Articles

13.4: Review Problems - Mathematics


1. Let (P_{n}(t)) be the vector space of polynomials of degree (n) or less, and (frac{d}{dt} colon P_{n}(t) o P_{n}(t)) be the derivative operator. ( extit{Recall from chapter 6 that the derivative operator is linear.

2. When writing a matrix for a linear transformation, we have seen that the choice of basis matters. In fact, even the order of the basis matters!

  1. Write all possible reorderings of the standard basis ((e_{1},e_{2},e_{3})) for (Re^{3}).
  2. Write each change of basis matrix between the standard basis and each of its reorderings. Make as many observations as you can about these matrices: what are their entries? Do you notice anything about how many of each type of entry appears in each row and column? What are their determinants? (Note: These matrices are known as ( extit{permutation matrices}).)
  3. Given (L:Re^{3} o Re^{3}) is linear and [Legin{pmatrix}xyzend{pmatrix}=egin{pmatrix}2y-z3x2z+x+yend{pmatrix}]

write the matrix (M) for (L) in the standard basis, and two reorderings of the standard basis. How are these matrices related?

3. Let $$X={heartsuit,clubsuit,spadesuit}, ,quad Y={*,star}, .$$ Write down two different ordered bases, (S,S') and (T,T') respectively, for each of the vector spaces (mathbb{R}^{X}) and (mathbb{R}^{Y}). Find the change of basis matrices (P) and (Q) that map these bases to one another. Now consider the map
$$
ell:Y o X, ,
$$
where (ell(*)=heartsuit) and (ell(star)=spadesuit). Show that (ell) can be used to define a linear transformation (L:mathbb{R}^{X} omathbb{R}^{Y}). Compute the matrices (M) and (M') of (L) in the bases (S,T) and then (S',T'). Use your change of basis matrices (P) and (Q) to check that (M'=Q^{-1}MP).

4. Recall that (tr MN = tr NM). Use this fact to show that the trace of a square matrix (M) does not depend not the basis you used to compute (M).

5. When is the (2 imes 2) matrix (egin{pmatrix}a & b c & dend{pmatrix}) diagonalizable? Include examples in your answer.

6. Show that similarity of matrices is an ( extit{equivalence relation}).

7. ( extit{Jordan form})
a) Can the matrix (egin{pmatrix}
lambda & 1
0 & lambda
end{pmatrix}) be diagonalized? Either diagonalize it or explain why this is impossible.

b) Can the matrix (egin{pmatrix}
lambda & 1 & 0
0 & lambda & 1
0 & 0 & lambda
end{pmatrix}) be diagonalized? Either diagonalize it or explain why this is impossible.

c) Can the (n imes n) matrix (egin{pmatrix}
lambda & 1 & 0 & cdots & 0 & 0
0 & lambda & 1 & cdots & 0 & 0
0 & 0 & lambda & cdots & 0 & 0
vdots & vdots & vdots & ddots & vdots & vdots
0 & 0 & 0 & cdots & lambda & 1
0 & & 0 & cdots & 0 & lambda
end{pmatrix}) be diagonalized? Either diagonalize it or explain why this is impossible.

( extit{Note:}) It turns out that every matrix is similar to a block matrix whose diagonal blocks look like diagonal matrices or the ones above and whose off-diagonal blocks are all zero. This is called the ( extit{Jordan form}) of the matrix and a (maximal) block that looks like
[
left(
egin{array}{ccccc}
lambda & 1 & 0&cdots & 0
0 & lambda & 1 & & 0
vdots & &ddots &ddots &
&&&lambda&1
0 &0 && 0 & lambda
end{array} ight)
]
is called a ( extit{Jordan (n)-cell}) or a ( extit{Jordan block}) where (n) is the size of the block.

8. Let (A) and (B) be commuting matrices (( extit{i.e.}), (AB = BA)) and suppose that (A) has an eigenvector (v) with eigenvalue (lambda). Show that (Bv) is also an eigenvector of (A) with eigenvalue (lambda). Additionally suppose that (A) is diagonalizable with distinct eigenvalues. What is the dimension of each eigenspace of (A)? Show that (v) is also an eigenvector of (B). Explain why this shows that (A) and (B) can be ( extit{simultaneously diagonalized}) (( extit{i.e.}) there is an ordered basis in which both their matrices are diagonal.)


13.4: Review Problems - Mathematics

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13.4: Review Problems - Mathematics

All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited.

Feature Papers represent the most advanced research with significant potential for high impact in the field. Feature Papers are submitted upon individual invitation or recommendation by the scientific editors and undergo peer review prior to publication.

The Feature Paper can be either an original research article, a substantial novel research study that often involves several techniques or approaches, or a comprehensive review paper with concise and precise updates on the latest progress in the field that systematically reviews the most exciting advances in scientific literature. This type of paper provides an outlook on future directions of research or possible applications.

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to authors, or important in this field. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.


Math 233 FALL 2012

1. Homework:
We will be using WebAssign for homework.

The access code (class key) to self-enroll is:

Section 1: WUSTL 0199 1768

Section 2: WUSTL 8806 1253

There are going to be 11 sets of homework assignments. The lowest homework grade will be dropped, and the remaining 11 will be counted towards your final course grade. The homework is due on Wednesdays 5pm (starting from Wednesday September 12).
No late homework will be accepted.

2. TA Information: The Teaching Assistant for this course is Brady Rocks ([email protected] ). He is at the Calculus Help desk (Lopata 323) 3-5pm Monday and Tuesday. There will be also a review session 5:30-7pm on every Thursday (except for the exam weeks) in Room 199 of Cupples I. For now, the review session will be like office hours, but if there is a certain topic you want to be reviewed, you can email Brady Rocks in advance.

3. Exams: There will be three evening exams during the semester, E1, E2, E3. There will also be a final F.
Solutions to the exams will be available the day after each exam.

Exam Date Location Time Solutions
E1 September 19 TBA 7-9PM Solutions
E2 October 17 TBA 7-9PM Solutions
E3 November 14 TBA 7-9PM Solutions
F December 13 TBA 3:30-5:30pm Solutions

Just before the exam, you can look up your exam room assignment on the web at this link: http://www.math.wustl.edu/seatlookup (the course number is 233 and the exam number for midterm one is 1)
You will be allowed to enter the exam room a few minutes before the starting time.

4. Grading Information:
The three evening exams, E1, E2, E3, and the final exam F, each out of 100, will be combined in the following way to obtain an exam total grade S, out of 100:

    S := (2E1 + 2E2 + 2E3 + 3F)/9

If your final grade F is larger than the minimum of E1, E2, E3, then the minimum of the midterm exams will be dropped out and will be replaced by the final exam score.

Your course grade will be based on your Webwork score HW, also scaled to 100, and your exam total grade S as follows:

    T := 0.85*S + 0.15*HW

For example, if a student obtains midterm exam scores of 86, 60, 74, a final exam score of 78, and a homework grade of 88 , then the 78 replaces the 60, and S is equal to 78.88.
The final grade will be 0.85*78.88 + 0.15*88 = 80.24.


Policy on missed exams. If you are unable to take one of the Exams E1, E2, or E3 for legitimate reasons (such as verified illness or serious family emergency), you will not be given a make-up exam. You first should talk to Dr. Blake Thornton and explain why you missed the exam if everything is in order, you will get an excused absence. Your grade for the missed exam will be calculated by a statistical procedure which uses your scores on the other taken exams including the Final. If you miss the Final exam, and for that receive an excused absence then you must take a make-up final exam (usually at the beginning of the spring semester).


13.4: Review Problems - Mathematics

1. Estimate the area of the region between (fleft( x ight) = - 2 + 4) the (x)-axis on (left[ <1,4> ight]) using (n = 6) and using,

  1. the right end points of the subintervals for the height of the rectangles,
  2. the left end points of the subintervals for the height of the rectangles and,
  3. the midpoints of the subintervals for the height of the rectangles.

Show All Solutions Hide All Solutions

a The right end points of the subintervals for the height of the rectangles. Show Solution

The widths of each of the subintervals for this problem are,

We don’t need to actually graph the function to do this problem. It would probably help to have a number line showing subintervals however. Here is that number line.

In this case we’re going to be using right end points of each of these subintervals to determine the height of each of the rectangles.

The area between the function and the (x)-axis is then approximately,

As we found in the previous part the widths of each of the subintervals are (Delta x = frac<1><2>).

Here is a copy of the number line showing the subintervals to help with the problem.

In this case we’re going to be using left end points of each of these subintervals to determine the height of each of the rectangles.

The area between the function and the (x)-axis is then approximately,

As we found in the first part the widths of each of the subintervals are (Delta x = frac<1><2>).

Here is a copy of the number line showing the subintervals to help with the problem.

In this case we’re going to be using midpoints of each of these subintervals to determine the height of each of the rectangles.


Common Core’s Newer Math

T he following sentences from the New York Times could have been written today in homage to the Common Core Standards Initiative, the recently adopted national standards for the teaching of mathematics and English-language arts in grades K–12.

“Instead of this old method, the educators would stress from the earliest grades the new concept of the unity of mathematics and an understanding of its structure, using techniques that have been developed since the turn of the century. . . . The new concepts must be taught in high school to prepare the students for the type of mathematics that they will find when they reach college.”

But the century in question here is the 20th, not the 21st. This article, written in 1961, is not about today’s Common Core, but about New Math, the program that was supposed to transform mathematics education by emphasizing concepts and theories rather than traditional computation. Instead, after a few short years of propagating ignorance of all things mathematical, New Math became the butt of jokes nationwide (the Peanuts comic strip took aim more than once) before it was unceremoniously abandoned.

Flash forward 50 years, and Common Core is today making the same promises: “The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.”

But what makes us think Common Core will live up to its hype? And how is it substantially different from New Math, as well as subsequent math programs such as Sequential Math, Math A/B, and the National Council of Teachers of Mathematics Standards? These have all failed America’s children — even though each program promised to transform them into young Einsteins and Aristotles.

The problem with Common Core is not that it provides standards, but that, despite its claims, there is a particular pedagogy that accompanies the standards. And this pedagogy is flawed, for, just as in New Math, from the youngest ages Common Core buries students in concepts at the expense of content.

Take, for example, my first-grade son’s Common Core math lesson in basic subtraction. Six- and seven-year-olds do not yet possess the ability to think abstractly their mathematics instruction, therefore, must employ concrete methodologies, explanations, and examples. But rather than, say, count on a number line or use objects, Common Core’s standards mandate teaching first-graders to “decompose” two-digit numbers in an effort to emphasize the concept of place value. Thus 13 – 4 is warped into 13 – 3 = 10 – 1 = 9. Decomposition is a useful skill for older children, but my first-grade son has no clue what it is about or how to do it. He can, however, memorize the answer to 13 – 4. But Common Core does not advocate that tried-and-true technique​ .

Common Core’s elevation of concept over computation continues in its place-value method for multiplying two-digit numbers, which is taught in fourth grade. Rather than multiply each digit of the number from right to left, Common Core requires students to multiply each place value so that they have to add four numbers, rather than two, as the final step in finding the product.

Common Core’s most distinctive feature is its insistence that “mathematically proficient students” express understanding of the underlying concepts behind math problems through verbal and written expression. No longer is it sufficient to solve a word problem or algebraic equation and “show your work” now the work is to be explained by way of written sentences.

I have seen this “writing imperative” first-hand in my sons’ first- and third-grade Common Core math classes. There is certainly space in their respective books for traditional computation, but the books devote enormous space to word problems that have to be answered verbally as well as numerically, some in sections called Write Math. The reason, we are told, is that the Common Core–driven state assessments will contain large numbers of word problems and spaces for students to explain their answers verbally. This prescription immediately dooms grammar-school students who have reading difficulties or are not fluent in English: The mathematical numbers that they could have grasped are now locked into sentences they cannot understand.

The most egregious manifestation of the “writing imperative” is the Four Corners and a Diamond graphic organizer that my sons’ school has implemented to help prepare for the writing portion of the state assessments. The “fourth corner” requires students to explain the problem and solution in multiple sentences. How all this writing helps them with math is yet to be demonstrated.

Hence Common Core looks terribly similar to the failed New Math program, which also emphasized “the why rather than the how, the fundamental concepts that unify the various specialties, from arithmetic to the calculus and beyond, rather than the mechanical manipulations and rule memorizations.” Common Core may not completely eschew the “how,” and it may not be obsessed with binary sets and matrices as New Math was, but it is likely to lose the “how” — the content — in its efforts to move the “why” — the concepts — into the foreground.

The problem is not that students, including those in the primary grades, should not be presented the basic concepts of mathematics — they should be. But there is a difference between learning basic concepts and expressing the intricacies of true mathematical proofs that Common Core desires. Mathematical concepts require a high aptitude for abstract thinking — a skill not possessed by young children and never attained by many. What will happen to students who already struggle with math when they not only are forced to explain what they do not understand, but are presented new material in abstract conceptual formats?

All students must learn to perform the basic mathematical operations of addition, subtraction, multiplication, and division in order to function well in society. Knowing why these operations work as they do is a great benefit, but it is not essential. And in mathematics, concepts are often grasped long after students have mastered content — not before.

In trying to learn both the “why” and the “how” in order to prepare for the state assessments, students will not fully grasp either: They will not receive the instructional time needed to learn how to do the operations because teachers will be forced to devote their precious few classroom minutes to explaining concepts, as the assessments require. The “how” of the basic operations, which need to be memorized and practiced over and over, will be insufficiently learned, since Common Core orders teachers to serve two masters.

The result is simple arithmetic: Instead of developing college- and career-ready students, we will have another generation of students who cannot even make change from a $5 bill, all courtesy of the latest set of bureaucrat-promoted standards that promise to save American education.

By giving concept priority over content, Common Core has failed to learn the history lesson from New Math. Students instructed according to Common Core standards will ultimately know neither the “why” nor the “how,” and we will eventually consign these standards to the ever-expanding dustbin of failed educational initiatives, until the next messianic program is unveiled.

And, of course, this doomed educational experiment, like its predecessors, has a high cost: our children’s ability to do math.

— David G. Bonagura Jr. is a teacher and writer in New York. He has written about education for Crisis, The Catholic Thing, The University Bookman, and the Wall Street Journal.


Announcements:

  • Announced 11/29/18: My last list of review material:
    • Week 9 Newsletter
    • Final Exam Facts Summary: A brief summary of everything we've done!
    • Taylor Series Quick Summary Sheet.
    • Final Review Worksheet and Solutions.
    • And remember that any and all of the exams in the math department old 126 final archive are good to study.
    • Just for your interest (won't be on your test): Old honors worksheet on some Taylor series application.
    • Week 8 Newsletter
    • Here is a detailed Overview of Taylor Notes 1, 2, and 3.
    • Here is a detailed Overview of Taylor Notes 4 and 5.
    • Here is a Taylor Series Fact Sheet.
    • Taylor Notes: This is the text for what we are covering now.
    • Week 7 Newsletter
    • Exam 2 Fact Sheet and Exam 2 Review
    • 15.2 and 15.3 Review: Try the problems in this review to practice reversing the order.
    • 15.2 Region Practice - Solutions: Use this to practice describing regions.
    • Even More Practice with 15-2 - Solutions
    • integrating powers of sine and cosine
    • Week 6 Newsletter
    • 14.4 and 14.7 Full Examples
    • 15.1 Review.
    • 15.2 Region Practice - Solutions
    • 15.2 and 15.3 Review.
    • Here is a list of integrals we know in one step. These integrals are the main ones from Math 125 that we quote in one step. For all others, we use some technique to simplify to one of these.
    • If you are worried about your integration skills, start with this warm up: Here are 24 very basic integrals (they only require simplification or substitution and should be very easy) and here are the solutions.
    • In section 15.3, you will need to be able to integrate trig functions well. Here are a few examples integrals involving powers of sine and cosine.
    • There may be a few problems that require other techniques. You are expected to know all integration techniques from Math 125. If you don't remember a technique here is more review: Trig Sub and identities and Summary of all methods See sections 7.1-7.5 of the book for more review.
    • Version 1
    • Version 2
    • Week 5 Newsletter.
    • Partial Derivatives Examples and a Review of Derivative Rules (including implicit differentiation).
    • 14.3 Review.
    • Here is an overview of max/min facts from calculus 1 and calculus 3.
    • Week 4 Newsletter
    • 10.3 Review (Polar Coordinates)
    • Trig Fact Sheet
    • 13.3, 13.4, and 14.1 Review (Read this for an example of how to do 14.1 HW problems)
    • Week 3 Newsletter
    • 13.1 Summary
    • 13.2 Summary
    • 13.3 Summary
    • 13.3 Practice
    • 13.4 Summary
    • Exam 1 Review
    • Exam 1 Fact Sheet
    • Week 2 Newsletter
    • 12.5 Summary
    • 12.5 Visuals and Derivations
    • 12.5 Flowcharts
    • 12.5 Practice Problems and full solutions: I strongly advise that you attempt these problems on your own in addition to the homework.
    • 12.6 Summary Notes: Please print these off for reference, we are discussing names and several 3D shapes and it is nice to have the pictures from these lecture notes for reference.
    • Conic Sections: This is an optional supplement that lets you know what conic sections are and it gives a visual of the shape of a hyperbola.
    • Week 1 Newsletter.
    • Overview of 12.1-12.4: quick overview.
    • Review of 12.1: more detailed review of this section with examples (also a full review of how to complete the square).
    • Review of 12.2: reference sheet of some basic vector facts you need.
    • Review of 12.3: summary of the dot product and its applications.
    • Review of 12.4: summary of the cross produce and its applications.
    • A supplemental discussion on vectors and physics: This is just for your own interest (most of this is not covered on the exam), but it may better help you see the applications in the vectors and vector operations we have been discussing in class.

    The only thing that students are required to purchase for this course is an access code for Webassign.

    If you took Math 124 and/or Math 125 at UW recently and purchased a Lifetime of the Edition (LOE) or multi-term access code for one of those courses, you do not need to purchase anything further.

    Otherwise, the cheapest option (and the one I recommend) is to follow the link this page and purchase "WebAssign Instant Access for Calculus, Multi-Term Courses, 1st Edition" for $87.00. This will come with an electronic version of the text, which I've found is sufficient for most students. If you want a hard copy of the text, you may purchase one at the bookstore (approximately $60 for a used copy). In addition, the Multi-Term Course code will work for Math 324 (which also uses Stewart's text and Webassign) or if you must retake Math 126 in a future quarter, as long as the 8th edition is in use. (Do note, however, that access is purchased for the text. Since Math 124, 125, 126, and 324 use the same text, the same code works for all these courses (provided it is multi-term access). If you take a course that uses a different textbook (such as Math 308), you will be required to purchase Webassign access for that textbook.)


    Go Math Grade 4 Answer Key Homework FL Chapter 13 Algebra: Perimeter and Area Review/Test

    Go Math Answer Key will give you the perfect answers with a clear explanation of every question in an easy way. Go Math explained clearly about Perimeter and Area of different Shapes by using images, indicating images with arrows, and numbers.

    Chapter: 13 – Review/Test

    Review/Test – Page No. 519

    Choose the best term from the box.

    Question 1.
    The number of square units needed to cover a flat surface is the ______________.
    _________

    Answer: Area
    The number of square units needed to cover a flat surface is the area.

    Question 2.
    The distance around a shape is the ____________.
    _________

    Answer: Perimeter
    The distance around a shape is the Perimeter.

    Find the area of the rectangle or combined rectangles.

    Question 3.

    A = _____ square feet

    Explanation:
    Given,
    length = 14 ft
    Width = 8 ft
    Area of the rectangle = l × w
    A = 14 ft × 8 ft
    A = 112 square feet
    Thus the area of the rectangle for the above figure is 112 square feet.

    Question 4.

    A = _____ square centimeters

    Explanation:
    Figure 1:
    S = 3 cm
    Area of the square = s × s
    A = 3 cm × 3 cm
    A = 9 square cm
    Figure 2:
    b = 11 cm
    h = 4 cm
    Area of the rectangle = b × h
    A = 11 cm × 4 cm
    A = 44 square cm
    Figure 3:
    b = 2 cm
    h = 6 cm
    Area of the rectangle = b × h
    A = 2 cm × 6 cm
    A = 12 square cm
    Area of the composite figure is 9 sq. cm + 44 sq. cm + 12 sq. cm = 65 square cm.

    Find the unknown measure of the rectangle.

    Question 5.

    A = _____ square yards

    Explanation:
    Figure 1:
    b = 12 yd
    h = 8 yd
    Area of the rectangle = b × h
    A = 12 yd × 8 yd
    A = 96 square yard.
    Figure 2:
    b = 16 yd
    h = 1 yd
    Area of the rectangle = b × h
    A = 16 yd × 1 yd
    A = 16 square yard.
    Figure 3:
    b = 4 yd
    h = 6 yd
    Area of the rectangle = b × h
    A = 4 yd × 6 yd
    A = 24 square yard.
    The area of the composite figure is 96 square yard + 16 square yard + 24 square yard = 136 square yard.

    Question 6.

    Perimeter = 60 meters
    width = _____ m

    Explanation:
    Given,
    Perimeter = 60 meters
    length = 18 m
    width = _____ m
    The perimeter of the rectangle = l + w + l + w
    P = 2l + 2w
    60 m = 2 × 18 m + 2w
    60 m – 36 m = 2w
    2w = 24
    w = 24/2
    w = 12 meters
    Thus the width of the above rectangle is 12 meters.

    Question 7.

    Area = 91 square feet
    height = _____ feet

    Explanation:
    Given,
    Area = 91 square feet
    base = 7 ft
    height = _____ feet
    Area of the rectangle = b × h
    91 sq ft = 7 ft × h
    h = 91/7
    h = 13 ft
    Thus the height of the above rectangle is 13 ft.

    Question 8.

    Area = 60 square inches
    base = _____ in.

    Explanation:
    Given,
    Area = 60 square inches
    height = 6 in
    base = _____ in.
    Area of the rectangle = b × h
    60 square inches = b × 6 in
    b = 60/6
    b = 10 inches
    Thus the base of the above rectangle is 10 inches.

    Question 9.
    What is the perimeter of a rectangle with a length of 13 feet and a width of 9 feet?
    P = _____ ft

    Explanation:
    Given,
    l = 13 ft
    w = 9 ft
    The perimeter of the rectangle = l + w + l + w
    P = 13 ft + 9 ft + 13 ft + 9 ft
    P = 44 ft
    Thus the perimeter of the rectangle is 44 ft.

    Review/Test – Page No. 520

    Fill in the bubble completely to show your answer.

    Question 10.
    Which pair of shapes has the same area?
    Options:
    a.
    b.
    c.
    d.

    Answer:

    Explanation:
    a.
    4 × 2 = 8
    3 × 3 = 9
    8 ≠ 9
    b.
    4 × 4 = 16
    3 × 5 = 15
    16 ≠ 15
    c.
    3 × 4 = 12
    2 × 6 = 12
    12 = 12
    Thus the correct answer is option c.

    Question 11.
    Jamie’s mom wants to enlarge her rectangular garden by adding a new rectangular section. The garden is now 96 square yards. What will the total area of the garden be after she adds the new section?
    Options:
    a. 84 square yards
    b. 96 square yards
    c. 180 square yards
    d. 192 square yards

    Explanation:
    Given that,
    Jamie’s mom wants to enlarge her rectangular garden by adding a new rectangular section. The garden is now 96 square yards.
    Add 96 square yards to the rectangular garden.
    96 square yards + 96 square yards = 192 square yards
    Thus the correct answer is option d.

    Question 12.
    A rectangular yoga studio has an area of 153 square feet. The width of the studio is 9 feet. What is the length of the studio?
    lenght = _____ ft

    Explanation:
    Given,
    A rectangular yoga studio has an area of 153 square feet.
    The width of the studio is 9 feet.
    Area of the rectangle = l × w
    153 square feet = l × 9 ft
    l = 153/9
    l = 17 ft
    Therefore the length of the studio is 17 feet.

    Review/Test – Page No. 521

    Fill in the bubble completely to show your answer.

    Question 13.
    Mr. Patterson had a rectangular deck with an area of 112 square feet built in his backyard. Which could be a diagram of Mr. Patterson’s deck?
    Options:
    a.
    b.
    c.
    d.

    Answer: c.

    Explanation:
    Given,
    Mr. Patterson had a rectangular deck with an area of 112 square feet built in his backyard.
    Area of the rectangle = l × w
    A = 28 ft × 4 ft
    A = 112 square feet
    Thus the correct answer is option c.

    Question 14.
    The town indoor pool is in a rectangular building. Marco is laying tile around the rectangular pool. How many square meters of tile will Marco need?

    Options:
    a. 96 square meters
    b. 252 square meters
    c. 572 square meters
    d. 892 square meters

    Explanation:
    The outer rectangle is
    l = 26 m
    w = 22 m
    Area of the rectangle = l × w
    A = 26 m × 22 m
    A = 572 square meters
    The inner rectangle is
    l = 20 m
    w = 16 m
    Area of the rectangle = l × w
    A = 20 m × 16 m
    A = 320 square meters
    Thus the square meters of tile will Marco need is 572 – 320 = 252 square meters.

    Review/Test – Page No. 522

    Question 15.
    A drawing of a high school pool is shown below.

    What is the area of the pool? Explain how you know.
    _______ square yards

    Explanation:
    Figure 1:
    l = 10 yd
    w = 15 yd
    Area of the rectangle = l × w
    A = 10 yd × 15 yd
    A = 150 square yard
    Figure 2:
    l = 15 yd
    w = 10 yd
    Area of the rectangle = l × w
    A = 15 yd × 10 yd
    A = 150 square yard
    Figure 3:
    l = 10 yd
    w = 5 yd
    Area of the rectangle = l × w
    A = 10 yd × 5 yd
    A = 50 square yard
    Area of the pool = 150 square yard + 150 square yard + 50 square yard = 350 square yard

    Question 16.
    Mr. Brown has 24 meters of fencing. He wants to build a rectangular pen for his rabbits.

    A. Draw two different rectangles that Mr. Brown could build. Use only whole numbers for the lengths of the sides of each rectangle. Label the length of each side.
    Type below:
    _________

    Answer:

    Question 16.
    B. Find the area in square meters of each rabbit pen you made in Part A. Show your work.
    Type below:
    _________

    Explanation:
    l = 10m
    w = 2m
    Area of the rectangle = l × w
    A = 10 m × 2 m
    A = 20 square meters
    Therefore the area in square meters of each rabbit pen is 20 square meters.

    Question 16.
    C. If you were Mr. Brown, which of the two pens above would you construct for your rabbits? Explain why.
    Type below:
    _________

    Answer: I would construct the second figure for the two rabbit pens.

    Conclusion:
    By following the Go Math Grade 4 Review Test solutions, students can quickly find the perimeter and area of shapes within a few minutes. Keep in touch with us to get the Go Math Grade 4 Answer Key Chapter 13 Algebra: Perimeter and Area.


    Mini-Math Mystery Activity 1st Grade - Addition within 20 Facts & Word Problems

    Print the single worksheet page, make as many copies as you need for your kids, and the activity is ready to go!

    Printable awards provided to give to your Detectives after solving the mystery.

    These Mini-Math Mysteries are different from my original Math Mystery Packets. They are designed as a condensed activity for a quick one-page practice of math skills. The elimination process is only based on answer matching with the images in this series. If you would like to try the other type of original Math Mysteries to see what’s included in those packets and the difference in design, please visit this FREE Multi-Grade Math Mystery Packet, The Case of the Super Bad Superhero.

    Add extra motivation with this Detective Rank Chart Download the Mystery Record & Rank Chart FREEBIE HERE

    For more ideas, activities, and resources, follow my store (by clicking the little green star) to stay updated on new releases. We can also stay connected via:

    You may also want to check out these other resources from my store:


    Free Common Core: 6th Grade Math Practice Tests

    Varsity Tutors&rsquo Learning Tools provide sixth grade students with a wide range of study materials to help improve their test-taking abilities. By visiting the Learning Tools interface on the website, students can familiarize themselves with a number of concepts to help improve their SIxth Grade Mathematics skills. Varsity Tutors offer free Common Core Sixth Grade Mathematics study material to help prepare any learner for the exam your state may use to evaluate students&rsquo Common Core curriculum mastery.

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    Watch the video: Visual Calculus 1: D = R T (November 2021).