Which statement is true? All decimals are rational. All rational numbers can be written as the quotient of Integers

The value for the given expression  [tex]\alpha ^{2} + \beta ^{2}[/tex] will be [tex](\alpha +\beta )^{2}-2\alpha \beta[/tex].

What are Algebraic Identities?

Algebraic identities are algebraic equations that hold true for any value of each of their variables. In factoring polynomials, they are also used.

As we know, [tex](a+b)^{2} = a^{2} +2ab + b^{2}[/tex]  then,

For the given question, [tex](\alpha +\beta )^{2} = \alpha ^{2} +2\alpha \beta + \beta ^{2}[/tex]

Now, subtracting [tex]2\alpha \beta[/tex] on both sides then, we get

or, [tex](\alpha +\beta )^{2}-2\alpha \beta = \alpha ^{2} +2\alpha \beta + \beta ^{2}-2\alpha \beta[/tex]

or, [tex](\alpha +\beta )^{2}-2\alpha \beta = \alpha ^{2} + \beta ^{2}\\[/tex]

or, [tex]\alpha ^{2} + \beta ^{2}=(\alpha +\beta )^{2}-2\alpha \beta[/tex]

That's how the value  [tex]-2\alpha \beta[/tex] comes.

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Each text will cost Hassan $0.09

A rate is a special ratio in which the two terms are in different units. It can also be defined as the amount of money that is charged for goods or services. For example, if a 12 malt can of costs 69¢, the rate is 69¢ for 12 malt. Then 1 malt will cost ¢69/12.

Similarly $5.31 is the cost of 59 text messages

Therefore a text message will cost $5.31/12=$0.09

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Answer:

x = 41.5

Step-by-step explanation:

the sum of the 3 angles in a triangle = 180°

sum the 3 angles and equate to 180

2x + 9 + x + 13 + x - 8 = 180 , that is

4x + 14 = 180 ( subtract 14 from both sides )

4x = 166 ( divide both sides by 4 )

x = 41.5

⇒The sum of the interior angles of a triangle is 180°

⇒Meaning (2x+9)°+(x-8)°+(x+13)°=180°

⇒Simplify the above equation to find the value of x.

⇒[tex]2x+x+x+9-8+13=180\\4x+14=180\\4x=180-14\\4x=166\\\frac{4x}{4} =\frac{166}{4} \\x=41.5[/tex]

∴ x=41.5

The set of ordered pairs that is a function is (d) (0, 0), (1, 1), (4, 2), (9, 3)

The list of options represents the given parameter

As a general rule, for an ordered pair to be a function;

The y values on the ordered pair must point to different x values

In (a) the y values 4, 5, 6 and 7 have the same x value of 3

So, it is not a function

In (b) the y values 5 and 8 have the same x value of 2

So, it is not a function

In (c) the y values -1 and 1 have the same x value of 1

So, it is not a function

In (d) all the y values have different x values

So, it is a function

Hence, the ordered pair that is a function is  (d)

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Answer:

212 feet fallen after 7 seconds

Step-by-step explanation:

You visit the Grand Canyon and drop a penny off the edge of the cliff. The distance the penny will fall is 32ft for the first second, 62ft the next second, 92ft the third second, and so on. How far will the penny have traveled after 7 seconds?

if 32 ft for 1 second

if 62 ft for 2 second

if 92 ft for 3 second

We see that the penny from second 1, falls an addition 30 ft/s.

equation: 32 + 30 (t - 1)

when t = 7 seconds:

32 + 30 (7 - 1)

32 + 30 (6)

32 + 180

= 212 feet fallen after 7 seconds

When Niecy combines the fruit she will have a total of Start Fraction 12 over 14 End Fraction

information form the question

Niecy has Nine-fourteenths of a cup of pears

Three-fourteenths of a cup of mangos

The problem is faced with addition of fractions

writing out the fractions in figures gives

Niecy has Nine-fourteenths of a cup of pears = 9 / 14

Three-fourteenths of a cup of mangos = 3 / 14

Addition of the fraction is done as follows

= 9 / 14 + 3 / 14

=  (9 + 3) / 14

= 12 / 14

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Answer:

Lol

Step-by-step explanation:

k

Using the concepts of equilateral triangle, we got that 107 square feet is the total surface area of an equilateral triangular pyramid with base side lengths of 6 feet and a height of 10 feet.

We have :

Base (b) = 6 feet

Slant height (l) = 10 feet

The lateral surface area is calculated using:

L = (3/2)× Length × Breadth

So, we have:

   L = (3/2) ×6 × 10

=>L=(3×3×10)

=>L=90 square feet.

The total surface area is calculated using:

T=Lateral surface area of the wall + area of equilateral triangle

=>T=L+ (√3/4)b²

So, we have:

T = 90 + (√3/4)×(6×6)

=>T=90+(3×3×√3)

=>T=90+9√3

=>T=107 square feet (approx)

Hence, the total surface area of an equilateral triangular pyramid with base side lengths of 6 feet and a height of 10 feet is 107 square feet.

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The area of a circle is 12πx-60π unit².

By examining the given figure, there are two circles i.e the small circular hole and the bigger circle. Since the small is a hole then we have to subtract its area from the area of the bigger circle.

Given:

radius of small circle(r₂) = x+2

radius of bigger circle(r₁) = x+8

The formula of Area(A) = πr²

Area of the figure = A₁ - A₂

                             =  πr₁² - πr₂²

                             =  π(x+8)²- π(x+2)²

                             = π(x²+16x+64) -  π(x²+4x+4)

                             =  x²π+16πx+64π - x²π-4πx-4π

                            =   12πx-60π unit squared

Therefore,  the area of the circle is 12πx-60π unit squared. So the 2nd option is the answer.

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The probability that 5 hearts are chosen if 8 cards are selected from a well-shuffled deck of 52 playing cards is  0.0156296302.

The number of choosing 8 cards out of 52 is = ⁵²C₈

Number of ways of choosing 5 cards of heart out of 13 cards and 3 out of remaining 39 cards = ¹³C₅ × ³⁹C₃

Probability of choosing 5 hearts =  ¹³C₅ × ³⁹C₃ /  ⁵²C₈

= 1287 ×9139 / 752538156

= 11761893 / 752538156

= 0.0156296302

A basic 52-card deck consists of 13 ranks in each of the four French suits: clubs, diamonds, hearts, and spades. Each card includes three court cards (face cards), King, Queen, and Jack, as well as reversible (double-headed) images. Each card also contains ten numeral cards or pip cards, from one to ten.

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Answer:

Step-by-step explanation:

Arithmetic Sequence

Answer:

  (a)  Arithmetic Sequence

Step-by-step explanation:

You want to know if the graph of a straight line represent an arithmetic, geometric, or neither sequence.

An arithmetic sequence is characterized by a common difference between terms that are 1 unit apart. A graph of an arithmetic sequence is points that lie on a straight line.

The graph shows an Arithmetic Sequence.

__

Additional comment

A graph of a geometric sequence would show points that lie on an exponential curve.

Answer:

5

Step-by-step explanation:

If the two events described in the questions are independent events, the probability of both happening next year is 28.67%.

Independent events refer to events whose occurrence is not interdependent. In other words, if the probability of occurrence of event A is not influenced by the probability of occurrence of event B, then A and B are independent events. Mathematically, independent events are represented as P(A│B) and is calculated by multiplying the individual probabilities of the two events. The formula of probability of two independent events is:

P(A│B) = P(A)*P(B)

Hence,

P(A│B) = 0.47*0.61

P(A│B) = 0.47*0.61

P(A│B) = 0.2867

The probability of two events happening next year is 28.67%.

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The probability that you select a black card and a red card is 26/51 as per the condition mentioned "choosing two cards (without replacement )".

The area of mathematics known as probability studies potential outcomes of events as well as their relative probabilities and distributions. Simply put, probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or likelihood of various outcomes. Statistics is the study of events that follow a probability distribution.

Here,

Since we have not mentioned the order,

so it can black-red or red-black,

=26/52*26/51+26/52*26/51

=2*26/52*26/51

=26/51

Assuming you choose two cards (without replacement), there is a 26/51 chance that you will choose a black card and a red card.

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The measure of ∠ YXR is 55 degrees.

Given that:-

m∠ XOY = 110 degrees

XR is a tangent to circle O at X.

We have to find the measure of ∠ YXR.

Draw the radii XO and OY to form the isosceles triangle XOY.

We know that the radius of the circle is perpendicular to the tangent.

Hence,  ∠OXR = 90 degrees.

Consider the triangle XOR, we know that XO and OY are the radii of the triangle and hence equal.

Let, ∠ OXY = OYX = x

Hence,

110 + x + x = 180 degrees

2x = 70 degrees

x = 35 degrees

Now, we know that,

∠ OXR = ∠ OXY + ∠ YXR

90 = 35 + ∠ YXR

∠ YXR = 90 - 35 = 55 degrees.

∠ YXR = 55 degrees.

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The line that has a slope of -2 and a y-intercept of -1 is shown in the graph below.

An equation of a line in slope-intercept form that is expressed as y = mx + b, can be graphed if we know the value of the slope (m), and the value of the y-intercept (b).

We are given the following:

To write the equation of the line in slope-intercept form, substitute m = -2 and b = -1 into y = mx + b:

y = -2x - 1

This means that the rise/run of the line, which is the slope, to be graphed would be equal to -2 while the line will intercept the y-axis at point -1, because the y-intercept given is -1.

The line with the stated features are shown in the graph below.

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The equation of line [L] would be y = (1/6)x + 9.

The general equation of a straight line is -

[y] = [m]x + [c]

where -

[m] → is slope of line which tells the unit rate of change of [y] with respect to [x].

[c] → is the y - intercept i.e. the point where the graph cuts the [y] axis.

We have a equation of line [k] is y + 2 = -6(x - 7). Line [L] includes the point (-6,8) and is perpendicular to line k.

We have the equation of line [K] as -

y + 2 = - 6(x - 7)

y = - 6(x - 7) - 2

y = - 6x + 42 - 2

y = - 6x + 40

Slope of line [L] will be -

m[L] = 1/6

Assume the equation to be -

y = (1/6)x + c

Line [L] includes the point (-6, 8), so we can write -

8 = (1/6) x (- 6) + c

8 + 1 = c

c = 9

So, the equation will be -

y = (1/6)x + 9

Therefore, the equation of line [L] would be y = (1/6)x + 9.

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The amount a container would take to spray 1 entire lawn is 7 litres.

Given that a container with 3½ liters of weed killer can spray 1/2 of a lawn.

We want to find the amount of liters to take a spray 1 entire lawn.

3½ liters of weed killer can spray 1/2,

liters take to spray 1 lawn is 3½/1/2=7 liters

Hence, the amount of liters it would take to spray 1 entire lawn when a container with 3½ liters of weed killer can spray of a lawn is 7 liters.

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The possible dimension of the triangular garden assuming it to be a right-angle triangle is (12,2),(8,3), and (4,6).

A triangle is a 3-sided shape that is occasionally referred to as a triangle. There are three sides and three angles in every triangle, some of which may be the same.

The area of a triangle is given as (1/2) × base × height.

The given area of the triangle = 12 feet².

Thus,

12 =  (1/2) × base × height

Base × height = 12 × 2

Base × height = 24

The combination of two quantities gives 24 by multiplying will be (12,2),(8,3), and (4,6).

For example, the triangle with dimensions (4,6) is attached below.

Hence "The possible dimension of the triangular garden assuming it to be a right-angle triangle is (12,2),(8,3), and (4,6)".

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The equation of the parabola with the vertex (3,5) in the form of f(x)=a(x-h)^2+k is f(x)=7(x-3)^2+5.

In the given question we have to find the equation of parabola in the form of f(x)=a(x-h)^2+k

The given function is f(x)=7x^2.

The given vertex = (3,5)

The equation of the parabola with the vertex of (h,k) is

y=a(x-h)^2+k

On comparing we get; a=7, h=3, k=5

Now putting the value

f(x)=7(x-3)^2+5

Hence, the equation of the parabola with the vertex (3,5) in the form of f(x)=a(x-h)^2+k is f(x)=7(x-3)^2+5.

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Answer:

86

Step-by-step explanation:

mean is calculated as

mean = [tex]\frac{sum}{count}[/tex]

given mean mark for class of 30 is 70 , then

[tex]\frac{sum}{30}[/tex] = 70 ( multiply both sides by 30 )

sum = 2100

given the mean mark for 20 boys is 62, then

[tex]\frac{sum}{20}[/tex] = 62 ( multiply both sides by 20 )

sum = 1240

the sum for the 10 girls is then 2100 - 1240 = 860

then mean score for the 10 girls is

mean = [tex]\frac{860}{10}[/tex] = 86

Answer:

15 coaches

Step-by-step explanation:

750 people go on a coach trip each coach can hold 50 people How many coaches are needed?

750/50 = 15 coaches

Answer:

750÷50 = 15

The answer is 15.

Because 750 people are going in a coach trip and each coaches can hold 50 people.

Answer:

x=22

Step-by-step explanation:

Let the unknown number be x

:- the power of x = x³

From the question

x³-15=7 and x²

x³=7+15

x³ * x²= 22

x= 22

Equation of the given data y = 1750x -20,000 and graph of the equation is attached.

As given in the question,

Data of the table :

Year :               1                 2                   3              4                5

Investment : $20,000      $21,750      $23,500    $25,250      $27,000

(x₁, y₁) = ( 1, 20,000)

(x₂, y₂) = ( 2, 21,750)

Equation of the given graph is :

( y - 20,000)/ (x -1) = (21,750 - 20,000)/ (2 -1)

⇒ ( y - 20,000)/ (x -1) = 1750

⇒ y = 1,750x -18,250

Graph of the equation  y = 1,750x -18,250 s represented by the graph.

Graph is attached.

Therefore, equation of the given data y = 1750x -20,000 and graph of the equation is attached.

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The result that we get after the long division is:

3[tex]x^{2}[/tex] + 9x + 71;

What is the long-division method?

Long division in mathematics is a strategy for breaking down complicated division problems into a series of simpler steps. It is the approach that division-based issues are often solved with. See the dividend, quotient, divisor, and remainder in the following long division.When splitting huge numbers, the task is divided into several sequential parts using the long division approach. The dividend is divided by the divisor, just as in conventional division problems, and the result is known as the quotient; occasionally, it also produces a remainder. In this post, we will learn more about the long division technique, including its processes and examples.

In the given question,

The expression we have:

[tex]$3 x^3+15 x^2-17 x+6$[/tex]; and this expression is been divided with x + 6;

we get:

At first, we will divide the expression with 3[tex]x^3[/tex]; we get:

9[tex]x^{2}[/tex] -17x +6;

Now again we will divide it by 9x: we get:

71x + 6;

Now, again dividing it by 71, we get:

-70.

So, The result that we get after the long division is:

3[tex]x^{2}[/tex] + 9x + 71;

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Answer:

33

Step-by-step explanation:

evaluate f(- 3) by substituting x = - 3 into f(x)

f(- 3) = 3(- 3)² - (- 3) + 5 = 3(9) + 9 + 5 = 27 + 14 = 41

evaluate f(4) by substituting x = 4 into f(x)

f(4) = 3(4)² - 4 + 5 = 3(16) + 1 = 48 + 1 = 49

Then

2f(- 3) - f(4) = 2(41) - 49 = 82 - 49 = 33

Line 1  4x³ + 2x² − 6x +3 is having error.

Line 1.

4x³ + 2x² − 6x +3

Line 2.

= 2x²(2x + 1) − 3(2x + 1)

= 4x³ + 2x² - 6x - 3

Line 3.

= 2x²(2x + 1) + ( − 3)(2x + 1)

= 4x³ + 2x² - 6x - 3

Line 4.

= (2x² − 3)(2x + 1)

= 4x³ + 2x² - 6x - 3

expect line 1 remaining all three are equal.

Therefore Line 1  4x³ + 2x² − 6x +3 is having error.

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The number of cans that are recycled is approximately 1 billion cans in 1 week.

Multiplication is the mathematical operation that is used to determine the product of two or more numbers. If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.

Given that  51.9 billion aluminum cans were recycled.

Therefore, the rate of recycled the can is;

51.9 billion / 1 year

We know that 1 year has 52 weeks

Then

51.9 billion / 1 year  x  1 year / 52 weeks

= 51.9 (1) billion / 1 (52 ) weeks

= 51.9 billion/ 52 weeks

= 1 billion/ 1 week

Hence, the number of cans that are recycled is approximately 1 billion cans in 1 week.

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