Page 1 of 5
Approximating Square Roots
BEFORE
Now
WHY?
You found square roots
of perfect squares.
You’ll approximate square
roots of numbers.
So you can find the falling speed
of a skydiver, as in Ex. 22.
In the Real World
Word Watch
irrational number, p. 541
real number, p. 541
Animals Dr. R. McNeill Alexander studies the motion
of animals. From his studies, he determined that
the maximum speed s, in feet per second, that
an animal can walk is
s 5.66l
where l is the animal’s leg length,
in feet. What is the maximum
walking speed for a giraffe with
a leg length of 11 feet? You’ll
find the answer in Example 3.
Evaluating Square Roots
You know how to evaluate
, 4
,
square roots like 1
because 1, 4, and 9
and 9
are perfect squares. But what
, 3
,
about square roots like 2
? The values of these
and 5
square roots fall between whole
numbers, as shown on the
number line below.
1
0
EXAMPLE
1
1
2
3 4 5
9
2
3
Approximating to a Whole Number
Approximate 11
to the nearest whole number.
Make a list of whole numbers that are perfect squares: 0, 1, 4, 9, 16, . . . .
9 < 11 < 16
9
< 11
< 16
3 < 11
<4
Identify perfect squares closest to 11.
Take positive square root of each number.
Evaluate square roots.
ANSWER Because 11 is closer to 9 than to 16, 11
is closer to 9
3.
So, to the nearest whole number, 11
≈ 3.
540
Chapter 11
Measurement and Area
Page 2 of 5
EXAMPLE
with
Solving
Once you find the
approximation of a square
root to the tenths’ place, you
can use the same method
to find the approximation
to the hundredths’ place,
thousandths’ place, and
so on.
2
Approximating to the Nearest Tenth
Approximate 11
to the nearest tenth.
You know from Example 1 that 11
is between 3 and 4.
Make a list of squares of 3.1, 3.2, . . . , 3.9. From the list,
is
you can see that 11 is between 3.32 and 3.42. So, 11
between 3.3 and 3.4.
ANSWER Because 11 is closer to 10.89 than to 11.56, 11
is
3.3. So, to the nearest tenth, 11
≈ 3.3.
closer to 10.89
3.12 9.61
3.22 10.24
3.32 10.89
3.42 11.56
3.52 12.25
Your turn now
Approximate the square root to the nearest whole
number and then to the nearest tenth.
1. 10
EXAMPLE
2. 22
3
3. 45
4. 115
Using Square Roots
You can use the approximation of 11
from Example 2 to estimate the
maximum walking speed of the giraffe described on the previous page.
s 5.66l
Write maximum walking speed formula.
5.6611
Substitute 11 for l.
≈ 5.66(3.3)
Use approximation of 11
to the nearest tenth.
≈ 19
Multiply.
ANSWER The maximum walking speed is about 19 feet per second.
Irrational Numbers The number 11
is an example of an irrational
number. An irrational number cannot be written as a quotient of
two integers, and the decimal form of an irrational number neither
terminates nor repeats. If n is a positive integer which is not a perfect
is an irrational number.
square, then n
The set of real numbers consists of all rational and irrational numbers.
The Venn diagram shows the relationships among numbers in the real
number system.
Real Numbers
Rational numbers
Integers
Irrational numbers
Whole numbers
Lesson 11.2
Approximating Square Roots
541
Page 3 of 5
EXAMPLE
4
Identifying Rational and Irrational Numbers
Tell whether the number is rational or irrational. Explain.
a. 2
1
b. 9
c. 169
d. 1.21121112. . .
Solution
with
Review
a. 2
is irrational because 2 is a positive integer but not a perfect
square.
Need help with rational
numbers? See p. 283.
1
1
1
b. 9 is rational because 9 9 , which is a quotient of integers.
c. 169
is rational because 169
13, which is an integer.
d. 1.21121112. . . is irrational because it neither terminates nor repeats.
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 715
Getting Ready to Practice
1. Vocabulary Copy and complete: A number that cannot be represented
as a quotient of two integers is called a(n) _?_ number.
Approximate the square root to the nearest whole number and then
to the nearest tenth.
2. 15
3. 23
4. 42
5. 131
Tell whether the number is rational or irrational. Explain your reasoning.
6. 2.6
7. 1600
8. 45
9. 115
10. Guided Problem Solving You buy 140 square feet of carpet to cover the
floor in a square bedroom. There is 12 square feet of carpet left over.
Approximate the side length of the bedroom floor to the nearest foot.
1 How many square feet of carpet did you use to cover the
bedroom floor?
2 The amount of carpet used falls between which two whole numbers
that are perfect squares?
3 The square root of which whole number is the better approximation
for the side length of the bedroom floor? What is the approximate
side length of the bedroom floor to the nearest foot?
542
Chapter 11
Measurement and Area
Page 4 of 5
Practice and Problem Solving
with
Example
1
2
3
4
Homework
Exercises
11–18
11–18, 21
22
23–30
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
Approximate the square root to the nearest whole number and
then to the nearest tenth.
11. 35
12. 89
13. 57
14. 63
15. 125
16. 188
17. 200
18. 253
19. Find the Error Describe
and correct the error in
to
approximating 29
the nearest whole number.
29 falls between 25 and
36. Because 29 is closer
to 25, 29
≈ 25.
20. Guess, Check, and Revise Guess a value of x that solves the equation
x 2 24. Square the value and compare it with 24. Revise your guess.
Repeat the process to approximate solutions to the nearest tenth.
21. Mural You use one gallon of white paint to apply a base coat of paint
on a square wall mural. The paint covers 350 square feet per gallon.
Approximate the side length of the mural to the nearest tenth of a foot.
Recreation
22. Skydiving A skydiver falls at a rate given by s 1.05w
where
s is the falling speed, in feet per second, and w is the total weight of
the skydiver with gear, in pounds. To the nearest tenth, what is the
approximate falling speed of a 150 pound man with 35 pounds of gear?
Tell whether the number is rational or irrational. Explain your reasoning.
23. 5.6537891. . . 24. 64
27. 21
3
28. 5 8
1
25. 11
7
26. 9
29. 1.3
7
5
30. 30.23233. . .
Approximate the square root to the nearest hundredth.
31. 87
32. 91
33. 140
34. 210
35. Critical Thinking Consider the square roots of the whole numbers
■
Skydiving
In 2000, there were
3.5 million skydives,
but only 317,741 people
who skydived. What is the
mean number of skydives
per person?
from 1 to 10. Are there more rational numbers or irrational numbers?
Explain your reasoning.
Use a number line to order the numbers from least to greatest.
36. 5
, 5, 9
, 1.5
17
37. 4.3, 4.3
, 17
, 3
27
38. 21
, 27
, 5, , 4.8
5
Algebra Solve the equation. Round solutions to the nearest hundredth.
39. 5x 2 65
40. 14x 2 123.2
41. 9x 2 3 48
42. Challenge If a 2b, then does a
2b
? Explain your reasoning.
Lesson 11.2
Approximating Square Roots
543
Page 5 of 5
Mixed Review
Simplify the expression. (Lesson 7.2)
43. 8(3 j) 4j
44. 7h 11 4h 45. 3(2t 3u)
46. 5r(r 6) 9
Solve the equation. (Lesson 11.1)
47. a 2 144
48. c 2 9 45
49. y 2 20 4 50. 5z 2 5 25
Basic Skills Find the greatest common factor of the numbers.
51. 64, 80
52. 28, 42
53. 18, 36, 66
54. 80, 96, 112
Test-Taking Practice
55. Multiple Choice A square kitchen floor has an area of 260 square feet.
INTERNET
To the nearest foot, what is the side length of the kitchen floor?
State Test Practice
CLASSZONE.COM
A. 13 ft
B. 14 ft
C. 16 ft
D. 17 ft
56. Multiple Choice To find the time it takes for a dropped object to hit
the ground, you can use the equation h 16t 2, where h is height, in
feet, and t is time, in seconds. If an object is dropped from a height of
48 feet, how long (to the nearest tenth of a second) does the object fall?
F. 1.7 sec
G. 1.8 sec
H. 2.4 sec
I. 5.7 sec
Who nose?
Evaluate the square roots. Then decipher the code
to find the answer to the riddle: Why can’t your
nose be 12 inches long?
3
4
3
2 10 9 11 12 2
7 11 13 6
2
9
8
5
5 15 2 14
16 7
7
5
A 81
B 9
C 100
D 36
E 4
F 256
H 225
I 64
L 169
N 196
O 49
S 144
T 25
U 121
W 16
544
Chapter 11
Measurement and Area