What is the relationship between the area of the rectangle and its sides?

We can factor using the rational root theorem by a rational number, both the leading coefficient and the constant term must be divisible by the fraction's denominator.

The rational root theorem, also known as the rational root test, is an algebraic principle that states that for a polynomial equation with one variable and integer coefficients to have a solution (root) that is a rational number, both the leading coefficient and the constant term must be divisible by the fraction's denominator.

A particularly helpful theorem is the theorem of rational roots.

It explains how to get a list of potential solutions for a polynomial function with integer or whole number coefficients by ranking the factors of the constant, or last term, over the factors of the coefficient of the leading term.

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The relationship between the volume of pyramid and prism is that when a pyramid and prism have the same base and height, volume of prism is 1/3rd the volume of prism.

The volume of the cylinder is 1/3rd the volume of the cone.

The volume of the sphere is 2/3rd the volume of the cylinder with same radius and height equal to the diameter.

Pyramid is said to have a polygonal base and flat triangular faces, which are connected to a common point called apex.

Prism is a combination of flat faces, identical bases and cross-sections being equal.

Cylinder has two parallel bases joined by curved surface through a fixed distance.

Cone has a circular base and a curved surface tapering till the apex.

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The required cost of hosta and ornamental grass is $6 and $3.

Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.

here,
let the cost of each hosta be x and ornamental grass be y,
According to the question,
For
Trevon,
5x + y = 33 - - - - - (1)

For, Scott,
x + 12y = 42 - - - - - - (2)

Solving equations 1 and 2 by substitution method,
x=6, y=3

Thus, the cost of hosta and ornamental grass is given as $6 and $3.

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43 in arithmetic stands for multiplying 4 by three. Suppose x =43; then, x+4*4*4.

If the power is positive, multiply the integer by itself that many times. If the power is negative, multiply the reciprocal of the integer by itself that many times. If the power is 0, the result is always 1.

No, 3:9 and 9:3 are not equivalent. 3^9=19683 and 9^3=729.

4^3=4*4*4\s =16*4\s =64

In the US and Canada, math is the word of choice. In the United Kingdom, Ireland, Australia, and other English-speaking countries, math is the preferred term. There isn't really a rational reason why math grew more popular in certain locations while maths was elsewhere.

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“3 more than 5” written as an algebraic expression is “3+5” which is equal to 8.

In a formula or equation involving mathematical operations, the representation of numbers by alphabet is referred to as algebra. Similar to how a statement explains the link between certain words, algebraic equations describe the connection between unknown variables.

We are given a mathematical statement and have to convert it into an algebraic expression using the information above.

3 more than 5 is obtained after adding 5 to the given number which is 3.

So 3 more than 5 is equal to 3+5=8.

Hence “3 more than 5” written as an algebraic expression is “3+5” which is equal to 8.

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The last expression is equivalent to √5³.

What equation contains an expression?

The equal to sign (=) is the best indicator of whether a given problem is an equation or an expression because it indicates that the problem is an equation.

                    It is only an expression if there isn't an equal to (=) sign present. It contains data that is used to demonstrate value, such as numbers, variables, and operators.

                   [tex]= ( 5^{1/8} * 5^{3/8} )[/tex]

                   [tex]= ( 5^{1/8} * 5^{3/8} ) \\= 5^{(1/8 + 3/8)}[/tex]

                    [tex]= 5^{4/8}[/tex]

                    [tex]= 5^{(1/2)}[/tex]

                   [tex]= [5^{(1/2)}]^{3}[/tex]

    Because of the "Power of a power property"

                                    = [tex]5^{3/2}[/tex]

The last expression is equivalent to √5³.

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The wax mixtures that will create the same shade of pink are

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

Red wax = 12 mL

White wax = 5 mL

When these parameters are represented as a ratio, we have the following representation

Ratio = Red wax : White wax

Substitute the known values in the above equation, so, we have the following representation

Red wax : White wax = 12 mL : 5 mL

Remove the units

So, we have

Red wax : White wax = 12 : 5

Multiply the ratio by 3

So, we have

Red wax : White wax = 36 : 15

Multiply the ratio by 5

So, we have

Red wax : White wax = 60 : 25

Hence, the wax mixtures are (d) and (e)

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Complete question

A pink crayon is made with 12 mL of red wax for every 5 mL of white wax. Which of the following wax mixtures will create the same shade of pink?

(A) 10 mL of red wax mixed wit 24 mL of white wax

(B) 48 mL of red mixed with 12 of white wax

(C) 13 mL of red mixed with 6 mL of white wax

(D) 36 mL of red mixed with 15 mL of white wax

(E) 60 mL of red mixed with 25 mL of white wax

Answer:-96

Step-by-step explanation:

The inequality that represents this scenario is 0.08B >= 57.

To get the promotion, Carolina needs to spend at least $75 on paintballs and the entrance fee. The entrance fee is a fixed cost of $18, so we need to find the number of paintballs that Carolina needs to buy in order to reach a total cost of $75 or more.

The cost of the paintballs is calculated using the formula 0.08B, where B is the number of paintballs that Carolina buys. We can use this formula to write an inequality that represents the scenario described in the problem.

The total cost of the paintballs and the entrance fee is calculated using the formula 0.08B + 18. This formula represents the total cost of the paintballs and the entrance fee for any number of paintballs that Carolina buys.

To find the number of paintballs that Carolina needs to buy in order to reach a total cost of $75 or more, we can set the total cost equal to $75 and solve for B:

0.08B + 18 >= 75

0.08B >= 57

B >= 712.5

The inequality 0.08B >= 57 represents the scenario described in the problem. It states that the number of paintballs that Carolina needs to buy in order to reach a total cost of $75 or more is greater than or equal to 712.5.

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The question is -

Carolina goes to a paintball field that charges an entrance fee of $18 and $0.08 per ball. The field has a promotion that says, "Get $10 off if you spend $75 or more!" Carolina wonders how many paintballs she needs to buy along with the entrance fee to get the promotion, Let B represent the number of paintballs that Carolina buys. What inequality describes this scenario?​

The inverse of a 2x2 identity matrix is also a 2x2 identity matrix. This is because the identity matrix is its own inverse.

An identity matrix is a square matrix with all primary diagonal members set to one and all other elements set to zero. The letter "In" or just "I" is used to indicate it. Any matrix will produce a given matrix if it is multiplied by the identity matrix. . The components of the provided matrix stay the same.

The 2x2 identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else. It has the following form:

[1 0]

[0 1]

The inverse of this matrix is also a 2x2 identity matrix with the same form:

[1 0]

[0 1]

Therefore, the inverse of a 2x2 identity matrix is also a 2x2 identity matrix.

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The mean of given numbers is 4.

To find: mean of given numbers.

Given data : 2, 3, 4, 5, 6

Average is the arithmetic mean. also called as mean .

Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers.

Here, sum of observations =20

Arithmetic mean refers to the average amount in a given group of data. It is defined as the summation of all the observation is the data which is divided by the number of observations in the data

Total number of observations=5

The number of observations provides information on the total number of values that are contained in the Dataset. This property is intended to provide an indication of the size of a Dataset.

Substituting values in above formula,

Mean=[tex]\frac{20}{5}[/tex]=4

Therefore, the mean is 4.

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On solving the provided question, we can say that the value of x = 5√3

and value of y = 10.

How to find the value of x and y?

To find the values of x and y, you will need to use

According to the given question:

Using the table of trig values, we find that

sin30°=1/2

So, this gives us:

1/2 = sin 30° = 5/y

1/2 = 5/y

(1xy)/2 = 5

y= 2×5 =10

Next, using the Pythagorean Theorem, we find that

5²+x²=10²

x²=100−25=75

Solve for x:

x=√75=√3×25=5√3

Complete question:

How do you find the value of X and Y. Y is the hypotenuse, 5 is the adjacent side and X is the opposite side from Y, and in the triangle there is 30 degrees where X and Y lines meet?

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Therefore , the factors of the given equation are (x + 4), (x - 5) and (x + 1)

The equal sign is used between numerical or changeable expressions to create an equation, such as 3x+5=11.

A number that can be entered as the variable's replacement in an equation serves as the solution.

Here,

The first term, x2, has a coefficient of 1 and is the starting term.

The coefficient of the middle word, +5x, is 5, thus

"The constant," the final word, is +4.

Step 1: Multiply the first term's coefficient by the formula 1 + 4 = 4.

Step 2 is to identify two factors of 4 whose sum equals the middle term's coefficient, which is 5.

     -4    +    -1    =    -5

     -2    +    -2    =    -4

     -1    +    -4    =    -5

        4          +            1      =    5    

Step 3: Rewrite the polynomial by separating the middle term into the two parts 1 and 4 x2 + 1x + 4x + 4 discovered in Step 2 above.

Step-4 : adding the first two terms while excluding similar elements

                   x • (x+1)

             Adding the last two terms together and finding commonalities:

                   4 • (x+1)

Step-5 : In step 4, add up the four terms:

                   (x+4)  •  (x+1)

            Which is the desired factorization

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Answer: c. 16:20

Step-by-step explanation:

30/5=6

24/4=6

Age demography is important that young people are poised to move into their prime reproductive years.

Demography is a measure of the characteristics or changes in population. Births, deaths, age , marriages, immigration, and periodic census records provide important information in making informed decisions about national policy.

Governments, businesses, and non-governmental organizations use demographic data to learn about population characteristics for many purposes, including policymaking and economic market research.

Age Demographics: In the demographics, it is defined as common to divide the population into three broad age groups, children and youth (under 15) and working-age population (15-64) and elderly population (Ages 65 and Over)

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Answer:

There were 74 hot dogs and 117 sodas sold.

Step-by-step explanation:

The following equation describes the conditions of the question:

[tex]191=x+(x+43)\\191=2x+43\\[/tex]

where x is the number of hot dogs sold.

Solve for x:

[tex]191=2x+43\\(191)-43=(2x+43)-43\\148=2x\\\frac{148}{2}=\frac{2x}{2}\\74=x[/tex]

There were 74 hot dogs sold and (74+43) or 117 sodas sold.

Answer:

8(11x-13)

Step-by-step explanation:

[tex]13(5x-8)+23x\\65x-104+23x\\88x-104\\8(11x-13)[/tex]

"If in two triangles, sides of one triangle are proportionate to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and consequently the two triangles are comparable," asserts the Side-Side-Side (SSS) criteria.

Take a look at the triangles below.

- We can see that all three pairs of the sides of these triangles are congruent. - This is also known as "side-side-side" or "SSS." According to the SSS criteria for triangle congruence, two triangles are congruent if they have three pairs of congruent sides.

SSS Congruence Rule Theorem

When one triangle's three sides are identical to the corresponding three sides of another triangle, two triangles are said to be congruent.

The aforementioned theorem will now be proven.

Given: [tex]\triangle A B C[/tex] and [tex]\triangle P Q R[/tex] such that AB=PQ, BC=QR and AC=PR.

To prove: [tex]\triangle A B C \cong \triangle PQR[/tex]

Construction: Let BC be the longest side of [tex]\triangle A B C[/tex] and so QR is the longest side of [tex]\triangle P Q R.[/tex]

Draw PS so that [tex]\angle R Q S=\angle C B A[/tex] and [tex]\angle Q R S=\angle B C A[/tex].

Join SQ and SR.

In [tex]\triangle ABC[/tex] and [tex]\triangle SQR[/tex],

BC=QR Given

[tex]\angle C B A=\angle R Q S[/tex] By construction

[tex]\angle B C A=\angle Q R S[/tex] By construction

[tex]\triangle A B C \cong \triangle SQR[/tex] By ASA congruence

[tex]\angle A=\angle S[/tex] By CPCTC

AB=SQ By CPCTC

AC=SR

Now, AB=PQ and AB=SQ => PQ=SQ

Similarly, AC=PR and AC=SR => PR=SR

Hence, [tex]\triangle ABC \cong \triangle PQR,[/tex]

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Answer:

9% decrease

Step-by-step explanation:

Hope this helps!

The equation of the line that passes through the given points will be equal to  11x -9y + 101 = 0.

It is possible to determine a line's direction and steepness by looking at its slope. Finding the slope between lines inside a coordinate plane can aid in anticipating if the lines are perpendicular, parallel, or none at all without physically using a compass.

As per the data in the question,

Use the equation of the line passing through the point (x₁, y₁) with slope m will be:

y - y₁ = m(x - x₁)

Slope, m = 11/9

And, the points are (-1, 10)

Now,

y - 10 = 11/9(x + 1)

9y - 90 = 11x + 11

11x - 9y + 11 + 90 = 0

11x -9y + 101 = 0

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The perimeter of triangle def similar to triangle abc is 22 cm.

Perimeter is the total distance around a object.

Note: The ratio of the perimeters of two similar figures is equal to the ratio of their corresponding side lengths.

To calculate the perimeter of triangle def, we use the formula below.

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1 and solve for P'

Hence, the perimeter of triangle def is 22 cm.

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Two or more shapes are said to be similar if they have common corresponding properties of sides and/ angles. Thus the error made is option D.

Similar shapes are two or more given shapes that have some common properties with respect to their corresponding sides or angles. So that if on comparing each corresponding sides or angles, and are related, then they are similar. Note that similarity does not imply congruency.

Comparing the sides and angles of the given triangles ΔLMN and ΔQRS, it can be seen that;

Angle Side Angle (ASA) similarity relation implies that the included side between two pairs of congruent angles are similar. Thus, the error made in the diagram is explained in option D.

Option D. The ASA Postulate cannot be used to prove the congruence of the two triangles as shown because the included sides between the two pairs of congruent angles are not marked as congruent in both triangles.

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This is an arithmetic sequence. In this sequence, the common difference is 6 (12 - 3/6 = 6).

A sequence is an ordered arrangement of objects or events which follow one another in a specific order. This order can be based on a mathematical rule or pattern, such as a sequence of numbers, or it can be based on physical or temporal characteristics of the objects or events, such as the order in which they appear in a movie or story. Sequences are useful for representing and analyzing data, as well as for visualizing relationships between objects. They can also be used to model problems, such as predicting the future or solving a puzzle.

This is an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a constant, called the common difference, to the preceding term. In this sequence, the common difference is 6 (12 - 3/6 = 6).

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Thousands, hundreds and tens are three examples of values and there are many more.

Value in mathematics is a number that represents the outcome of a computation or function. In the aforementioned example, you may inform your teacher that 5 + 6 Equals 30 or that x + y = 9 if x = 6 and y = 3. A variable or constant can also be referred to as a value.

The number 4 is in this case in the tens column. As a result, the place value of the number four is tens or tens.

Thousands, hundreds and tens are three examples of values and there are many more like ones etc. Moreover after solving any problem for unknown variable the resulting numerical solution can also be termed as value.

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The AAA (angle-angle-angle) similarity theorem states that two triangles' corresponding angles are identical if and only if their corresponding sides are proportionate.

Knowing two angles, such as 32° and 64°, we may calculate the following angle, which is 84°, since 180-(32+64)=84. (This is commonly referred to as the AAA Postulate—which is valid in every way, although two angles are more than enough.) Working backwards will help you understand the proposition better.

The sides and angles of a set of triangles must be equal for them to be congruent. The sides of a triangle with all respective angles equal, i.e. AAA condition, may or may not be equal.

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Answer:

A. 4,500 + 50g = 6,000

Step-by-step explanation:

25(18) + 50(g) = 6000

4500 + 50g = 6000

The correct option is A.

The 17th hole would be 255 yards long.

What are ratio and proportion?

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other.

hole = c yards

1.5 / 40 = 8.5 /c

1.5 * c = 45 * 8.5

1.5 * c = 382.5

c = 255 yards

Hence, the 17th hole would be 255 yards long.

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The answer is True. If two vertical angles are congruent, then two lines cut by a transversal will be parallel.

Vertical angles are angles that are opposite each other and share the same vertex. They are congruent angles, meaning that they have the same measure. When two angles are congruent, the sides of the angles are parallel. This means that the two lines cut by a transversal will also be parallel.

The formula for this is that if two angles are congruent, then the lines cut by a transversal are parallel. This is also known as the Angle-Angle-Angle (AAA) Postulate. This postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. This means that the lines cut by a transversal are parallel.

In other words, if two angles of a triangle are congruent, then the two sides of the triangle that make those angles are parallel. This is because the two angles are congruent, meaning that they have the same measure. Therefore, if two vertical angles are congruent, then two lines cut by a transversal will be parallel.

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