Prove that the angle between any two diagonals of a cube is given by cos=1/3​​

The product of three consecutive odd numbers is an odd number.

An odd number is a number that is not divisible by 2, therefore also can not be divided exactly into pairs. Numbers that are deemed odd number end with the number 1, 3, 5, 7, or 9. When divided by 2, odd numbers leave a remainder of 1.

When three consecutive odd numbers are multiplied, the end product would be another odd number. For example:

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The best finish of the construction of TU :

Set the compass at length RS. Then place point U on the lower segment at a distance RS from Point T.

What is a Line Segment?

A line segment is a line with a fixed endpoint.

It has a definite length.

We have to construct TU a copy of RS using a compass and a straightedge.

Construction of TU:

Set the compass at length RS. Then place point U on the lower segment at a distance RS from Point T.

Hence, the best finish of the construction of TU :

Set the compass at length RS. Then place point U on the lower segment at a distance RS from Point T.

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The vertex of the parabola equation is (5, 43) and the parabola standard form is y = 2x² + 20x + 43.

What is Parabola?

A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed-line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola.

We have,

The parabola standard from f(x) = a(x-h)² + k

f(x) = 2x² + 20x + 43

The general equation of a parabola is y = a(x - h)² + k or x = a(y - k)² +h, where (h,k) denotes the vertex.

Consider an equation y = 2x² + 20x + 43. For this parabola, a = 2 , b = 20 and c = 43. which is a parabola.

Vertex: (h, k)

h = -b/2a

= 20/(2 × 2)

= 5

k = f(h)

= f(5) = 2(5)² + 20(5) + 43

= 50 + 100 + 43

Thus vertex is (5, 43)

Hence, the vertex of the parabola equation is (5, 43) and the parabola standard form is y = 2x² + 20x + 43.

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The ratio of red grapes to blueberries in simplest form is 3/4.

What is ratio?

A ratio in mathematics demonstrates how many times one number is present in another. For instance, if a bowl of fruit contains eight oranges and six lemons, the ratio of oranges to lemons is eight to six. The ratio of oranges to the total amount of fruit is 8:14, and the ratio of lemons to oranges is 6:8.

Given:

Matthew made a fruit salad, in which the ratio of blueberries to red grapes is 8:6.

We have to find the ratio of red grapes to blueberries in simplest form.

Let there are 6 red grapes and 8 blueberries.

So, the ratio is

Red grapes / blueberries = 6/ 8

Reduces the ratio,

6 / 8 = (2 x 3) / (2 x 4) = 3 /4

Hence, the ratio of red grapes to blueberries in simplest form is 3/4.

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The group of things from which data are taken for a statistical research is referred to as a population in statistics. A data pool for study might consist of numerous things, many individuals, etc.

large-scale numerical data collection and analysis, especially with the aim of extrapolating proportions in the total from those in a representative sample.

The study and development of techniques for gathering, processing, interpreting, and presenting empirical data are the focus of the science of statistics. It could be a collection of things, a gathering of people, etc. It serves as the study's data set.

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The four types of Transformations - Rotation, Translation, Reflection, Dilation.

A point, line, or geometric figure can be transformed in one of four ways, each of which affects the shape and/or location of the object. Pre-Image refers to the object's initial shape, and Image, after transformation, refers to the object's ultimate shape and location.

Transformations may be divided into four categories: translation, rotation, reflection, and expansion. You must be able to carry out each change and recognize which ones have already been done.

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Translation, rotation, reflection, and dilation are the four basic forms of transformations.

By rotating, reflecting, or translating a shape on a coordinate plane, a transformation is made. A fresh viewpoint on geometry known as transformational geometry was put forth by Felix Klein in the 19th century.

The transformation is f: X X, which stands for a function, f, that maps to itself. The transformation causes the pre-image X to change into the picture X. Translation, rotation, reflection, and dilation are only a few examples of the transformations that can be used alone or in combination. A function is translated when it is moved in one direction, rotated when it is spun around a point, reflected when it is turned into the opposite of itself, and dilated when it is scaled. Two-dimensional geometrical figures can move about a coordinate plane via mathematical transformations.

Hence, Translation, rotation, reflection, and dilation are the four basic forms of transformations.

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(a) The vector sum A+B = (a(1)+b(1))i + (a(2)+b(2))j.

(b) The difference vector A-B = (a(1)-b(1))i + (a(2)-b(2))j.

(c) The square of the magnitude of the sum vector |A + B| = (a(1)+b(1))^2 + (a(2)+b(2))^2.

In the given question, two vectors A and B, expressed in unit vector notation as A = a(1)i+a(2)j and B = b(1)i+b(2)j.

A = a(1)i+a(2)j

B = b(1)i+b(2)j

(a) We have to find the vector sum A + B.

A+B = (a(1)i+a(2)j) + (b(1)i+b(2)j)

A+B = a(1)i+a(2)j + b(1)i+b(2)j

A+B = (a(1)+b(1))i + (a(2)+b(2))j

(b) We have to find the difference vector A - B.

A-B = (a(1)i+a(2)j) - (b(1)i+b(2)j)

A-B = a(1)i+a(2)j - b(1)i-b(2)j

A-B = (a(1)-b(1))i + (a(2)-b(2))j

(c) We have to find the square of the magnitude of the sum vector |A + B|.

|A + B| = [√{(a(1)+b(1))^2 + (a(2)+b(2))^2}]^2

|A + B| = (a(1)+b(1))^2 + (a(2)+b(2))^2

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The right question is:

Consider two vectors A and B, expressed in unit vector notation as A = ai+aj and B = bi+byj.

33% Part (a) Enter an expression for the vector sum A + B in terms of quantities shown in the expressions above A+B.

33% Part (b) Enter an expression for the difference vector A-B in terms of quantities shown in the expressions

33% Part (c) Enter an expression for the square of the magnitude of the sum vector |A + B| in terms of quantity above.

The domain and range of the given quadratic function are respectively; Domain = -3 < x < 7; Range is; 3 ≤ x ≤ ∞

The domain of a function is defined as the set of all possible input values for which the function is possible.

Meanwhile, the range is defined as the set of all possible output values for every possible input value that makes the function possible.

Now, from the graph, the range will be values on the y-axis while the domain will be values on the x-axis.

Now, from the given graph we see that the curve is approaching the lines x = -3 and x = 7. Thus, the domain of this function is;

Domain = -3 < x < 7

Meanwhile the range keeps increasing to positive infinity and as such we can say that;

Range is; 3 ≤ x ≤ ∞

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Answer:

ur mom

Step-by-step explanation:

haha lol;)

2x² 7x 0 is not a quadratic equation.

A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The standard form of quadratic equation is ax^2+bx+c

Standard Form: y = a x 2 + b x + c y=ax^2+bx+c y=ax2+bx+c.

Factored Form: y = a ( x − r 1 ) ( x − r 2 ) y=a(x-r_1)(x-r_2) y=a(x−r1)(x−r2)

Vertex Form: y = a ( x − h ) 2 + k y=a(x-h)^2+k y=a(x−h)2+k.

Comparing given  equation 2x² 7x 0 with standard form of quadratic equation is , we are not able to find values of a,b,c.

Hence,2x² 7x 0 is not a quadratic equation.

2x² 7x 0 is not a quadratic equation.

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Step-by-step explanation:

the domain of a function is the interval or set of all valid x (input) values.

the range of a function is the interval or set of all valid y (function result) values.

we see in the graph that we get for y all continuous values between 0 (max) and -9 (min).

both end points are filled, so we need to include them into the interval of valid values (via the <= or >= signs).

so the range is

-9 <= y <= 0

Answer:

x = 6

y = 2

(6,2)

Step-by-step explanation:

THE SOLUTION WORK IS ATTACHED

The AC is 18 feet long.

Triangles are primarily involved in the mid-point theorem. According to this rule, if the midpoints of any two triangle sides are connected, the line formed is parallel to the third side and equal to half of the third side. Contrarily, if a parallel line is drawn from one side's midpoint to any other, it will bisect the third side and be equal to half the side to which it is parallel.

As shown in the diagram,

AM = MB, CN = NB.

The midpoints of the sides AB and CB, respectively, are M and N.

So, according to the midpoint theorem,

MN ║ AC,

The alternative interior angle theorem demonstrates that

∠BMM ≅∠BAC

∠BNM ≅∠BCA

According to the AA similarity postulate,

ΔBMM ≅∠BAC

The characteristic of identical triangles

[tex]\frac{BM}{BA} = \frac{MN}{AC}\\ \frac{BM}{BM+MA} = \frac{MN}{AC}\\ \frac{4}{4+4} = \frac{9}{AC}\\ \frac{4}{8} =\frac{9}{AC} \\[/tex]

4AC=72

AC=18 ft

As a result, AC is 18 feet long.

The complete question is:-

What is the length of the AC?

3 ft

4 ft

9 ft

18 ft

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The roots of the quadratic equation are real and distinct

What is Quadratic Equation?

A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

where ( b² - 4ac ) is the discriminant

when ( b² - 4ac ) is positive, we get two real solutions

when discriminant is zero we get just one real solution (both answers are the same)

when discriminant is negative we get a pair of complex solutions

Given data ,

Let the quadratic equation be represented as A

Now , the value of A is

2x² + 4x - 7 = 0   be equation (1)

On simplifying the equation , we get

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

Substituting the values in the equation , we get

x = [ -4 ± √ ( ( 4 )² - 4 ( 2 ) ( -7 ) ) ] / 2 ( 2 )

On simplifying the equation , we get

x = [ -4 ± √ ( 16 + 56 ) ] / 4

x = [ -4 ± 6√2 ] / 4

x = ( -2 + 3√2 ) / 2

And ,

x = ( -2 + 3 ) / √2

Therefore , the roots are x = ( -2 + 3√2 ) / 2 and x = ( -2 + 3 ) / √2

Hence , the roots of the equations are real and distinct

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Answer:

0.2

Step-by-step explanation:

0.5*0.4=0.2

The required length of the RT is given as 38 units, in the triangle.

The triangle is a geometric shape that includes 3 sides and the sum of the interior angle should not be greater than 180°

here,

Given figure show is of the triangle where UV, VW, and UW are midsegments,
The property of triangles states that,
The length of the side is half of the line joining the midpoints of the adjacent of the triangle.
2VW = RT
RT = 2[19]
RT = 38

Thus, In the triangle, the required length of the RT is stated as 38 units.

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An object is moved rigidly when it is transferred from one location to another without affecting its size or shape.

A "matching" figure is one that can be placed perfectly on another. The bread has the same size and form when it is stacked. Match refers to objects that are precisely the same size and form. If one of two geometric figures can be consistently overlaid on the other, they are said to be congruent or to be in a congruent relationship.

When an object is transferred from one place to another without changing in size or shape, it is said to be moved rigidly. Two shapes can only match if a precise movement results in a correspondence between them.

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If the degree of the polynomial function is an even number and the leading coefficient is negative, the graph will have an even number of turning points and will be concave down. The graph will also have a single x-intercept at the origin, and the y-intercept will be negative.

A graph of polynomial functions with an even degree and negative leading coefficient will have an even number of turning points and will be concave down. A turning point is the point on a curve where the curve changes direction, and a concave down shape means that the curve is curved down from left to right. The graph will have a single x-intercept at the origin, which is the point where the graph crosses the x-axis. The y-intercept of the graph will also be negative, which is the point where the graph crosses the y-axis. This graph will have a negative slope and will decrease as it moves away from the origin. The graph will approach the x-axis but will not touch it, creating a U-shaped curve. The graph will also have a maximum or minimum value, depending on the specific values of the polynomial function.

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Let f be the function given by f (x) = e + cos x - 1. the value of f' (2)  is 0.0349

The given function is f (x) = e + cos x - 1,

f' is the derivative of the given function, and the derivative is sort for below

f (x) = e + cos x - 1

f' (x) = 0 + -sin x - 0

f' (x) = -sin x

for f' (2) we substitute the x = 2 in the derivative

f' (x) = -sin x

f' (2) = -sin 2

f' (2) = -sin 2

f' (2) = -0.0349

hence f' (2) = -0.0349

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The correct options will be:

y = x + 6 and 3x - 3y = -18 .... Infinitely many solutions.

y = -2x + 5 and 2x + y = -7 .... No solution.

y = -4x + 11 and -6x + y = 11 ... One solution.

Solving equations

y = -4x + 11 ....... equation i

-6x + y = 11 ...... equation ii

Put equation i into ii.

-6x + y = 11

-6x + (-4x + 11) = 11

-6x - 4x + 11 = 11

-6x - 4x = 11 - 11

-10x = 0

x = 0

Therefore,

x = 0 and y = 11.

This gives a solution for each variable.

y = -2x + 5 ..... i

2x + y = -7 ..... ii

Put equation i into ii

2x + y = -7

2x + (-2x + 5) = -7

2x - 2x = -7 - 5

0 = -12

This indicates no solution.

y = x + 6 ..... i

3x - 3y = -18 ..... ii

Put equation i into ii

3x - 3y = -18

3x - 3(x + 6) = -18

3x - 3x - 18 = -18

3x - 3x = - 18 + 18

0 = 0

This implies that there are infinitely many solutions.

The complete question is:

How many solutions will each system of linear equations have? Match the systems with the correct number of solutions.

y = x + 6 and 3x - 3y = -18

no solution

y = -2x + 5 and 2x + y = -7

Ano

infinitely many solutions

y = --4x + 11 and 6x + y = 11

one solution

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So  in the traingle PQR shown in the figure below PQ,QR,PR is the side of the traingle .

we have to prove that ∠QPR = ∠PQR = ∠ PRQ.

According to the definition of equilateral triangle all sides length of equilateral traingle are equal.

so PQ=QR=PR

Statement 1:

Angles opposite to equal sides QR and PR are equal so .

=>   ∠QPR = ∠PQR

Statement 2:

Angles opposite to equal sides PR and PQ are equal so .

  =>   ∠PQR = ∠ PRQ.

combining statement 1 and statement 2.

=>  ∠QPR = ∠PQR = ∠ PRQ and hence all three angles are same [Proved]

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Answer:

m<1+m<2+m<3=180°

90°+26°+m<3=180°

116°+m<3=180°

m<3=180°-116°

m<3=64°

m<3=64

Answer:

Yes

Step-by-step explanation:

We want to find the multiples of 9

9*1 = 9

9*2 = 18

9*3 = 27

9*4 = 36

9*5 = 45

9*6 =54

Since 9*6 = 54

54 is a multiple of 9

Answer with step-by-step explanation:

OR

Answer:

s=2h

Step-by-step explanation:

it takes 2 hours for 1 inch of snow so 1snow=

Answer:

Fair Dice. In mathematics we say "fair dice" when we mean that there is an equally likely chance of landing on any face. Most people think of these little cubes. when we say "dice" ... ... but all of the Platonic Solids can make fair dice! Because the faces are all the same, there is an equal chance of landing on any face.

Step-by-step explanation:

Answer:

whole numbers are numbers including zero, positive and no fractions

Step-by-step explanation:

0, 8, 89, 98, 999999999999, 57, 3, 5

Step-by-step explanation:

-6 - (-5.5) = -0.5

-5.5 - (-5) = -0.5

rate of change is 0.5. Points, if plotted, would be angled up towards the right so slope is positive.

y-intercept occurs when x  = 0. From the chart this is when y = -6

Answer:

y = 0.5x - 6

Answer: [tex]\frac{-7}{8}[/tex]

Step-by-step explanation:

    First, we will need to find the slope of line r.

[tex]\displaystyle \frac{y_{2} -y_{1} }{x_{2} -x_{1}} =\frac{9-1}{8-1}=\frac{8}{7}[/tex]

    Next, we know that perpendicular slopes are negative reciprocals of one another. To find the slope of line s, we will find the negative reciprocal of line r's slope.

[tex]\displaystyle \frac{8}{7} \rightarrow \frac{-7}{8}[/tex]

There are 37 elves are helping Santa by solving both the equations x + y=58 and 4x+2y=158 the concept of equations.

Equations are mathematical expressions that have two algebraic expressions on either side of an equals (=) sign. This relationship is illustrated by the left and right expressions being equal to one another. L.H.S = R.H.S is the first term in every mathematical equation.

According to the question we assume the number of reindeer are x and number of elves are y.

Total 37 elves are helping Santa. There are a total of 58 reindeer and elves helping Santa get ready for Christmas hence the equation of given condition is as follows:

x + y=58

Each reindeer has 4 legs and each elf has 2 legs counted 158 legs when all the reindeer and elves were in Santa’s workshop helping Santa. Hence the equation of given condition is as follows:

4x+2y=158

By solving both the above equation multiplying first equation by 2

2x+2y=116

Subtract both the above equation

2x=42

x=21

Hence the number of reindeer are x=21

Inserting value of x in first equation gives y=37

The number of elves is y=37

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Answer:

A fair number cube is a cube with six sides, each containing a different number.

Step-by-step explanation:

six sidesA fair number cube is a cube with six sideseach containing a different number. What is the probability that the number cube turns up a multiple of 3? 1/3Let E be the event of getting a multiple of 3. Hence the probability of getting a multiple of 3 when a die is thrown is 1/3. Cube Numbers – Numberblocks 1 to 8000!