What is the meaning of "only under the assumption that at least one set X exists"?

The value of the unknown side to the nearest hundredth is 6.78

Cosine rule states that; if a, b,c are the sides of a triangle and A,B,C are the opposite angles of the sides, then

c² = a² + b² -2ab CosC

Cosine rule is used when all the sides or two of the sides are given. The angle opposite to side c is angle C.

c² = 4² + 10² - 2(4)(10)cos 29

c² = 16 +100 - 80cos29

c² = 116 -69.97

c² = 46.03

c = √ 46.03

c = 6.78 ( nearest hundredth).

Therefore the value of c is 6.78

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The area of the rectangle is 1296 square feet.

Let's denote the width of the rectangle as "w" and its length as "4w" since the length is four times the width.

The formula for the perimeter of a rectangle is given by:

P = 2(l + w), where P is the perimeter, l is the length, and w is the width.

Given that the perimeter of the rectangle is 180 ft, we can write the equation as:

180 = 2(4w + w)

Simplifying the equation:

180 = 2(5w)

180 = 10w

w = 18

So, the width of the rectangle is 18 ft.

Now, we can find the length of the rectangle:

Length = 4w = 4(18) = 72 ft

The area of a rectangle is given by the formula:

[tex]A = l \times w,[/tex]

where A is the area,

l is the length,

and w is the width.

Substituting the values we found:

Area [tex]= 72 ft \times 18 ft = 1296 ft^2[/tex]

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Answer:

At the time the ball was thrown, it was 8 feet above the ground.

h'(x) = -32x + 64 = 0, so x = 2

The ball reaches its maximum height after 2 seconds.

The solution of this equation is z = 3.

We are given that;

1-3+4z = 1

Now,

To solve for z, we need to isolate it on one side of the equation. We can do this by following these steps:

Subtract −31​

from both sides to eliminate it from the left side.

Multiply both sides by z to eliminate the fraction on the left side.

Divide both sides by 4 to isolate z on the left side.

Let’s see how this works:

−31​+z4​=1

−31​+z4​=1

z4​=1+31​

z4​=34​

4=4/3 ​z

z=3*4​/4​

z=4/4×3​

z=3

Therefore, by the equation answer will be z = 3.

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Subtraction and Division

Inverse operations help find the solution to equations.

Defining Inverse Operations

Firstly, let's define an operation. An operation in math is a function that can manipulate a value. Inverse operations are operations that are opposite operations that undo each other. For example, addition and subtraction are inverse operations because subtraction undoes addition. Multiplication and division are also inverse operations.

Solving the Equation

The equation 16 = 3x + 1 involves both addition and multiplication. So, to solve this, we can use the inverse operations of subtraction and division. First, subtract 1 from both sides.

Then, divide both sides by 3.

This shows that by using subtraction and division, we can undo the addition and multiplication used in the equation. This allows us to find the value of x.

Step-by-step explanation:

7! = 5040 ways

Given statement solution is :- The correct solutions to the quadratic equation  [tex]3(x-7)^2 = 27[/tex]  are x = 10 and x = 4.

Diego made an error in the step where he simplified [tex](x-7)^2 = 9 to x^2[/tex] - 72 = 9. The correct expansion of [tex](x-7)^2 is x^2 - 14x + 49[/tex], not [tex]x^2 - 72.[/tex]Therefore, his subsequent steps and solution are incorrect.

Mai made an error in the step where she simplified [tex](x-7)^2 = 9[/tex] to x - 7 = 9. The square root of 9 is ±3, so the correct equation should be x - 7 = ±3, not x - 7 = 9. Therefore, her solution is incorrect.

To solve the equation correctly, we start with the equation [tex]3(x-7)^2 = 27.[/tex]

Dividing both sides by 3, we have:

[tex](x-7)^2 = 9[/tex]

Taking the square root of both sides, we get:

x - 7 = ±3

Now, we need to solve for x by adding 7 to both sides:

x = 7 ± 3

This gives us two possible solutions:

x = 7 + 3 = 10

x = 7 - 3 = 4

So, the correct solutions to the quadratic equation [tex]3(x-7)^2 = 27[/tex] are x = 10 and x = 4.

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The cross-section that results from slicing a cube through the center vertically, perpendicular to the base, is both a rectangle and a square.

When a cube is sliced through the center vertically, perpendicular to the base, the resulting cross-section will be a rectangle and a square.

Let's consider the properties of a cube. A cube is a three-dimensional solid with six square faces. Each face of the cube is congruent and perpendicular to the adjacent faces. The edges of the cube are all equal in length, and the angles between the faces are all right angles (90 degrees).

When we slice the cube vertically through the center perpendicular to the base, we cut through the cube in such a way that the cross-section obtained is a plane figure. This cross-section will have the same shape as the face of the cube that was cut.

In this case, since the cube has square faces, the cross-section will also have the same shape. Therefore, a square is one of the shapes that apply to the resulting cross-section.

Additionally, since the slice is made through the centre of the cube, the resulting cross-section will pass through the midpoints of two opposite sides of the square face. As a result, the cross-section will have equal side lengths and parallel sides, making it a rectangle.

Final answer:

Therefore, the cross-section that results from slicing a cube through the center vertically, perpendicular to the base, is both a rectangle and a square. These shapes capture the properties of the cube's face and the nature of the cut.

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The completely factorized forms are:

1.1) 2x + 4y - 6z (no further factorization)

[tex]1.2) 2pq(5p^5q - 2p^2 + 1)[/tex]

1.3) (m + n)(m + n - 5p)

1.4) 7c(4 - a) + d(-4 + a)

We have,

Let's factorize each expression completely:

1.1)

2x + 4y - 6z

There are no common factors among the terms, so we can't factorize it further.

1.2)

[tex]10p^6q^2 - 4p^3q^2 + 2pq[/tex]

The common factor among the terms is 2pq, so we can factor it out:

[tex]2pq (5p^5q - 2p^2 + 1)[/tex]

1.3)

(m + n)² - 5p(m + n)

Using the distributive property, we can expand the squared term:

(m + n)(m + n) - 5p(m + n)

Now we can factor out the common factor (m + n):

(m + n)(m + n - 5p)

1.4)

4(7c - d) + a(d - 7c)

Using the distributive property, we can expand the terms:

28c - 4d + ad - 7ac

Rearranging the terms:

(28c - 7ac) + (-4d + ad)

Factoring out common factors:

7c(4 - a) + d(-4 + a)

Thus,

The completely factorized forms are:

1.1) 2x + 4y - 6z (no further factorization)

[tex]1.2) 2pq(5p^5q - 2p^2 + 1)[/tex]

1.3) (m + n)(m + n - 5p)

1.4) 7c(4 - a) + d(-4 + a)

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There are 19 nickels, 15 quarters, and 18 dimes in the collection.

Let's assume that there are n number of nickels in the collection.There are four fewer quarters than nickels, so the number of quarters will be n - 4.

There are 3 more dimes than quarters, so the number of dimes will be (n - 4) + 3 = n - 1.

The value of each nickel is $0.05, the value of each dime is $0.10, and the value of each quarter is $0.25.Using this information, we can form the following equation to represent the total value of the collection:

0.05n + 0.10(n - 1) + 0.25(n - 4) = 6.5Simplifying this equation, we get:0.05n + 0.10n - 0.10 + 0.25n - 1 = 6.50.40n - 1.10 = 6.50

Adding 1.10 to both sides, we get:0.40n = 7.60Dividing both sides by 0.40, we get:n = 19

Therefore, there are 19 nickels in the collection.There are four fewer quarters than nickels, so the number of quarters will be n - 4 = 19 - 4 = 15.

There are 3 more dimes than quarters, so the number of dimes will be (n - 4) + 3 = 15 + 3 = 18.

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Answer:

42

Step-by-step explanation:

Rectangle: A = 6 x 5 = 30

Triangle: A = 1/2(6 x 4) = 12

Area of figure: 30 + 12 = 42

Answer:

area of rectangle+area of triangle= area of figure

6×5 + 1/2×4×6=30+24= 42

or

given figure is a trapezium

so we know are of trapezium= 1/2×sum of parallel sides×height

= 1/2×( 5+9)×6

=42

Answer:  A    -6x² + 2x -4

Step-by-step explanation:

[tex]\frac{-12x^{3}+ 4x^{2} -8x}{2x}[/tex]                             >Divide each of the top terms by 2x[tex]=\frac{-12x^{3}}{2x} +\frac{4x^{2} }{2x} -\frac{8x}{2x}[/tex]

= -6x² + 2x -4

A triangular prism is a three-dimensional geometric shape that consists of two triangular bases and three rectangular faces connecting them. It is a polyhedron with six faces, nine edges, and six vertices.

The two triangular bases of a triangular prism are congruent and parallel to each other. The rectangular faces are perpendicular to the triangular bases, and their lateral edges connect the corresponding vertices of the triangular bases.

The triangular bases are identical and parallel to each other. Each base has three vertices, three edges, and one face.There are three rectangular lateral faces connecting the corresponding vertices of the triangular bases. Each lateral face has two edges and one face.

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John's z-score is 0.75.

John's score of 79 on the math test corresponds to a z-score of 0.75 and a percentile of 77.82%.

To get John's z-score, we will use the formula: z = (x - μ) / σ

Data

x = John's score (79)

μ = mean score (73)

σ = standard deviation (8)

z = (79 - 73) / 8

z = 6 / 8

z = 0.75.

To find John's percentile, we can use a standard normal distribution table or calculator.

The percentile represents the percentage of scores that fall below a given value. Using  standard normal distribution table, we find that a z-score of 0.75 equals to percentile of 77.82%.

Full question:

The scores on a math test have a mean of 73 points and a standard deviation of 8 points. John scores a 79 on the test. Convert this score to a z-score and a percentile..

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The total tax due for a married couple who have a combined income of $110,000 is $16,079

From the question, we have the following parameters that can be used in our computation:

Income = $110,000

Also, we have the table of values

From the tax table, we have the category of tax to be paid by the couple to be

Tax = 8907 + 22% * excess of 77400

So, we have

Tax = 8907 + 22% * (110,000 - 77400)

Evaluate

Tax = 16079

Hence, the tax amount is $16,079

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y = 3x - 1: consistent, independent

2y - 6x = 4: consistent, independent

2y = -x + 6: consistent, independent

4y + 2x = 12: consistent, independent

y = 2x + 3: consistent, independent

x + y = -9: consistent, dependent

y = 3x - 1 and 2y = -x + 6: inconsistent

Let's match each system on the left with the appropriate words that describe the system on the right:

y = 3x - 1: consistent, independent

This system represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

2y - 6x = 4: consistent, independent

Similar to the first system, this represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

2y = -x + 6: consistent, independent

This system also represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

4y + 2x = 12: consistent, independent

Like the previous systems, this represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

y = 2x + 3: consistent, independent

This system represents a linear equation with a unique solution. The equations are not multiples of each other, and they intersect at a single point.

x + y = -9: consistent, dependent

This system represents a linear equation with infinitely many solutions. The equations are multiples of each other, meaning they represent the same line. Therefore, any point on the line satisfies both equations.

y = 3x - 1 and 2y = -x + 6: inconsistent

This system represents inconsistent equations that do not have a common solution. The lines represented by the equations are parallel and do not intersect.

To summarize the matches:

y = 3x - 1: consistent, independent

2y - 6x = 4: consistent, independent

2y = -x + 6: consistent, independent

4y + 2x = 12: consistent, independent

y = 2x + 3: consistent, independent

x + y = -9: consistent, dependent

y = 3x - 1 and 2y = -x + 6: inconsistent

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Since all three points satisfy the equation 2x + 2y = 24 when their coordinates are substituted, we can conclude that the points (4,8), (2,10), and (5,7) indeed lie on the line represented by the equation 2x + 2y = 24.

To determine if the points (4,8), (2,10), and (5,7) lie on the line 2x + 2y = 24, we can substitute the x and y values of each point into the equation and check if the equation holds true.

For the point (4,8):

2(4) + 2(8) = 8 + 16 = 24, which satisfies the equation.

For the point (2,10):

2(2) + 2(10) = 4 + 20 = 24, which also satisfies the equation.

For the point (5,7):

2(5) + 2(7) = 10 + 14 = 24, which satisfies the equation as well.

Since all three points satisfy the equation 2x + 2y = 24 when their coordinates are substituted, we can conclude that the points (4,8), (2,10), and (5,7) indeed lie on the line represented by the equation 2x + 2y = 24.

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Answer:

         m∠RQS = 102°

         m∠TQS = 78°

Step-by-step explanation:

    A straight angle is equal to 180 degrees. We will create an equation to solve for x.

         9x° + 3° + 7x° + 1° = 180°

         16x° + 4° = 180°

         16x° = 176°

         x = 11

    Next, we will substitute this value into the expressions representing the angles.

         m∠RQS = 9x° + 3° = 9(11)° + 3° = 102°

         m∠TQS = 7x° + 1° = 7(11)° + 1° = 78°

Answer:

42 = 23 + j

Step-by-step explanation:

The equation representing the given sentence would be:

42 = 23 + j

Here, the variable "j" represents Janelle's age.

The possible lengths of the tree lot are 25 + 5√5 yards and 25 - 5√5 yards.

Let's denote the length of the rectangular tree lot as "l" and the width as "w".

We know that the perimeter of a rectangle is given by the formula:

Perimeter = 2(l + w)

Given that the perimeter of the tree lot must be 100 yards, we can write the equation as:

2(l + w) = 100

Next, we know that the area of a rectangle is given by the formula:

Area = l × w

Given that the area of the tree lot must be at least 500 square yards, we can write the inequality as:

l × w ≥ 500

Now, let's solve the equations simultaneously to find the possible lengths of the tree lot.

Perimeter equation:

2(l + w) = 100

l + w = 50

w = 50 - l

Area inequality:

l × w ≥ 500

Substituting the value of w from the perimeter equation into the area inequality, we have:

l × (50 - l) ≥ 500

50l - l^2 ≥ 500

l^2 - 50l + 500 ≥ 0

Now, we need to find the values of l that satisfy the inequality. Since the coefficient of the squared term is positive, the graph of this quadratic opens upward. This means that the values of l that satisfy the inequality will be either the entire range of possible values or a portion of it.

To find the possible lengths, we can either factor the quadratic or use the quadratic formula. Let's use the quadratic formula:

l = (-(-50) ± √((-50)^2 - 4(1)(500))) / (2(1))

l = (50 ± √(2500 - 2000)) / 2

l = (50 ± √500) / 2

l = (50 ± 10√5) / 2

l = 25 ± 5√5

Therefore, the possible lengths of the tree lot are 25 + 5√5 yards and 25 - 5√5 yards.

In summary, the possible lengths of the rectangular tree lot are 25 + 5√5 yards and 25 - 5√5 yards, respectively.

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Answer:

Look Below

Step-by-step explanation:

The statement "definition of finiteness uses the notion of a natural number" means that when defining what it means for a set or collection to be finite, the concept of a natural number is employed.

In mathematics, a set is said to be finite if it can be put into a one-to-one correspondence with a specific subset of the natural numbers. This subset typically starts with the number 1 and includes a finite sequence of consecutive natural numbers, depending on the size of the set.

For example, consider a set A with three elements: A = {a, b, c}. To show that A is finite, we can establish a one-to-one correspondence between the elements of A and the set {1, 2, 3} as follows: a ↔ 1, b ↔ 2, c ↔ 3. This mapping shows that each element of A corresponds to a unique natural number, and vice versa. Since there is a bijection (a one-to-one correspondence) between the elements of A and a subset of the natural numbers, we conclude that A is finite.

Thus, in the definition of finiteness, the notion of a natural number is used to provide a precise and rigorous characterization of what it means for a set to be finite. It allows us to establish a clear distinction between finite sets and infinite sets, which do not have a one-to-one correspondence with any subset of the natural numbers.

The general form of the solutions of the recurrence relation is

[tex]a_n = (A + B n + C n^2 + D n^3) 1^n + (E + F n + G n^2) (-2)^n + (H + I n) 3^n + (J + K n) (-4)^n,[/tex]

We are given that;

Series=1, 1, 1, 1, -2, -2, -2, 3, 3, -4

Now,

In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought.

If the characteristic equation has repeated roots, then the general solution is modified by multiplying each term by a polynomial in n whose degree is equal to the multiplicity of the root minus one. For example, if r is a root of multiplicity m, then the corresponding term in the general solution is

[tex]a_n = (A + B n + C n^2 + … + Z n^(m-1)) r^n,[/tex]

where A, B, C, …, Z are arbitrary constants.

In this question, the characteristic equation has roots 1, 1, 1, 1, -2, -2, -2, 3, 3, -4. The root 1 has multiplicity 4, the root -2 has multiplicity 3, and the roots 3 and -4 have multiplicity 2 each.

[tex]a_n = (A + B n + C n^2 + D n^3) 1^n + (E + F n + G n^2) (-2)^n + (H + I n) 3^n + (J + K n) (-4)^n,[/tex]

where A, B, C, D, E, F, G, H, I, J and K are arbitrary constants.

Therefore, by the equation answer will be [tex]a_n = (A + B n + C n^2 + D n^3) 1^n + (E + F n + G n^2) (-2)^n + (H + I n) 3^n + (J + K n) (-4)^n,[/tex]

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The algebraic expression for the sentence in this problem is given as follows:

n - (x + 8).

The sentence in this problem is given as follows:

"A number n minus the sum of the number and eight".

The sum of an unknown number x and eight is given as follows:

x + 8.

The subtraction of the number n by the expression is given as follows:

n - (x + 8).

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Answer:

$1250

Step-by-step explanation:

A = P + Prt

1337.5 = P + P(0.04)(21/12)

1337.5 = 1.07P

P = 1250

Answer: $1250

Answer:

y = -1/2x

Step-by-step explanation:

Currently, y = 2x - 5 is in slope-intercept form, whose general equation is y = mx + b, where

The slopes of perpendicular lines are negative reciprocals of each other.  We can show this with the following formula;

m2 = -1 / m1, where

Finding the slope of the other line:

Since 2 is the slope of y = 2x - 5, we can plug in 2 for m1 in perpendicular slope formula to find m2, the slope of the other line:

m2 = -1 / 2

m2 = -1/2

Thus, the slope of the other line is m = -1/2.

Finding the y-intercept of the other line:

We can find the y-intercept of the other line (i.e., b in the slope-intercept equation) by plugging in (-2, 1) for x and y and -1/2 for m in the slope-intercept form:

1 = -1/2(-2) + b

1 = 1 + b

0 = b

Thus, y = -1/2x is the line that passes through (-2, 1) and is perpendicular to y = 2x - 5.

Answer:

5

She must therefore earn 5 points on the last test to achieve a mean score of 4.

Step-by-step explanation:

The sum of the data divided by the total number of data represents the mean, which is the average of a set of data.

The expression that shows the volume of the prism is x³ - x² - 6.

The prism above is a rectangular prism. The volume of the prism can be found as follows:

The volume of the prism can be found as follows:

volume of the rectangular prism = lwh

where

Therefore,

volume of the rectangular prism = (x - 3)(x)(x + 2)

volume of the rectangular prism = (x² - 3x)(x + 2)

volume of the rectangular prism = x³ + 2x² - 3x² - 6

volume of the rectangular prism = x³ - x² - 6

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Given statement solution is :- The area (A) of the base of the cylinder, you need to use the formula for the area of a circle, which is A = [tex]πr^2[/tex], where π is approximately 3.14 and r is the radius of the base.

To calculate the area (A) of the base of the cylinder, you need to use the formula for the area of a circle, which is A = [tex]πr^2[/tex], where π is approximately 3.14 and r is the radius of the base.

Given that the radius (r) is 10 cm, the area of the base (ABase) can be calculated as follows:

ABase =[tex]πr^2[/tex]

ABase = 3.14 * [tex](10 cm)^2[/tex]

ABase = 3.14 * [tex]100 cm^2[/tex]

ABase = 314 [tex]cm^2[/tex]

The area of the base is [tex]314 cm^2.[/tex]

To calculate the volume (V) of the cylinder, you need to multiply the area of the base (ABase) by the height (h):

V = ABase * h

V = 314 [tex]cm^2[/tex] * 30 cm

V = 9420 [tex]cm^3[/tex]

The volume of the cylinder is 9420 [tex]cm^3.[/tex]

However, there seems to be some confusion in the provided information. The given values for the area of the base (ABase) and the volume (V) are not clear. The values "- 20 gm" and "cm cm?" are not meaningful units for area or volume.

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Answer:

x+4y=7

Step-by-step explanation: