the triangle below js equilateral. find the length of side x in the simplest radical form with a rational denominator

Problem 22

The smaller inner circle has a radius of r = 5. That will be plugged into the circumference formula below.

C = 2pi*r

C = 2pi*5

C = 10pi

This is the exact distance around the circle in terms of pi. Think of it like the amount of fencing you need to enclose the circle.

If you want an approximate perimeter, then use something like pi = 3.14

So 10pi = 10*3.14 = 31.4 cm is the approximate perimeter aka circumference of the smaller circle.

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

The larger circle is the same idea. The radius is r = 5+5 = 10 cm

C = 2pi*r

C = 2pi*10

C = 20pi .... exact circumference

C = 20*3.14

C = 62.8 ... approximate circumference

Use more decimal digits of pi to get a more accurate value.

===================================================

Problem 23

The smaller circle has a diameter of 9 ft, which means the radius is r = 9/2 = 4.5 ft

C = 2pi*r

C = 2pi*4.5

C = 9pi .... exact circumference

C = 9*3.14

C = 28.26 .... approximate circumference

The units of the circumference are in feet

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

The larger circle has radius r = 4.5+2.5 = 7 ft

C = 2pi*r

C = 2pi*7

C = 14pi .... exact circumference

C = 14*3.14

C = 43.96 .... approximate circumference

===================================================

Problem 24

The larger circle has radius 22. This radius is the diameter of the smaller circle.

So the smaller circle has radius 22/2 = 11

C = 2pi*r

C = 2pi*11

C = 11pi .... exact circumference

C = 11*3.14

C = 34.54 .... approximate circumference

The circumference units are "meters".

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

Now onto the larger circle

C = 2pi*r

C = 2pi*22

C = 44pi .... exact circumference

C = 44*3.14

C = 138.16 .... approximate circumference

In general, the maximum order of the spectrum that can be observed with a diffraction grating is limited by the grating equation.

This equation states that mλ = d(sinθi - sinθr), where m is the order of the diffraction, λ is the wavelength of light, d is the spacing between the grating lines, θi is the angle of incidence, and θr is the angle of diffraction. Since d is fixed and θi is typically kept constant, the maximum order of the spectrum that can be observed is limited by the angle of diffraction (θr). This angle can be calculated using the equation θr = sin-1[(mλ/d)+sinθi]. Thus, the maximum order of the spectrum is limited by the value of θr, which is determined by the wavelength of light and the spacing between the grating lines.

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The family spent $2004 per month on groceries and eating out.

The amount spent in five months is given as follows: $10,020.

The amounts are obtained applying the proportions in the context of this problem.

The monthly budget is given as follows:

$6012.

1/3 of that is spent in groceries and eating out, hence the amount is of:

1/3 x 6012 = $2004.

Hence in five months the amount spent is going to be of:

5 x 2004 = $10,020.

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The partials ∂y/∂x = 1 and ∂z/∂x = -1, so we get the Reciprocal Theorem.

Reciprocity Theorem: Let F(x, y, z) be a continuous function of x, y, and z, then we have

∂F/∂x = (∂F/∂y)x - (∂F/∂z)y

This is proven by writing the condition of F(x, y, z) = 0 into explicit forms x-x(y,z) or z-z(x,y). Then from the chain rule, we have

∂F/∂x = (∂F/∂y)(∂y/∂x) + (∂F/∂z)(∂z/∂x)

And since x = x(y,z) and z = z(x,y), the partials ∂y/∂x = -1 and ∂z/∂x = 1, so we get the Reciprocity Theorem.

Reciprocal Theorem: Let F(x, y, z) be a continuous function of x, y, and z, then we have

(∂F/∂y)x + (∂F/∂z)y = F(x, y, z)

This is proven by writing the condition of F(x, y, z) = 0 into explicit forms x-x(y,x) or z-z(x,y). Then from the chain rule, we have

∂F/∂x = (∂F/∂y)(∂y/∂x) + (∂F/∂z)(∂z/∂x)

And since x = x(y,x) and z = z(x,y), the partials ∂y/∂x = 1 and ∂z/∂x = -1, so we get the Reciprocal Theorem.

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Comparing rounding and compatible number, rounding gives a closer estimate than compatible number.

While compatible numbers are those that are simple to multiply mentally, rounding entails reducing one or both of the numbers to a close "friendly" number.

Estimating the result of 48 x 123 can be done using compatible integers and rounding.

Compatible numbers make it simpler to find an estimate calculation because they are close in value to the real number. A multiplication issue is presented here. By rounding the values to the closest hundreds place, you can obtain an approximation of the product. 400 and 100 are equivalent to 395 and 103.

1) Using compatible numbers :

48 = 50

123 = 100

50 × 100 = 5000

2) By rounding numbers :

48 = 50

123 = 120

50 × 120 = 6000

6000 is closer to the actual value of the product than 5000 ;

Therefore, rounding gives a closer estimate than compatible number.

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Answer: The simplified expression is 6x - 23

Step-by-step explanation: To simplify the expression (12x-13)-2(3x+5) we will follow the order of operations:

First, we simplify the inner parentheses: 2(3x+5) = 6x + 10

Next, we substitute this simplified version back into the original expression: (12x-13)-(6x+10)

Now we can simplify the expression by combining like terms: 12x-13-6x-10 = 6x - 23

So the simplified expression is 6x - 23

Answer:6x-23 this your answer to simplify the expression I hope this helps have a good day

Step-by-step explanation:

(12x-13)-2(3x+5)

Remove unnecessary parentheses

12x-13-2(3x+5)

then do this

distribute -2 through the parentheses

12x-13-6x-10

Collect like terms which something like this

6x-13-10

then last thing you do is

Calculate the difference which is like this

6x=23

Answer:

2

Step-by-step explanation:

The division of the value 8.8x10^9 and 2.2 x10^-3 in standard form is 4*10^12.

The concept that will be used here is division. One of the four fundamental arithmetic operations, or how numbers are combined to create new numbers, is division. The other operations are multiplication, addition, and subtraction.

Division in mathematics is the process of dividing an amount into equal parts. For instance, we may split a group of 20 people into four groups of 5, five groups of 4, and so on.

To simplify the values 8.8x10^9/2.2 x10^-3, the we can write as

8.8/2.2 *10^9*10^3

=4*10^12

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complete question:

simplify 8.8x109/2.2 x10-3

Answer: If Hue can form 2 rows of a given length with 3 chairs left over, we can represent this as:

2x + 3 = chairs in that row length

Also, if she can form 8 rows of that same length if she gets 15 more chairs, we can represent this as:

8x + 15 = chairs in that row length

Since the number of chairs in that row length is the same in both equations, we can set them equal to each other:

2x + 3 = 8x + 15

To solve for x, we can subtract 3 from both sides:

2x = 8x + 12

Then we can subtract 2x from both sides:

6x = 12

Finally, we can divide both sides by 6:

x = 2

So the number of chairs in that row length is 2x + 3 = 2(2) + 3 = 7.

Step-by-step explanation:

To prove that triangles ABD and ΔABC are congruent by SSS, what needs to be true is: A. BC ≅ BD.

For two triangles to be congruent to each other by the SSS Congruence Theorem, the triangles must have three pairs of corresponding sides that are congruent to each other.

In the image given, we can deduce that the two triangles have the following properties:

AB ≅ AB (reflexive property)

AC ≅ AD (given)

This means that have two pairs of corresponding congruent sides. Therefore, in order to prove that ΔABD ≅ ΔABC by SSS, what needs to be true is: A. BC ≅ BD.

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m+n is equal to (x+y+z)^2 + n x, y, and z are real numbers.

First, we can rewrite x+y+z as n(x+y+z)^2.

Then, we can substitute x^2+y^2+z^2 into the equation and solve for m^2. We get m^2 = (x+y+z)^3 - 3xyz, which can be simplified to

m^2 = (x+y+z)(x^2+y^2+z^2-xyz).

We can substitute the original equation back in to get

m^2 = (x+y+z)\frac{m^2}{n}.

Then, we can divide both sides of the equation by (x+y+z) to get m^2 = \frac{m^2}{n}. Finally, we can solve for m+n by multiplying both sides of the equation by n, which gives us m+n = m^2.

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The linear function with a slope of 1/3 and intercept of 7 is defined as follows:

y=1/3x+7.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

For which the parameters are given as follows:

Considering a slope of 1/3 and intercept of 7, the values of the parameters are given as follows:

m = 1/3, b = 7.

Hence the function is defined as follows:

y = 1/3x + 7.

The problems asks for the linear function with a slope of 1/3 and intercept of 7.

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They marked it up to 2979 since 993*3(aka 200%). You then multiply 2979 by 1.14 to get the total of $3396.06

Answer:

$3396.06

Step-by-step explanation:

Markup is what a store adds to the price they paid in order to come up with a selling price.

store selling price = cost to store + markup

Store purchase price: $993

Store markup of 200%: 200% of $993 = 2 × $993 = $1986

Store selling price: $993 + $1986 = $2979

14% tax: 14% of $2979 = 0.14 × $2979 = $417.06

Total selling price including markup & sales tax: $2979 + $417.06 = $3396.06

Answer: $3396.06

A transformation is applied to the input vector v by the function f. This transformation might include extending, rotating, or reflecting the vector in a certain way, depending on the precise shape of the function.

We may enter certain fixed vectors into the function, c, and watch as f transforms them to better understand its behavior (c). It may also be helpful to see the modified vectors as images or diagrams to have a better understanding of how the function behaves.

The term "vector" is used informally to describe constituents of particular vector spaces or some values that cannot be described by a single integer.

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Answer:

24.5 days i believe

The required number of pieces cut from 22 metal rods each 21 in are 242 pieces.

A mixed fraction is one that is represented by both its quotient and remainder. For example, 2 1/3 is a mixed fraction, where 2 is the quotient, 1 is the remainder. An amalgam of a whole number and a legal fraction is a mixed fraction.

We have,

1(4/5) = 9/5 ,

So,

21/9/5

105/9

11(6/9)

Now,the waste is 6/9

Number of pieces are 11(22)

= 242 pieces

Thus, required number of pieces are 242.

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Answer:

(mn+5)(mn-5)

Step-by-step explanation:

This is a difference of squares as m^2n^2 = (mn)^2. So it cna be turned into the form a^2-b^2 = (a+b)(a-b) or in our case (mn+5)(mn-5).

Answer:x=13

Step-by-step explanation:

The statement 'A vector b is a linear combination of the columns of a matrix A is and only if the equation Ax = b has at least one solution.' is True.

We first thought of a matrix as a rectangular array of numbers. When the number of rows is m and columns is n, we say that the dimensions of the matrix are m×n.

A vector is most simply thought of as a matrix with a single column. There are two operations we can perform with vectors: scalar multiplication and vector addition. Both of these operations have geometric meaning.

Given a set of vectors and a set of scalars we call weights, we can create a linear combination using scalar multiplication and vector addition.

The product of two matrices can be seen as the result of taking linear combinations of their rows and columns. This way of interpreting matrix multiplication often helps to understand important results in matrix algebra.

The statement 'A vector b is a linear combination of the columns of a matrix A is and only if the equation Ax = b has at least one solution.' is True.

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The numeric value of the expression for this problem is given as follows:

[tex]\sqrt{2} + 1[/tex]

The expression for this problem is defined as follows:

[tex]\frac{4 - 3\sqrt{2}}{(\sqrt[4]{2} - \sqrt[4]{8})^2}[/tex]

Applying the square of the subtraction in the denominator, we have that:

[tex]{(\sqrt[4]{2} - \sqrt[4]{8})^2 = \sqrt[4]{4} - 2\sqrt[4]{2}\sqrt[4]{8} + \sqrt[4]{64}[/tex]

[tex]{(\sqrt[4]{2} - \sqrt[4]{8})^2 = 4^{\frac{1}{4}} - 2\sqrt[4]{16} + \sqrt[4]{4^3}[/tex]

[tex]{(\sqrt[4]{2} - \sqrt[4]{8})^2 = (2^2)^{\frac{1}{4}} - 2 \times 2 + 3[/tex]

[tex]{(\sqrt[4]{2} - \sqrt[4]{8})^2 = \sqrt{2} - 1[/tex]

Hence the simplified expression is given as follows:

[tex]\frac{4 - 3\sqrt{2}}{\sqrt{2} - 1}[/tex]

Rationalizing the denominator, we have that:

[tex]\frac{4 - 3\sqrt{2}}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1}[/tex]

Applying the subtraction of perfect square, the denominator simplifies to one, hence the value of the expression is given by the numerator as follows:

[tex](4 - 3\sqrt{2})(\sqrt{2} + 1) = 4\sqrt{2} + 4 - 3 \times 2 - 3\sqrt{2}[/tex]

[tex][tex](4 - 3\sqrt{2})(\sqrt{2} + 1) = \sqrt{2} + 1[/tex]

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Indicated by a trend line is a linear relationship. Y = mx + b, where x is the independent variable and b is the constant, is the equation for a linear connection.

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According to the pie chart given -

a) The worker saved GH¢ 20.00.

b) 2/5 fraction of worker's salary was spent on food.

c) The worker spent GH¢ 120.00 on food.

What is a pie chart?

One sort of graph that illustrates the information in the circular graph is a pie chart. It is a sort of graphical representation of data where the slices of pie depict the relative sizes of the data. A list of numerical and categorical variables is necessary for a pie chart. Pie in this context refers to the entire thing, and slices to its component portions.

The rent charges is = GH¢ 80.00

The savings can be obtained by ratio method -

Savings is depicted by 24° and rent by 96° in the pie chart.

So, the ratio will be 24/96.

Then the savings value will be -

(24/96) × 80

(1/4) × 80

20

Therefore, the savings value is GH¢ 20.00.

The total angle of pie chart is 360°.

Savings is depicted by 24°, rent by 96°, entertainment by 48° and clothing by 48° in the pie chart.

The angle for food will be -

360° - (24° + 96° + 48° + 48°)

360° - 216°

144°

Now the fraction that is for food -

144/360

2/5

Therefore, food covers 2/5 part of pie chart.

The amount spent on food can be obtained by formula of angle sector -

(144/96) × 80

1.5 × 80

120

Therefore, money for food was GH¢ 120.00.

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The degree measure of the smaller angle formed by the hands of a clock at 10 O'clock is 60°.

The degree measure of a full circle is 360 degrees. The clock face is divided into 12 equal parts, or 30-degree increments, for the hours. At 10 o'clock, the hour hand would point to the 10 on the clock face, and the minute hand would point to the 12.

This makes them 1/6th of a circle apart, and (1/6) * 360° = 60°.

The angle between the two hands would be formed by the distance between the 10 and the 12, which is 60 degrees.

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An example of such a function can be f(x)  { 1  x∈Q

                                                                      -1   x∈ R\Q}

A function in mathematics is like a machine that takes input and gives output. There is always a limit for output also called the range or image of the function.

A function has two values output and input. The limit within which input is defined is called the domain or pre-image of the function.

A function is represented as f(x) i.e., f of x or function of x. The input and output have a relation defined for them

A function must follow two rules they are:

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The alternative hypothesis for this test would be that the new dog food extends the average lifespan of dogs beyond 10 years.

Mathematically, the alternative hypothesis can be stated as:

Ha: μ > 10, where μ is the population mean of the lifespan of dogs on the new dog food.

In this scenario, the null hypothesis is that the average lifespan of dogs on the new dog food is equal to 10 years, which is the commonly accepted average lifespan of dogs. However, the purpose of the study is to determine if the new dog food can extend the lifespan of dogs beyond this average.

Thus, the alternative hypothesis is that the average lifespan of dogs on the new dog food is greater than 10 years. This is represented mathematically as Ha: μ > 10, where μ is the population mean of the lifespan of dogs on the new dog food.

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Note: The row echelon form of a matrix is a way of putting it in a specific form, in which all the leading entry of a non-zero row is to the right of the leading entry of the row above it, and all the entries below a leading entry are zero. A pivot is the leading entry of a non-zero row. In a matrix in row echelon form, the number of pivots is equal to the rank of the matrix, which is the number of linearly independent rows or columns. A matrix with a pivot in every column is said to be in reduced row echelon form, and it is unique for a given matrix.

A linear transformation is a function that maps a vector from one vector space to another vector space. The matrix associated with this transformation is a way of representing the linear transformation using a matrix. The row echelon form of the matrix is a specific way of arranging the matrix, where all the leading entry of a non-zero row is to the right of the leading entry of the row above it, and all the entries below a leading entry are zero. The properties of the transformation can be determined by analyzing the row echelon form of the matrix.

For example, if the row echelon form of the matrix has a pivot in every column, it means that the transformation is one-to-one (meaning that the function assigns a unique output for each input). On the other hand, if the row echelon form of the matrix has a pivot in every row, it means that the transformation is onto(meaning that the function's range is equal to the output vector space). If there are zero rows in the row echelon form of the matrix, it means that the transformation is not one-to-one.

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Answer: a) To calculate the total rate of return if Josh invests the same amount in each company, we need to add up the individual rates of return from each company and divide by the number of companies.

The total rate of return for the four companies is: (7.62% + -4.17% + -0.83% + 3.70%) / 4 = 2.52%

So the total rate of return if Josh invests the same amount in each company is 2.52%.

b) To calculate the total rate of return if Josh invests three times as much in Stiller Co. as in the other companies, we need to multiply the rate of return for Stiller Co. by 3 and add it to the rates of return for the other companies.

The total rate of return for the four companies is: (-4.17% x 3) + 7.62% + -0.83% + 3.70% = -7.32%

So the total rate of return if Josh invests three times as much in Stiller Co. as in the other companies is -7.32%.

It's important to note that the answer may change if the investment amount on each company is different.

Step-by-step explanation:

Strategic planning is an essential process for any organization that wants to succeed and achieve long-term success. It involves identifying long-term organizational goals, strategies, and resources that will help the organization reach its objectives.

This process requires a comprehensive assessment of the organization's current and future needs, including its strengths and weaknesses, opportunities and threats, and the external environment. Once identified, the organization must develop and implement a plan to achieve its goals and objectives. This involves setting objectives, developing and implementing strategies and tactics, and allocating resources in a way that will maximize the organization's chances of success.

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The like terms are listed as follows:

-y, 7y and y/2.

Like terms are terms that share these two features:

If two terms are like terms, then they can be either added or subtracted.

Hence the first option is the only option in which all the three terms are like terms, as:

The remaining options have terms with different letters(second and third), or different exponents (fourth), hence they are not like terms.

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