A carpenter has a box of nails of various
different lengths. You decide to practice your
weighted averaging skills to figure out the
average length of a nail in the box. You grab
two handfuls of nails and count out the
number of each type of nail. You record your
data in the table below.
Sample
Type
Short nail
Medium nail
Long nail
Number Abundance
of Nails
(%)
67
18
10
[?]
Nail Length
(cm)
2.5
5.0
7.5
What is the percent abundance of the
medium nails in your sample?

In scientific notation the distance of the Barnard’s star from Earth, is given as 3.5 × 10^13

Scientific notation is a way of writing very large or very small numbers in a more compact form using powers of 10. In scientific notation, a number is expressed as a product of a coefficient and 10 raised to a power. The coefficient must be greater than or equal to 1 but less than 10. The power of 10 indicates the number of places the decimal point is moved to obtain the original number.

For 35,000,000,000,000

you have to start counting the digits from the right. The decimal point would occur just before the first digit (that is the number 3)

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Answer:

12×2, 3×8, 4×6

Step-by-step explanation:

To find the area of a parallelogram, the formula is A = bh.

1. Height: 6 ft

Base: 2 ft

6 × 2 = 12 ft

So it does not work.

2. Height: 12 ft

Base: 2 ft

12 × 2 = 24 ft

So it does work.

3. Height: 10 ft

Base: 14 ft

14 × 10 = 140 ft

So it does not work.

4. Height: 3 ft

Base: 4 ft

3 × 4 = 12 ft

So it does not work.

5. Height: 3 ft

Base: 8 ft

3 × 8 = 24 ft

So it does work.

6. Height: 4 ft

Base: 6 ft

4 × 6 = 24 ft

So it does work.

Answer:

57

57

123

123

57

57

123

that's all.

Answer:

m<1 = 57°

m<2 = m<1 = 57°

m<3 = x = 123°

m<4 = x = 123°

m<5 = m<1 = 57°

m<6 = m<5 = 57°

m<7 = m<4 = 123°

Step-by-step explanation:

[tex]{ \tt{m \angle 1 + x = 180 \degree}} \\ { \colorbox{silver}{corresponding \: angles}} \\ { \tt{m \angle 1 = 180 - 123}} \\ { \tt{ \underline{ \: m \angle 1 = 57 \degree \: }}}[/tex]

Answer:

350.3 g

Step-by-step explanation:

310 * 113% = 310 * 1.13 = 350.3 g

Hope this helps!

Answer:

Step-by-step explanation:

According to the problem the answer is cos 9 = sin 56.3.

Cos is a mathematical function that returns the ratio of the side adjacent to a given angle of a right triangle to the hypotenuse of the triangle. It is one of the most commonly used trigonometric functions and is used in various mathematical calculations such as finding angles, distances, and wave lengths. Cos is often written as cosine and is typically abbreviated as "cos." It can also be written as COS or csc. Cos is typically used in conjunction with sin and tan to measure angles and distances in a triangle. Cos can also be used to calculate wave lengths in physics and to determine the magnitude of a vector in vector calculus. Cos is an important concept in mathematics and is used in many areas of science, engineering, and technology.

Use the identity sin(90°-x) = cos x:

sin 56.3 = cos (90° - 56.3)

cos 9 = cos (90° - 56.3)

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For cos 9°, angle lies between 0° and 90° (First Quadrant). Since cosine function is positive and in the first quadrant, thus cos 9° value = 0.9876883. Also Cos 9 = √(1-sin²(9°)).

Cos is a mathematical function that returns the ratio of the side adjacent to a given angle of a right triangle to the hypotenuse of the triangle. It is one of the most commonly used trigonometric functions and is used in various mathematical calculations such as finding angles, distances, and wave lengths. Cos is often written as cosine and is typically abbreviated as "cos." It can also be written as COS or csc.

The value of cos 9 degrees in decimal is 0.987688340, since Cos 9 degrees can also be expressed using the equivalent of the given angle (9 degrees) in radians (0.15707)

Using degree to radian conversion, θ in radians is:

= θ in degrees × (pi/180°)

⇒ 9 degrees

= 9° × (π/180°) rad

= π/20 or 0.1570

∴ cos 9° = cos(0.1570)

= 0.9876883

Since the cosine is the periodic function, we can represent cos 9° as,

So, cos 9 degrees = cos(9° + n × 360°).

⇒ cos 9°

= cos 369°

= cos 729°, and so on.

Cos 9 = √(1-sin²(9°))

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If a relation R is an equivalence relation, then R will be circular.

If a relation R is an equivalence relation, then it must satisfy three properties: reflexivity, symmetry, and transitivity.

(i) Reflexivity: For any element "a" in the set, "aRa" must be true.

(ii) Symmetry: For any elements "a" and "b" in the set, if "aRb" then "bRa" must be true.

(iii) Transitivity: For any elements "a", "b", and "c" in the set, if aRb and bRc then "aRc" must be true.

A Circular relation is where "aRb" and "bRc" imply "cRa".

This is a stronger condition than the transitivity property of an equivalence relation, which only requires that "aRb" and "bRc" together imply "aRc".

Therefore, a relation that is an equivalence-relation have to be circular.

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The given question is incomplete, the complete question is

A relation R is called circular if "aRb" and "bRc" imply that "cRa". if R is an equivalence relation, then would R be circular?

The required probability of W being greater than 6 is 0.065 or 6.5%.

A probability value represents the likelihood that an occurrence will take place. In terms of percentage notation, it is stated as a number between 0 and 1, or between 0% and 100%. The higher the probability, the more probable it is that the event will take place.

The complement of an event A is the event that A does not occur. In this case, the event A is "W ≤ 6", and its complement is "W > 6".

We are given that P(W ≤ 6) = 0.935. Using the complement rule of probability, we can find the probability of W > 6 as follows:

[tex]$$P(W > 6) = 1 - P(W \leq 6)$$[/tex]

Substituting the given value, we have:

P(W > 6) = 1 - 0.935 = 0.065

Therefore, the probability of W being greater than 6 is 0.065 or 6.5%.

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The inequality is X > 0 and a word sentence represent the graph is  X  the graph of a number line with an open circle on zero and an arrow pointing to the right.

The inequality X > 0 represents the graph of a number line with an open circle on zero to left and an arrow pointing to the right. This means that any value of X that is greater than zero is a valid solution for the inequality.

In other words, X can be any positive number, such as 1, 2, 3, and so on. However, X cannot be zero or any negative number, as those values do not satisfy the inequality. Therefore, the word sentence that represents this inequality is "X is greater than zero."

This means that X must be a positive number, and it can be any value that is greater than zero.

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The solution to the system of equations is (x, y) = (-4, 12) or (3, 5).

simultaneous equations, or system of equations Two or more equations in algebra must be solved concurrently (i.e., the solution must satisfy all the equations in the system). The number of equations must match the number of unknowns for a system to have a distinct solution. Even then, a solution is not assured.

We can solve this system of equations by substitution or elimination. Here, we will use substitution:

Substitute y in the first equation with its expression from the second equation:

x + (x^2 - 4) = 8

Now, we have a quadratic equation in x:

x^2 + x - 12 = 0

Factor the quadratic equation:

(x + 4)(x - 3) = 0

So, either x + 4 = 0 or x - 3 = 0:

x = -4 or x = 3

Substitute each value of x into either equation to find the corresponding value of y:

If x = -4, then y = (-4)^2 - 4 = 12

If x = 3, then y = 3^2 - 4 = 5

Therefore, the solution to the system of equations is (x, y) = (-4, 12) or (3, 5).

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The least important value to today's college students according to the yearly values survey is Political involvement. So the option B is correct.

The survey found that only 32% of college students believed that political involvement was important. This is significantly lower than the other values surveyed such as Financial Success (86%) and Community Involvement (71%). As such, it can be concluded that Political Involvement is the least important value to today's college students.

This may be due to a variety of factors, such as a lack of understanding of the political system, a lack of interest in politics, or a sense that individual political action will not make a difference.

It is important to recognize, however, that political involvement is still an important part of civic engagement and an essential part of the democratic process. As such, college students should strive to become more informed about politics and to find ways to get involved in the political process.

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The complete question is:

Which of the following values, which is least important to today's college students according to the yearly values survey?

A. Career success

B. Political involvement

C. Healthy lifestyle

D. Social media presence

The width of the blanket to the nearest tenth of an inch is approximately 67.4 inches.

What is width in rectangle ?

In a rectangle, the width refers to the measurement of the shorter side of the rectangle, which is perpendicular to its length.

Let x be the width of the blanket in inches. Then the length of the blanket is x + 18 inches.

The area of the blanket is given by:

Area = Length x Width

5760 sq inches = (x + 18) inches * x inches

Expanding the right-hand side and solving for x, we get:

[tex]x^2 + 18x - 5760 = 0[/tex]

We can solve this quadratic equation using the quadratic formula:

[tex]x = (-b \pm \sqrt{(b^2 - 4ac)}) / 2a[/tex]

where a = 1, b = 18, and c = -5760. Plugging in these values, we get:

[tex]x = (-18 \pm \sqrt{(18^2 - 4(1)(-5760))}) / 2(1)[/tex]

[tex]x = (-18 \pm \sqrt{(18^2 + 23040)}) / 2[/tex]

x ≈ 67.42 or x ≈ -85.42

Since the width cannot be negative, the width of the blanket to the nearest tenth of an inch is approximately 67.4 inches.

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Here's a graph of the cost function C = 25000p / (100 - p), using graphing utility. The graph shows that the cost function is continuous and defined for all values in its domain.

The domain of the function is restricted to 0 ≤ p ≤ 100, since it doesn't make sense to supply recycling bins to more than 100% of the population, or to a negative percentage of the population.

As p approaches 100%, the cost of supplying recycling bins becomes very large, since it becomes increasingly expensive to provide bins to a large portion of the population. As p approaches 0%, the cost also approaches 0, since there are fewer bins needed for a smaller population.

The function has a vertical asymptote at p = 100, since the denominator of the fraction approaches 0 as p approaches 100 from the left. This means that the cost becomes infinitely large as the percentage of the population approaches 100%.

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The scale factor of dilation of the shaded in gray to the shaded in black is 3

Given that

The preimage = shaded in gray

The image = shaded in black

From the graph, we have the following values on the image and the preimage

The preimage = shaded in gray = 4

The image = shaded in black = 12

The scale factor of dilation is then calculated as

Scale factor = shaded in black/shaded in gray

So, we have

Scale factor = 12/4

Evaluate

Scale factor = 3

Hence, the scale factor dilation is 3

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Answer: A matrix A of dimension n x n is called invertible if and only if there exists another matrix B of the same dimension, such that AB = BA = I, where I is the identity matrix of the same order.

Step-by-step explanation:

hope it helped! <3

Answer:

Step-by-step explanation:

7x - 2 = 5x + 14

2x - 2 = 14

2x = 16

x = 8

The total number of miles from Sharon's house to her mother's house would be = 60 miles.

The total amount of time it takes Sharon to reach the mother's house at constant speed = 180 minutes

The speed she used after the storm = 20 mile/hr

Therefore the miles she needs to cover before she reaches the mother's house = ?

That is;

20 miles = 1hr or 60 mins

X miles = 180 minutes

make X the subject of formula;

X miles = 20×180/60

= 3600/60

= 60 miles.

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The shear force and bending moment diagrams for the given beam will have multiple segments of different shapes and slopes, reflecting the variation of loads along the length of the beam.

To construct the shear and bending diagrams for the given beam, we need to analyze the beam for the different sections where the load is applied. We can break down the beam into five sections:

Leftmost section (0 ≤ x ≤ L/4)

Second section (L/4 < x ≤ L/2)

Third section (L/2 < x ≤ 5L/12)

Fourth section (5L/12 < x ≤ 7L/12)

Rightmost section (7L/12 < x ≤ L)

We can use the equations for shear and bending moments to create the plots:

For section 1: 0 ≤ x ≤ L/4

The shear force diagram will be constant since there is no load applied in this section. The bending moment diagram will be a sloping line, which will be zero at x = 0 and will increase linearly with x as we move toward the right end of the section.

For section 2: L/4 < x ≤ L/2

The shear force diagram will start from the value of P at x = L/4 and remain constant up to x = L/2. The bending moment diagram will be a parabolic curve, which will be zero at x = L/4 and L/2 and will reach a maximum value at the midpoint of the section.

For section 3: L/2 < x ≤ 5L/12

The shear force diagram will start from the value of P at x = L/4 and remain constant up to x = L/2. At x = 5L/12, a load of P/3 is added, causing the shear force to increase suddenly. The bending moment diagram will be a cubic curve, which will be zero at x = L/4 and L/2, and will have a local minimum at x = 5L/12.

For section 4: 5L/12 < x ≤ 7L/12

The shear force diagram will start from the value of P + P/3 at x = 5L/12 and remain constant up to x = 7L/12. The bending moment diagram will be a cubic curve, which will be zero at x = L/4 and L/2, and will have a local maximum at x = 7L/12.

For section 5: 7L/12 < x ≤ L

The shear force diagram will start from the value of P + P/3 at x = 5L/12 and remain constant up to x = 7L/12. At x = L/3, a load of P/3 is added, causing the shear force to decrease suddenly. The bending moment diagram will be a parabolic curve, which will be zero at x = L/4 and L/2 and will reach a minimum value at the midpoint of the section.

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Some of the tips which is used to conduct a survey are:

Conducting a survey is an important tool for gathering information from a large and diverse group of people. Its allows allow researchers to obtain data from a representative sample of the population, which can then be used to make informed decisions, identify trends, and measure changes over time.

Also important, its can provide insight into people's attitudes, beliefs, and behaviors, which can be valuable in developing marketing strategies, designing programs and policies, and making important decisions.

A paragraph in a designed questionnaire which can remind participants why your doing the survey and that their information will kept confidential is as follows: "Thank you for taking the time to participate in this survey. The purpose of this questionnaire is to gather information about your experiences and opinions on a particular topic. Your participation is crucial to help us gain insights and make informed decisions. All of the information provided in this survey will be kept strictly confidential and will only be used for research purposes".

Full question "In a designed questionnaire, remind participants why your doing the survey and that their information will kept confidential".

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The measurement of angle PCD in equilateral triangle  is 67.38°.

Let $s$ be the side length of the square $abcd$, and let $O$ be the center of the square. Then $OP = s/\sqrt{2}$.

Since triangle $ABP$ is equilateral, we have $\angle ABP = 60^\circ$ and $AP=BP=s$. Let $E$ be the foot of the perpendicular from $P$ to $AB$. Then $AE=BE=s/2$ and $EP=s\sqrt{3}/2$.

Since $EP$ is perpendicular to $AB$ and $AB$ is parallel to $CD$, we have $\angle PCD = \angle ACE$.

Since $OP$ is perpendicular to $AB$, we have $\angle OEP = \angle AEP = 30^\circ$. Also, $OE=OP/\sqrt{2}=s/2$.

Using the Pythagorean theorem in triangle $OCE$, we have

CE= [tex]\sqrt{(OE)^{2} + (OC)^{2}} = \sqrt{(s/2)^{2} + s^{2}} = s\sqrt{5}/2[/tex]

Therefore, $\sin \angle ACE = CE/CP = \sqrt{5}/2$. Thus,

∠PCD = ∠ACE = arcsin(√5/2) ≈ 67.38°

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The quοtient is [tex]6x^2 - 8xy + 17y^2 - 34y + 70[/tex], and the remainder is [tex]8y^3 - 40.[/tex]

A polynomial is a mathematical expression made up of variables (like x, y, and z) and coefficients (constants multiplied by the variables), as well as addition, subtraction, and multiplication operators. The variables and coefficients are raised to n-negative integer powers, also known as degrees.

We can use long division to divide fractions. Divide 6x³ + 4xy² + 3y³ + 13xy + 8 by x + 2 as shown:

        [tex]6x^2 - 8xy + 17y^2 - 34y + 70[/tex]

  -------------------------------------

[tex]x + 2 | 6x^3 + 4xy^2 + 3y^3 + 13xy + 8[/tex]

       [tex]- 6x^3 - 12x^2[/tex]

       ---------------

           [tex]12x^2 + 4xy^2[/tex]

            [tex]- 12x^2 - 24x[/tex]

            --------------

                    [tex]4xy^2 + 24x[/tex]

                    [tex]- 4xy^2 - 8y^3[/tex]

                    --------------

                              [tex]24x - 8y^3 + 8[/tex]

                              [tex]- 24x - 48[/tex]

                              ------------

                                       [tex]8y^3 - 40[/tex]

Therefοre, the quοtient is [tex]6x^2 - 8xy + 17y^2 - 34y + 70[/tex], and the remainder is [tex]8y^3 - 40[/tex].

Thus, we can express the οriginal pοlynοmial as:

[tex]6x^3+4xy^2+3y^3+13xy+8[/tex]

[tex]= (x + 2)(6x^2 - 8xy + 17y^2 - 34y + 70) + (8y^3 - 40) \div (x + 2)[/tex]

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The answer is 41 minutes.

explaination:

540 secs = 9 minutes

2/3 of 60 mins= 40 mins

(to find a fraction of a number you divide it by the denominator and times it after by the numerator)

9+40= 49

homework: 90 mins

reading and making lunch: 49 minutes

we want the difference so:

90-49= 41

Roberto spent 41 more minutes doing his homework than reading and making lunch.

Therefore, the coefficient of variation for the given data is approximately 24.71%.

A box plot, also known as a box-and-whisker plot, is a graphical representation of a data set that shows the distribution of the data using quartiles. It is a standardized way of displaying the distribution of data based on the five-number summary: minimum, first quartile, median, third quartile, and maximum. The box represents the middle 50% of the data, with the median marked by a line inside the box. The whiskers extend from the box to the minimum and maximum values, or to a certain range if there are outliers. Box plots are useful for comparing the distributions of different data sets and identifying potential outliers.

Here,

1. Stem-and-Leaf Plot:

A stem-and-leaf plot is a way to display data that separates the tens digit of each number from the ones digit. Here is the stem-and-leaf plot for the given data:

3 | 9 9

4 | 2 2 6 6

5 | 1 4 4 6

6 | 1 4 5 5 5 5 5 5

7 | 0 2 3 4

8 | 1 2 2 3 5

9 | 3 4

In this plot, the stem represents the tens digit and the leaves represent the ones digit.

2. Coefficient of Variation:

The coefficient of variation is a measure of the relative variability of a data set. It is calculated as the ratio of the standard deviation to the mean, expressed as a percentage. Here is how to calculate the coefficient of variation for the given data:

Calculate the mean of the data:

 Mean = (83+64+51+46+82+91+73+82+65+61+74+64+75+81+94+65+42+81+56+61+72+65+54+39+70+93+42+46+54+72)/20 = 68.25

Calculate the standard deviation of the data:

 Standard deviation = sqrt((1/20) * ((83-68.25)^2 + (64-68.25)^2 + ... + (72-68.25)^2))

Standard deviation ≈ 16.88

Calculate the coefficient of variation:

Coefficient of variation = (Standard deviation / Mean) * 100

Coefficient of variation ≈ 24.71%

3. Box Plot:

A box plot is a way to visualize the distribution of data. It displays the minimum value, first quartile (Q1), median, third quartile (Q3), and maximum value of the data. Here is the box plot for the given data:

  |              +----------+

94 |                        |

  |              +----------+

93 |              |

  |        +-----+-----+

82 |        |           |

  |        |           |

81 |        |           |

  |  +-----+           |

80 |  |                 |

  |  |                 |

79 |  |                 |

  |  |                 |

78 |  |                 |

  |  |                 |

77 |  |                 |

  |  |                 |

76 |  |                 |

  |  |                 |

75 |  |                 |

  |  |                 |

74 |  |                 |

  |  |                 |

73 |  |                 |

  |  |                 |

72 |  +-----------------+

  |                    

  +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+

     1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19 20

In this plot, the horizontal line inside the box represents the median, the bottom and top edges of the box represent the first and third quartiles (Q1 and Q3), respectively, and the vertical lines extending from the box represent the minimum and maximum values, excluding outliers.

4. To find the 85th percentile, we need to arrange the data in order from smallest to largest:

39, 42, 42, 46, 46, 51, 54, 56, 61, 61, 64, 64, 65, 65, 65, 70, 72, 72, 73, 74, 75, 81, 81, 82, 82, 83, 91, 93, 94

There are a total of 20 scores, so the 85th percentile would be the score at the 0.85(20) = 17th position:

85th percentile = 72

To find the 2nd decile, we first need to calculate the number of scores in each decile. Since there are 20 scores, each decile would have 2 scores. The 2nd decile would be the score at the 0.2(20) = 4th position:

2nd decile = 46

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Complete question:

Please, I need help with this, it's about statistics. Thank you very much. The grades of 20 students who took an admission exam to a certain faculty, using a scale of 0 to 100, were: 83 64 51 46 82 91 73 82 65 61 74 64 75 81 94 65 42 81 56 61 72 65 54 39 70 93 42 46 54 72

• Make: stem-and-leaf plot

• Calculate: coefficient of variation

• Create a box plot

• 85th percentile, 2nd decile.

Step-by-step explanation:

If the polynomial 3x^3+ax^2-11x+3 is exactly divisible by (x – 1), then find the value of a. Hence, factorize the polynomial. can anyone help

The function for this problem is given as follows:

[tex]f(x) = \sqrt{57 - x}[/tex]

The derivative of the function is given as follows:

[tex]f^{\prime}(x) = -\frac{1}{2\sqrt{57 - x}}[/tex]

The slope of the tangent line at point (8,7) is given by the numeric value of the derivative of f(x) at x = 8, hence:

[tex]f^{\prime}(8) = -\frac{1}{2\sqrt{57 - 8}}[/tex]

m = f'(8) = -1/14.

Hence the equation of the tangent line, which is a linear function, is given as follows:

y = -x/14 + b.

When x = 8, y = 7, hence the intercept b is obtained as follows:

7 = -8/14 + b

b = 7 + 8/14

b = 49/7 + 4/7

b = 53/7.

Hence the equation is given as follows:

y = -x/14 + 53/7.

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The solutions to the given equation in the interval $[0,2\pi)$ are:

[tex]$x=\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$$[/tex]

The given equation is:

[tex]$$\cos(2x) + \sin(x) = 0$$[/tex]

We can write [tex]$\cos(2x)$[/tex] as [tex]$1-2\sin^2(x)$[/tex]using double angle formula for cosine. Substituting this in the given equation, we get:

[tex]$$1-2\sin^2(x) + \sin(x) = 0$$[/tex]

Rearranging and factoring, we get:

[tex]$$(2\sin(x) - 1)(\sin(x) + 1) = 0$$[/tex]

So, either [tex]$2\sin(x) - 1 = 0$[/tex] or  [tex]$\sin(x) + 1 = 0$[/tex].

Solving [tex]$2\sin(x) - 1 = 0$[/tex], we get:

[tex]$$\sin(x) = \frac{1}{2}$$[/tex]

which has solutions [tex]x=\frac{\pi}{6}$[/tex] and $[tex]x=\frac{\pi}{6}$[/tex] in the interval [tex]$[0,2\pi)$[/tex].

Solving [tex]$\sin(x) + 1 = 0$[/tex], we get:

[tex]$$\sin(x) = -1$$[/tex]

which has solution [tex]x=\frac{3\pi}{2}$[/tex]in the interval [tex]$[0,2\pi)$[/tex]

Therefore, the solutions to the given equation in the interval $[0,2\pi)$ are:

[tex]$x=\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$$[/tex]

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To make a rectangle with a rotation symmetry of order 2, first draw the rectangle on a piece of paper. Then draw a cross in the middle of one of the squares in the rectangle. This is the square that needs to be coloured blue in order to create the rotation symmetry. Finally, colour the square with the cross blue to create the rotation symmetry.

v = u1 - 2u2 + u3 This means that v can be expressed as a linear combination of u1, u2, and u3. Specifically, we can take one copy of u1, subtract two copies of u2, and add one copy of u3 to obtain v.

To write v as a linear combination of u1, u2, and u3, we need to find constants a, b, and c such that v = au1 + bu2 + c*u3.

We can set up a system of equations to solve for these constants:

3a - b - 2c = -1

2a - 3b + c = 7

8a - b - 3c = 2

We can solve this system using row reduction, which yields:

a = 1

b = -2

c = 1

Therefore, we have:

v = u1 - 2u2 + u3

This means that v can be expressed as a linear combination of u1, u2, and u3. Specifically, we can take one copy of u1, subtract two copies of u2, and add one copy of u3 to obtain v.

It's worth noting that not every vector can be expressed as a linear combination of a given set of vectors. However, in this case, we were able to find constants that satisfy the equation v = au1 + bu2 + c*u3, so it is possible to write v as a linear combination of u1, u2, and u3.

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