
trigonometric ratio.
h
Substitute 61 for the angle measure
and 30 for the adjacent side.
tan A = adjacent
tan 61° = 30
30(tan 61°) = h
54 h
54 + 6 = 60
The tree is about 60 ft tall.
11-7
Multiply each side by 30.
Use a calculator or a table.
Add 6 to account for the height
of Greg’s eyes from the ground.
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).
tan 3° = 1.5
d
d • tan 3° = 1.5
Choose an appropriate trigonometric ratio.
Multiply each side by d.
11-7
Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
(continued)
1.5
d • tan 3°
= tan 3°
tan 3°
1.5
Divide each side by tan 3°.
d = tan 3°
Simplify.
d
Use a calculator.
28.6
The airplane is about 28.6 mi from the airport.
11-7
Quick Check
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?
137 ft
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?
57 m
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)?
48 mi
11-7