Two parallel lines are cut by a transversal as shown below.
Suppose m4=124°.

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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The equation of line d that is perpendicular to line c is: y = 7/3x - 2/3.

Perpendicular lines have slope values, whose product is equal to -1, that is, they are negative reciprocals to each other. An equation in slope-intercept form is often written for a line as y = mx + b, where m = slope and b = y-intercept.

The slope of y - 3/7x + 4 is -3/7. The negative reciprocal of -3/7 is 7/3. Therefore, the slope (m) of the perpendicular line d is 7/3.

Substitute m = 7/3 and (x, y) = (2, 4) into y = mx + b to find the value of b (y-intercept of the line):

4 = 7/3(2) + b

4 = 14/3 + b

-14/3 + 4 = b

(-14 + 12)/3 = b

-2/3 = b

b = -2/3.

To write the equation of line d, substitute m = 7/3 and b = -2/3 into y = mx + b:

y = 7/3x - 2/3

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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 length of the shortest ladder that will reach from the ground over the fence to the wall of the building is 11.31 feet

Let L stand for the ladder's overall length.

Right angled triangle ΔAGB contains the following:

L² = h² +(x+4)²

( Since by using Pythagorean Theorem)

Triangles AGB and CDB also have similarities.

As a result, the corresponding sides' ratio is equal.

Hence, we have:

h/4 = (x+4)/x

h = 4(x+4)/x

As a result, when we enter the value of h in equation (1), we obtain:

L² = h² +(x+4)²

L² = (4(x+4)/x)² +(x+4)²

L² = (16(x+4)²/x²)+(x+4)²

L² = (x+4)²(16/x² + 1)

We now have to reduce L.

Therefore, we employ the differentiation method.

We distinguish in relation to x as follows:

L² = (x+4)²(16/x² + 1)

2L dL/dx =2(x+4)(16/x² + 1) + (x+4)² (-32/x³)

2L dL/dx =2(x+4){ (16/x² + 1) + (x+4) (-16/x³)

2L dL/dx =2(x+4){ (16/x² + 1 -16/x²-64/x³)

2L dL/dx =2(x+4){ (1-64/x³)

when the derivative is zero we have:

2(x+4){ (1-64/x³) = 0

x= -4 and x³ = 64 and x =∛64 = 4

But x can't be negative.

Hence, we have:

x = 4

After solving equation (2) with this value of x, we get the following result:

L² = (x+4)²(16/x² + 1)

L² = 128

L = 11.313 feet

Hence,

L = 11.313 feet

which on rounding to two decimal places is: 11.31 feet

As a result, the shortest ladder's length, measured from the ground to the building's wall over the fence, is 11.31 feet.

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Answer:

x ≥ -10

Step-by-step explanation:

Given equation,

→ 5x + 13 ≥ -37

Now the value of x will be,

→ 5x + 13 ≥ -37

→ 5x ≥ -37 - 13

→ x ≥ -50/5

→ [ x ≥ -10 ]

Hence, the value of x is -10.

y = 0.1x + 1 is the linear function for pressure based on the depth in meters

Given that:  At 30 meters in depth the water pressure measures 4 (atm)

At 50 meters in depth under water the pressure is 6 (atm)

We will consider the two given data values as our points 1 and 2 respectively. And that is what we are going to use to create a linear function for pressure

Let the y and x represent the pressure and depth respectively

point 1 (30, 4) and point 2 (50, 6)

slope(m) = y2-y1 / x2-x1

where  x1 =30, y1 = 4 and x2 = 50, y2 = 6

slope(m) = 6-4 / 50-30 = 2/20 = 1/10

Linear function are of the form y-y1 = m(x-x1). Thus:

y-4 = 1/10 (x-30)

y-4 = 1/10(x) - 3

y = 1/10(x) + 1 = 0.1x + 1

Therefore, the linear function for pressure based on the depth in meters is y = 0.1x + 1

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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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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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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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Answer:

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

Answer:

Difference = 6 inches

Step-by-step explanation:

Given information,

→ Meg measured the length of some pieces of the wire.

Now we have to,

→ find the difference between the longest and shortest piece of wire.

Then the difference will be,

→ Longest wire - Shortest wire

→ 9 - 3

→ 6 inches

Hence, the difference is 6 inches.

Answer:

5

Step-by-step explanation:

Answer: 1.8 movies per day.

Step-by-step explanation: 8 1/3=25/3. One day would be 3/25. This means that each day, a college students watched 3/25 of 8 movies. Do 3/25x15. This makes 1.8 movies watched per day.

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:

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

Answer:

  5 5/6 yards

Step-by-step explanation:

You want the perimeter of a rug that is 2 yards by 11/12 yards.

The formula for the perimeter of a rectangle is ...

  P = 2(L+W)

Using the given values, we find the perimeter to be ...

  P = 2(2 yd + 11/12 yd) = 4 22/12 yd = 5 5/6 yd

The perimeter of the rug is 5 5/6 yards.

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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Answer:

9x + 16

Step-by-step explanation:

Let's simplify step-by-step.

5(x+2)−3(x−2)+7x

Distribute:

=(5)(x)+(5)(2)+(−3)(x)+(−3)(−2)+7x

=5x+10+−3x+6+7x

Combine Like Terms:

=5x+10+−3x+6+7x

=(5x+−3x+7x)+(10+6)

=9x+16

Please give me brainiest!

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

a) a/5 + 3 = 8

Step 1 : Minus both sides by 3

a/5 + 3 - 3 = 8 - 3 -> a/5 = 5

Step 2 : Multiply both sides by 5

a/5 x 5 = 5 x 5 -> a = 25.

b) (3b)/7 - 1 = 5

Step 1 : Add both sides by 1

(3b)/y - 1 +1 = 5 + 1 -> (3b)/7 = 6

Step 2 : Multiply both sides by 7

(3b)/7 x 7 = 6 x 7 -> 3b = 42

Step 3 : Divide both sides by 3

3b : 3 = 42 : 3 -> b = 14.

Answer:

7.2461538

Step-by-step explanation:

Answer: 7.24615384615

Step-by-step explanation: I had a test that had this question.

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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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

The table of solutions should be completed based on the given equation (y = 2x²) as follows:

x                    y_

-2                  8

-1                   2

0                   0

1                    2

2                   8

Also, a graph of the solution to the given equation is shown in the image attached below.

In Mathematics, a graph can be defined as a type of chart that is commonly used for the graphical representation of data points on both the horizontal and vertical lines of a cartesian coordinate, which are the x-coordinate and y-coordinate.

Next, we would complete the table based on the given equation as follows:

When x = -2, the value of point y is given by:

y = 2x²

y = 2(-2)²

y = 2 × 4

y = 8

When x = -1, the value of point y is given by:

y = 2x²

y = 2(-1)²

y = 2 × 1

y = 2

When x = 0, the value of point y is given by:

y = 2x²

y = 2(0)²

y = 2 × 0

y = 0

When x = 1, the value of point y is given by:

y = 2x²

y = 2(1)²

y = 2 × 1

y = 2

When x = 2, the value of point y is given by:

y = 2x²

y = 2(2)²

y = 2 × 4

y = 8

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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:

Lol

Step-by-step explanation:

k

Answer: x=-1, y=2, z=-3

Step-by-step explanation:

[1]    2x + 3y - 4z = 16

  [2]    -x + 2y - z = 8

  [3]    2x - y - 2z = 2

Solve by Substitution :

Solve equation [3] for the variable  y

 [3]    y = 2x - 2z - 2

Plug this in for variable  y  in equation [1]

    2x + 3•(2x-2z-2) - 4z = 16

    8x - 10z = 22

Plug this in for variable  y  in equation [2]

    -x + 2•(2x-2z-2) - z = 8

      3x - 5z = 12

Solve equation [2] for the variable  x

 3x = 5z + 12

     x = 5z/3 + 4

Plug this in for variable  x  in equation [1]

   8•(5z/3+4) - 10z = 22

    10z/3 = -10

    10z = -30

Solve equation for the variable  z

    10z = - 30

   z = - 3

By now we know this much :

   x = 5z/3+4

   y = 2x-2z-2

   z = -3

Use the  z  value to solve for  x

   x = (5/3)(-3)+4 = -1

Use the  x  and  z  values to solve for  y

 y = 2(-1)-2(-3)-2 = 2

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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The true statement about the change of scale is 1 cm represented 5 m in the first scale, but now 1 cm represents 4 m in the second scale.

A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).

Given that, Tran planned a rectangular pool and made a scale drawing using cm as the unit of measurement. He originally planned for the length of the pool to be 40 m but decided to change it to 32 m. If the length of the pool in his scale drawing is 8 cm,

The scale factor of drawing to original pool was 1/5 before changing the length but when length changed to 32 m scale factor now is 1/4

Hence, The true statement about the change of scale is 1 cm represented 5 m in the first scale, but now 1 cm represents 4 m in the second scale.

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