Chapter 01 Homework Assignment

Due: 11:59pm on Sunday, January 25, 2015

To understand how points are awarded, read the Grading Policy for this assignment.

Welcome!

Mastering presents homework items assigned by your instructor and *works with you *to answer them. Homework items

typically have an introduction, possibly figures, and one or more parts for you to answer.

Type of help offered

Mastering tells you immediately whether or not your answers are correct. Usually, you will have multiple

chances to arrive at the correct answer. Your instructor will determine how many tries you have available.

In many items, hints are available to help you if you get stuck. If you don’t need the hints to solve the

problem, you can still use them for review later on.

If you submit an incorrect answer, Mastering often responds with specific, helpful feedback.

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let you know and give you another chance.

These exercises were chosen specifically to lead you through the key features of Mastering and are not intended to test

your knowledge of any specific subject material. Therefore, on this item you will not be penalized for using hints and

submitting incorrect answers. In fact, you should submit incorrect answers and use the hints to see what happens!

Part A

How many squares are in this grid ? Note that the

figure link lets you know that a figure goes along with this

part. This figure is available to the left.

Enter your answer as a number in the box below and

then submit your answer by clicking Submit.

ANSWER:

g

Number of squares = 5

Correct

What you are reading now is called a “followup

comment.” These comments typically offer more information or

provide an interpretation of the answer you just obtained.

Before you move on to a slightly more challenging question, have a quick look at the other buttons available

around the answer box.

**Give Up **allows you to complete the question if you can’t solve it on your own. Your instructor

controls whether or not the correct answer is displayed to you.

**My Answers **brings up a new window that lists all of the answers you have submitted for this

question, along with any helpful feedback you received for incorrect submissions.

Grading

See the help file available by clicking the **Help **link at the top right corner, if you want to know more about how grading

works in general. Here is the most important information you’ll need.

You must complete every part to get credit for an item. To complete a part, either answer the main part question correctly

or click the **Give Up **button.

In a graded homework item, each part counts equally toward your score on the overall item. If you get full credit on each

part, you will receive full credit for the problem. You *may *lose a fraction of the credit for a part when you submit an

incorrect answer. Whether you do lose credit and how much you lose are set by your instructor. However, you won’t lose

credit for most types of formatting mistakes or for submitting a blank answer.

As you might expect, you will receive no credit for a part if you use the **Give Up **button. If you just can’t figure out a

question, there is a way to get partial credit by using hints, as the following part will illustrate.

Part B

What is the magic number?

Note that there is a figure also associated with this part.

However, the figure for Part A may still be visible on the

left. To view the figure associated with Part B, click on the

figure link. A new figure should appear on the left.

You could try to guess the magic number but you

would probably use up all your tries before getting

the answer. Notice the *new *Hints button underneath

the answer box for this question. Clicking this button

will open up a list of hints that will guide you to the

correct number.

**Hint 1. **Different types of hints and their impact on grading

Notice that there are three hints for this question. You are not required to use all of the hints or to use them in

order. Each hint has a tagline that describes its contents. Based on the tagline you can decide whether or not

a particular hint will be useful to you.

There are two kinds of hints. Some hints, such as Hint 2 below, just provide you with information. Other hints,

such as Hint 3 below, give you an opportunity to answer a simpler question that is related to the main

question you are solving. These hints either have questions in the tagline or tell you to do something (e.g.,

Find…, Determine…, Identify…, etc.). There are two ways that this type of hint can help you:

Answering the simpler question gives you a chance to check that you are on the right track.

If you correctly answer the simpler question, you will receive partial credit for the part even if

you are unable to answer the main question.

Your instructor *may *choose to give you a bonus for not using hints or to deduct a small penalty for using

hints. If you are stuck, *using the hints will usually result in a higher score than simply trying to guess *because

you *may *lose fewer points for opening a hint than for getting the answer to the main question incorrect. There

is a more detailed explanation of how hints are graded in the help available by clicking the **Help **link at the top

right corner of your screen in the main Mastering window. In this problem, however, you will not lose any

credit for using the hints.

Now, open up the second hint for some help finding the magic number.

**Hint 2. **How to approach the problem

Although you could try to guess the magic number you would most likely exhaust your tries before getting the

correct answer. To help you, the magic number is , where is a number between 1 and 10.

**Hint 3. **What is ?

Recall that the previous hint stated that the magic number is , where is a number between 1 and

10. Specifically is an even number between 1 and 10. Try to guess the value of .

You may submit as many guesses as you need. Enter each guess into the answer box that follows.

ANSWER:

Correct

Now that you have determined , compute to find the magic number.

ANSWER:

Correct

Your instructor *may *choose to give you a bonus for not using hints or to deduct a small penalty for using hints.

If you are stuck, *using the hints will usually result in a higher score than simply trying to guess *because you

*may *lose fewer points for opening a hint than for getting the answer to the main question incorrect. Note that

you are never required to use the hints; if you want to figure the question out on your own, go ahead!

Notice that a new button, **Review Part**, appears when you correctly answer a part with hints. This button allows

you to review all of the hints for that part, even if you didn’t need them to get the answer. This is a useful way to

review the question when studying for a test. *You do not lose any credit for reviewing the hints *after you have

answered the question. If you didn’t look at all of the hints while answering the last question, you should read

through them now for some important information about hints and hint grading.

g4 4

4

g4 4

4 4

4 = 4

4 g

magic number = 60

Part C

Multiplechoice

questions have a special grading rule determined by your instructor. Assume that your instructor has

decided to grade these questions in the following way: If you submit an incorrect answer to a multiplechoice

question with options, you will lose of the credit for that question. Just like the similar multiplechoice

penalty on most standardized tests, this rule is necessary to prevent random guessing.

If a multiplechoice

question has five answer choices and you submit one wrong answer before getting the question

correct, how much credit will you *lose *for that part of the question?

ANSWER:

Correct

Your instructor may choose not to deduct of the credit for a multiplechoice

question with options.

To see how your instructor is grading you, click your instructor’s **Grading Policy **on your assignment page.

If you click on the **Continue **button before finishing all the Parts, you will see a message reminding you that you

need to complete each Part to get credit. If you have completed the item, clicking **Continue **will take you to the

next item on the Assignment. At any time you may click on the **Provide Feedback **link to access a survey page

without losing your work.

Once you have completed an item, you may access your score from the assignment. Your score will display

below the item title.

Significant Figures Conceptual Question

In the parts that follow select whether the number presented in statement A is greater than, less than, or equal to the

number presented in statement B. Be sure to follow all of the rules concerning significant figures.

Part A

Statement A: 2.567 , to two significant figures.

Statement B: 2.567 , to three significant figures.

Determine the correct relationship between the statements.

**Hint 1. **Rounding and significant figures

Rounding to a different number of significant figures changes a number. For example, consider the number

3.4536. This number has five significant figures. The following table illustrates the result of rounding this

number to different numbers of significant figures:

* *Ã

100

50

33

25

20

*Ã *

LN

LN

Four significant figures 3.454

Three significant figures 3.45

Two significant figures 3.5

One significant figure 3

Notice that, when rounding 3.4536 to one significant figure, since 0.4536 is less than 0.5, the result is 3, even

though if you first rounded to two significant figures (3.5), the result would be 4.

ANSWER:

Correct

Part B

Statement A: (2.567 + 3.146 ), to two significant figures.

Statement B: (2.567 , to two significant figures) + (3.146 , to two significant figures).

Determine the correct relationship between the statements.

ANSWER:

Correct

Evaluate statement A as follows: (2.567 + 3.146 ) = 5.713 to two significant figures is 5.7 .

Statement B evaluates as 2.6 + 3.1 = 5.7 . Therefore, the two statements are equal.

Part C

Statement A: Area of a rectangle with measured length = 2.536 and width = 1.4 .

Statement B: Area of a rectangle with measured length = 2.536 and width = 1.41 .

Since you are not told specific numbers of significant figures to round to, you must use the rules for multiplying

numbers while respecting significant figures. If you need a reminder, consult the hint.

Determine the correct relationship between the statements.

Statement A is

greater than

less than

equal to

Statement B.

LN LN

LN LN

Statement A is

greater than

less than

equal to

Statement B.

LN LN LN LN

LN LN LN

N N

N N

**Hint 1. **Significant figures and multiplication

When you multiply two numbers, the result should be rounded to the number of significant figures in the less

accurate of the two numbers. For instance, if you multiply 2.413 (four significant figures) times 3.81 (three

significant figures), the result should have three significant figures: . Similarly,

, when significant figures are respected (i.e., 15.328646 rounded to one significant

figure).

ANSWER:

Correct

Evaluate statement A as follows: (2.536 ) (1.4 ) = 3.5504 to two significant figures is 3.6 .

Statement B evaluates as (2.536 ) (1.41 ) = 3.57576 to three significant figures is 3.58 .

Therefore, statement A is greater than statement B.

Vector ComponentsReview

Learning Goal:

To introduce you to vectors and the use of sine and cosine for a triangle when resolving components.

Vectors are an important part of the language of science, mathematics, and engineering. They are used to discuss

multivariable calculus, electrical circuits with oscillating currents, stress and strain in structures and materials, and flows

of atmospheres and fluids, and they have many other applications. Resolving a vector into components is a precursor to

computing things with or about a vector quantity. Because position, velocity, acceleration, force, momentum, and angular

momentum are all vector quantities, resolving vectors into components is *the most important skill *required in a

mechanics course.

The figure shows the components of , and , along the *x *and *y *axes of the coordinate system, respectively. The

components of a vector depend on the coordinate system’s orientation, the key being the angle between the vector and

the coordinate axes, often designated .

g –

g –

Statement A is

greater than

less than

equal to

Statement B.

N N N N

N N N N

. 4 5

J

Part A

The figure shows the standard way of measuring the

angle. is measured *to *the vector *from *the *x *axis, and

counterclockwise is positive.

Express and in terms of the length of the

vector and the angle , with the components

separated by a comma.

ANSWER:

J

4 5

J

4 , 5 = DPTJ TJOJ

Correct

In principle, you can determine the components of *any *vector with these expressions. If lies in one of the

other quadrants of the plane, will be an angle larger than 90 degrees (or in radians) and and

will have the appropriate signs and values.

Unfortunately this way of representing , though mathematically correct, leads to equations that must be

simplified using trig identities such as

and

.

These must be used to reduce all trig functions present in your equations to either or . Unless

you perform this followup step flawlessly, you will fail to recoginze that

,

and your equations will not simplify so that you can progress further toward a solution. Therefore, it is best to

express all components in terms of either or , with between 0 and 90 degrees (or 0 and

in radians), and determine the signs of the trig functions by knowing in which quadrant the vector lies.

Part B

When you resolve a vector into components, the components *must have the form *or . The

signs depend on which quadrant the vector lies in, and there will be one component with and the other with

.

In real problems the optimal coordinate system is often rotated so that the *x *axis is not horizontal. Furthermore, most

vectors will not lie in the first quadrant. To assign the sine and cosine correctly for vectors at arbitrary angles, you

must figure out which angle is and then properly reorient the definitional triangle.

As an example, consider the vector shown in the diagram labeled “tilted axes,” where you know the angle

between and the *y *axis.

Which of the various ways of orienting the definitional

triangle must be used to resolve into components in

the tilted coordinate system shown? (In the figures, the

hypotenuse is orange, the side adjacent to is red, and

the side opposite is yellow.)

.

J R DPT J

TJO J

.

TJO È ] – ÃTJO ]

DPT È ] – ÃTJO ]

TJO ] DPT ]

TJO È ]DPT È Ã] –

TJO ] DPT ]] R

. ]]. DPT J ]]. TJO J

TJO J

DPT J

J

. J

.

.

J

Indicate the number of the figure with the correct orientation.

**Hint 1. **Recommended procedure for resolving a vector into components

First figure out the sines and cosines of , then figure out the signs from the quadrant the vector is in and

write in the signs.

**Hint 2. **Finding the trigonometric functions

Sine and cosine are defined according to the following convention, with the key lengths shown in green: The

hypotenuse has unit length, the side adjacent to has length , and the side opposite has length

. The colors are chosen to remind you that the vector sum of the two orthogonal sides is the vector

whose magnitude is the hypotenuse; red + yellow = orange.

J

J DPT J

TJO J

ANSWER:

Correct

Part C

Choose the correct procedure for determining the components of a vector in a given coordinate system from this list:

ANSWER:

Correct

Part D

The space around a coordinate system is conventionally divided into four numbered *quadrants *depending on the

1

2

3

4

Align the adjacent side of a right triangle with the vector and the hypotenuse along a coordinate direction

with as the included angle.

Align the hypotenuse of a right triangle with the vector and an adjacent side along a coordinate direction

with as the included angle.

Align the opposite side of a right triangle with the vector and the hypotenuse along a coordinate direction

with as the included angle.

Align the hypotenuse of a right triangle with the vector and the opposite side along a coordinate direction

with as the included angle.

J

J

J

J

signs of the *x *and *y *coordinates . Consider the following

conditions:

A. ,

B. ,

C. ,

D. ,

Which of these lettered conditions are true in which the

numbered quadrants shown in ?

Write the answer in the following way: If A were true in the third quadrant, B in the second, C in the first, and

D in the fourth, enter “3, 2, 1, 4” as your response.

ANSWER:

Correct

Part E

Now find the components and of in the tilted coordinate system of **Part B**.

Express your answer in terms of the length of the vector and the angle , with the components separated

by a comma.

ANSWER:

Correct

Exercise 1.12

4 5

4 5

4 5

4 5

1,4,2,3

4 5 .

J

4 , 5 = ÃTJOJDPTJ

Part A

The recommended daily allowance (RDA) of the trace metal magnesium is 410 for males. Express this

quantity in .

Express your answer using two significant figures.

ANSWER:

Correct

Part B

For adults, the RDA of the amino acid lysine is 12 per of body weight. How many grams per day should a 73

adult receive?

Express your answer using two significant figures.

ANSWER:

Correct

Part C

A typical multivitamin tablet can contain 2.0 of vitamin B2 (riboflavin), and the RDA is 0.0030 . How many

such tablets should a person take each day to get the proper amount of this vitamin, assuming that he gets none

from any other sources?

Express your answer as an integer.

ANSWER:

Correct

Part D

The RDA for the trace element selenium is 0.000070 . Express this dose in .

Express your answer using two significant figures.

ANSWER:

NHEBZ

NHEBZ

4.1×105 NHEBZ

NH LH

LH

0.88 HEBZ

NH HEBZ

2 tablets

HEBZ NHEBZ

Correct

Exercise 1.7

Part A

How many years older will you be 1.00 billion seconds from now? (Assume a 365day

year.)

ANSWER:

Correct

Exercise 1.24

You are using water to dilute small amounts of chemicals in the laboratory, drop by drop.

Part A

How many drops of water are in a 1.0 bottle? (*Hint: *Start by estimating the diameter of a drop of water.)

ANSWER:

Correct

± Vector Addition and Subtraction

In general it is best to conceptualize vectors as arrows in space, and then to make calculations with them using their

components. (You must first specify a coordinate system in order to find the components of each arrow.) This problem

gives you some practice with the components.

Let vectors , , and . Calculate the following, and express your answers as

7.0×10−2 NHEBZ

31.7 years

–

drops of water

drops of water

drops of water

drops of water

Õ

Õ

Õ

Õ

.- Ã .- Ã .-

ordered triplets of values separated by commas.

Part A

ANSWER:

Correct

Part B

ANSWER:

Correct

Part C

ANSWER:

Correct

Part D

ANSWER:

Correct

Part E

ANSWER:

.Ã. = 3,5,

4

.Ã. = 5,4,0

Ã..Ã. = 6,4,3

.Ã. = 3,

2,

11

Ã..Ã. = 11,14,8

Correct

Part F

ANSWER:

Correct

Tracking a Plane

A radar station, located at the origin of *xz *plane, as shown in the figure , detects an airplane coming straight at the station

from the east. At first observation (point A), the position of the

airplane relative to the origin is . The position vector

has a magnitude of 360 and is located at exactly 40

above the horizon. The airplane is tracked for

another 123 in the vertical eastwest

plane for 5.0 ,

until it has passed directly over the station and reached point

B. The position of point B relative to the origin is (the

magnitude of is 880 ). The contact points are shown in

the diagram, where the *x *axis represents the ground and the

positive *z *direction is upward.

Part A

Define the displacement of the airplane while the radar was tracking it: . What are the

components of ?

Express in meters as an ordered pair, separating the *x *and *z *components with a comma, to two

significant figures.

**Hint 1. **How to approach the problem

Keep in mind that .According to the rules of vector addition and subtraction, the *x*

component of is .

**Hint 2. **Finding the components of

.Ã .Ã. = 17,12,

6

.

.

N

EFHSFFT

EFHSFFT T

.

. N

. – Ã . .

.#”

.#”

. – Ã

.

.

.

#” – Ã .

4 .

4 .

4

.

. ? ?

What are the components of in the and directions?

Express your answer in meters as an ordered pair, separating the *x *and *z *values with commas, to

three significant figures.

ANSWER:

**Hint 3. **Finding the components of

What are the components of in the and directions?

Express your answer in meters as an ordered pair, separating the *x *and *z *components with a comma,

to three significant figures.

ANSWER:

ANSWER:

Correct

± Vector Dot Product

Let vectors , , and .

Calculate the following:

Part A

**Hint 1. **Remember the dot product equation

If and , then

.

ANSWER:

. % ? ‘?

4 , = 276,231 6 N

.

.

% ? ‘?

4 , = 842,257

6 N

. = 1100,26

#” N

.- Ã .- Ã .- ÃÃ

.- 4 5 6 .- 4 5 6

.ø.- 44 55 66

.ø . = 10

Correct

Part B

What is the angle between and ?

Express your answer using one significant figure.

**Hint 1. **Remember the definition of dot products

, where is the angle between and .

ANSWER:

Correct

Part C

ANSWER:

Correct

Part D

ANSWER:

Correct

Part E

Which of the following can be computed?

**Hint 1. **Dot product operator

The dot product operates only on two vectors. The dot product of a vector and a scalar is not defined.

J“# . .

.ø .- ].] ].] DPT JJ . .

J“# = 2 SBEJBOT

.ø . = 30

.ø . = 30

ANSWER:

Correct

and are different vectors with lengths and respectively. Find the following:

Part F

Express your answer in terms of

**Hint 1. **What is the angle between a vector and itself?

The angle between a vector and itself is 0.

**Hint 2. **Remember the definition of dot products

, where is the angle between and .

ANSWER:

Correct

Part G

If and are perpendicular,

**Hint 1. **What is the angle between perpendicular vectors?

The angle between vectors that are perpendicular is equal to radians or 90 degrees.

ANSWER:

.ø .ø .

.ø .ø .

.ø ..

ø .

.

.

.ø .- ].] ].] DPT JJ . .

= ø .

.

.

.

R

= ø .

.

Correct

Part H

If and are parallel,

Express your answer in terms of and .

**Hint 1. **What is the angle between parallel vectors?

The angle between vectors that are parallel is equal to 0.

ANSWER:

Correct

Finding the Cross Product

The figure shows two vectors and separated by an angle .

You are given that , , and

.

Part A

Express as an ordered triplet of values, separated by commas.

ANSWER:

.

.

= ø .

.

. . J56

.- .-

.g.- .

.

Correct

Part B

Find the magnitude of .

ANSWER:

Correct

Part C

Find the sine of the angle between and .

ANSWER:

Correct

Exercise 1.40

In each case, find the *x*and

*y*components

of vector .

Part A

4.60 6.30

Enter your answers numerically separated by a comma. Express your answers using three significant

figures.

ANSWER:

Correct

.= 0,0,10

.

] ]. = 10

. .

TJO J = 0.707

.

– . Ã % ?

& ?

4 , 5 = 4.60,6.30

Part B

13.2 8.91

Enter your answers numerically separated by a comma. Express your answers using three significant

figures.

ANSWER:

Correct

Part C

15.0 22.4

Enter your answers numerically separated by a comma. Express your answers using three significant

figures.

ANSWER:

Correct

Part D

, where 4 6

Enter your answers numerically separated by a comma. Express your answers using one significant figure.

ANSWER:

Correct

Problem 1.81

While following a treasure map, you start at an old oak tree. You first walk 825 directly south, then turn and walk 1.25

at 30.0 west of north, and finally walk 1.00 at 40.0 north of east, where you find the treasure: a biography of

Isaac Newton!

Part A

To return to the old oak tree, in what direction should you head ? Use components to solve this problem.

– . Ã & ?

% ?

4 , 5 = 8.91,13.2

– Ã . % ?

& ?

4 , 5 = 15.0,22.4

– . ? – ? Ã % ?

& ?

4 , 5 = 20,30

N

LN È LN È

ANSWER:

Correct

Part B

To return to the old oak tree, how far will you walk? Use components to solve this problem.

ANSWER:

Correct

Score Summary:

Your score on this assignment is 63.1%.

You received 6.94 out of a possible total of 11 points.

= 8.90 J È west of south

= 911 N