Evaluate the following expression
6^-3

The identity 1/csc x+1 - 1/csc x-1 = [tex]-2tan^2 x[/tex] is verified and correct.

To verify the identity:

[tex]\\\frac{1}{csc x+1} - \frac{1}{csc x-1} = -2tan^2 x[/tex]

Starting with the reciprocal identity, we can say:

csc x = [tex]\frac{1}{sin x}[/tex]

So we have:

[tex]1/(1/sin x + 1) - 1/(1/sin x - 1) = -2tan^2 x[/tex]

We need to identify a common denominator in order to simplify the left side of the equation. The common denominator is:

[tex](1/sin x + 1)(1/sin x - 1) = (1 - sin x)/(sin x)^2[/tex]

As a result, we can change the left side of the equation to read:

[tex][(1 - sin x)/(sin x)^2] [(sin x - 1)/(sin x + 1)] - [(1 - sin x)/(sin x)^2] [(sin x + 1)/(sin x - 1)][/tex]

Simplifying this expression by multiplying the numerators and denominators, we get:

[tex](1 - sin x)(sin x - 1) - (1 - sin x)(sin x + 1) / (sin x + 1)(sin x - 1)(sin x)^2[/tex]

Expanding the brackets and simplifying, we get:

[tex]-(2sin^2 x - 2sin x) / (sin x + 1)(sin x - 1)(sin x)^2[/tex]

Factor out -2sin x from the numerator:

[tex]-2sin x(sin x - 1) / (sin x + 1)(sin x - 1)(sin x)^2[/tex]

Simplifying, we get:

[tex]-2sin x / (sin x + 1)(sin x)^2[/tex]

Now, we can use the identity:

[tex]tan^2 x = sec^2 x - 1 = (1/cos^2 x) - 1 = sin^2 x / (1 - sin^2 x)[/tex]

Simplifying, we get:

[tex]sin^2 x = tan^2 x (1 - tan^2 x)[/tex]

When we add this to the initial equation, we obtain:

[tex]-2sin x / (sin x + 1)(sin x)^2 = -2tan^2 x(sin x)/(sin x + 1)[/tex]

Now, we can use the identity:

sin x / (sin x + 1) = 1 - 1/(sin x + 1)

Simplifying, we get:

[tex]-2tan^2 x(sin x)/(sin x + 1) = -2tan^2 x + 2tan^2 x / (sin x + 1)[/tex]

When we add this to the initial equation, we obtain:[tex]-2tan^2 x + 2tan^2 x / (sin x + 1) = -2tan^2 x[/tex]

Simplifying, we get:

[tex]-2tan^2 x = -2tan^2 x[/tex]

Therefore, the identity is verified.

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Answer:

(4x - 5) years old

Step-by-step explanation:

Nick = x years old

Matt = (3 x X)

=3x years old

Oscar = (x - 5) years old

total age = x + 3x + (x - 5)

= (4x - 5) years old

According to the quadrilaterals, the proof perpendicularity and congruence are stated below.

Proof #7:

Statement | Reasons

XY | ZW | Given

XW bisects ZY | Given

ZR ≅ RY | Definition of segment bisector

∠XRY ≅ ∠RW | Alternate interior angles theorem

ΔXRY ≅ ΔWRZ | AAS ≅ theorem

∠XYR ≅ ∠WZR | Definition of segment bisector and corresponding parts of congruent triangles

Proof #8:

Statements | Reasons

EF ≅ HL | Given

∠PER ≅ ∠PHE | Given

∠EPF and ∠HPL are right angles | Definition of perpendicular lines

EP ≅ PH | Definition of perpendicular bisector

AEFP ≅ AHLP | SAS ≅ theorem

ΔEFP ≅ ΔHLP | Base angles converse theorem

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Image transcribed:

A Proof #7

Given: XY || ZW

XW bisects ZY

Prove: ΔXRY ≅ ΔWRZ

Statements | Reasons

1.                | 1.

2.                | 2.

3.                | 3.

4.                | 4.

5.                | 5.

6.                | 6.

XY || ZW,   Alternate Int. ∠ Theorem,   AAS ≅ Theorem

∠XRY ≅ ∠RW,    Def. of Segment Bisector,   ZR ≅ RY

ΔXRY ≅ ΔWRZ,    ∠XYR ≅ ∠WZR,    XW bisects ZY

ASA ≅ Theorem,   Given,     Vertical Angles ≅ Theorem

A Proof # 8

Given: EL ⊥ FH, ∠PEH ≅ ∠PHE

EF ≅ HL

Prove: ΔEFP ≅ ΔHLP

Statements | Reasons

1.                | 1.

2.                | 2.

3.                | 3.

4.                | 4.

5.                | 5.

6.                | 6.

EF ≅ HL,    AEFP ≅ AHLP,    EP ≅ PH,    ∠PER ≅ ∠PHE

HL ≅ Theorem,      SAS ≅ Theorem,      Base Angles Converse Theorem

Definition of ⊥ Lines,     ∠EPF and ∠HPL are right angles

EL ⊥ FH,        Given

a) The percentage of residents who liked the local parks out of those surveyed is 30%.

b) The percentage of the residents who liked the school system out of those surveyed is 60%.

The percentage refers to a portion of a whole value or quantity, expressed in percentage terms.

The percentage is a ratio, which compares a value of interest with the whole, and is computed by multiplying the quotient of the division operation between the particular value and the whole value by 100.

The total number of Plana residents surveyed = 240

The number of residents who responded that they liked the local parks = 72

The percentage of residents who liked the local parks = 30% (72 ÷ 240 x 100)

The number of residents who responded that they liked the school system = 144

The percentage of residents who liked the school system = 60% (144 ÷ 240 x 100).

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As a result, the company produced number of rods are **56** rods.

What exactly is radius?

A radius is a section of a straight line in geometry that connects a circle's or sphere's center to its edge or surface Its length is divided by the circle's diameter by two.

Each rod has a diameter of 6 centimeters, hence the radius is 3 centimeters. Each rod has a 20-centimeter height.

The following formula can be used to determine a cylinder's volume:

V = πr²h

where V stands for volume, r for radius, which is equal to half of the diameter, and h for height.

Adding these values to the formula yields the following results:

V = π(3)²(20) (20)

V = 180π

The volume of each rod is therefore 180 cubic centimeters.

30520.8 cubic centimeters of steel were used to make how many rods?

To find out,

divide the total volume of steel by the volume of each rod:

30520.8 / (180π) ≈ **56** rods

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Answer:

To find the slope of a line perpendicular to another line, we need to first find the slope of the given line. The equation of the given line is x + y = 3. We can rewrite this equation in slope-intercept form (y = mx + b) by solving for y: y = -x + 3 So the slope of the given line is -1. To find the slope of a line perpendicular to this line, we know that it will have a slope that is the negative reciprocal of -1, which is 1. Therefore, the slope of a line perpendicular to the line whose equation is x + y = 3 is 1.

Answer: 1

Step-by-step explanation:

The given equation x + y = 3 can be rearranged to slope-intercept form, which is y = -x + 3.

To find the slope of this line, we can see that the coefficient of x is -1. Therefore, the slope of the line is -1.

To find the slope of a line perpendicular to this line, we need to take the negative reciprocal of the slope of the given line.

The negative reciprocal of -1 is 1/1 or simply 1. Therefore, the slope of a line perpendicular to the line x + y = 3 is 1.

The pοlynοmial functiοn with a fifth degree, 2 as a zerο οf multiplicity 4, −3 as the οnly οther zerο, and a leading cοefficient οf 2 is:

[tex]f(x) = 2(x-2)^4(x+3) = 2x^5 - 32x^4 + 192x^3 - 384x^2 + 216x + 1080[/tex]

What is Polynomial Function?

A pοlynοmial functiοn is a mathematical functiοn οf the fοrm [tex]f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0[/tex], where the coefficients[tex]a_n, a_{n-1}, ..., a_1, a_0[/tex] are constants, x is the variable, and n is a non-negative integer. It is a functiοn that can be graphed as a smοοth curve with nο breaks οr jumps.

If 2 is a zerο οf multiplicity 4, then the pοlynοmial functiοn must have the factοr[tex](x-2)^4[/tex].

If −3 is the οnly οther zerο, then the pοlynοmial functiοn must alsο have the factοr (x+3).

The pοlynοmial functiοn with these properties and a leading cοefficient οf 2 can be written as:

[tex]f(x) = 2(x-2)^4(x+3)[/tex]

Expanding this polynomial gives:

[tex]f(x) = 2x^5 - 32x^4 + 192x^3 - 384x^2 + 216x + 1080[/tex]

Therefore, the polynomial function with a fifth-degree, 2 as a zero of multiplicity 4, −3 as the only other zero, and a leading coefficient of 2 is:

[tex]f(x) = 2(x-2)^4(x+3) = 2x^5 - 32x^4 + 192x^3 - 384x^2 + 216x + 1080[/tex]

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Therefore, the distance between the pilot and the home is roughly 5,348.2 feet.

The height of the balloon divided by the neighboring side yields the tangent of the angle of depression (the distance from the balloon to the house).

Tan(32°) is therefore equal to 2828/x,

where x is the distance between the balloon and the home.

The answer to the x equation is

x = 2828/tan(32°)

Value of tan 32° = 0.6610060414

x=  5,348.2 feet.

The height of the balloon divided by the neighboring side yields the tangent of the angle of depression (the distance from the balloon to the house).

Tan(45°) is therefore equal to 2828/x, where x is the distance between the balloon and the home.

The answer to the x-problem is x = 2828/tan(45°) 2828 feet.

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After plotting the points A (-5, -2), B (-3, -2), and D (-5, -4) in the 4th quadrant as shown in the attached graph, we can see that to make the square we have to plot point C, at (-3, -4).

Simply said, the graph is a structured representation of the data. Because of this, it is simpler for us to understand the facts. Data is the term used to describe the numerical information obtained through observation.

The word data is derived from the Latin word datum, which meaning "something delivered."

After developing a research question, data are progressively obtained through observation. The information is then organised, categorised, and visually shown.

Here in the question,

After plotting the points A (-5, -2), B (-3, -2), and D (-5, -4) in the 4th quadrant as shown in the attached graph, we can see that to make the square we have to plot point C, at (-3, -4).

And the coordinate rule that will rotate the rectangle to the fourth quadrant, with vertex C’ at point (4, -3) is known as reflection.

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Answer:

14°

Step-by-step explanation:

Let a be the angle of depression.

Set your calculator to degree mode.

[tex]tan(a) = \frac{2000}{8020} [/tex]

[tex] a = {tan}^{ - 1} \frac{2000}{8020} = 14[/tex]

So a = 14°

Answer: 14°

Step-by-step explanation:

tan(x) = 8020/2000

x = tan^-1 (8020/2000)

x = 75.99 ≈ 76°

Angle of Dep = 90 - 76

Angle of Dep = 14°

The height of the cliff diver after 1.25 seconds is 35 feet.  

The concept used is evaluating a function at a given input or value. In this case, we are evaluating the height function H(t) at t = 1.25 to find the height of the cliff diver after 1.25 seconds. The formula for the height function is given as H(t) = -16t^2 + 32t + 20.

Here we have

A cliff diver's height (in feet) after t seconds is given by the model

H(t)=-16t² +32t +20  

Given time t = 1.25 seconds

To find the height after 1.25 seconds, we simply need to evaluate H(1.25) using the given formula:

H(1.25) = -16(1.25)² + 32(1.25) + 20

On Simplifying the expression we get:

H(1.25) = -16(1.5625) + 40 + 20

H(1.25) = -25 + 60

H(1.25) = 35

Therefore,

The height of the cliff diver after 1.25 seconds is 35 feet.  

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Answer: Let the total number of students in the class be x.

Then the number of boys in the class is 60% of x, or 0.6x.

And the number of girls in the class is 16.

We can write an equation based on the information given:

0.6x + 16 = x

Solving for x:

0.4x = 16

x = 40

Therefore, there are 0.6x = 24 boys in the class.

Step-by-step explanation:

Answer:

The circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9) is (-4, -8).

Step-by-step explanation:

To find the coordinates of the circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9), we can use the following steps:

Step 1: Find the midpoint of two sides

We first find the midpoint of two sides of the triangle. Let's take sides AB and BC:

Midpoint of AB: ((-7 + (-1))/2, (-1 + (-1))/2) = (-4, -1)

Midpoint of BC: ((-1 + (-7))/2, (-1 + (-9))/2) = (-4, -5)

Step 2: Find the slope of two sides

Next, we find the slope of the two sides AB and BC:

Slope of AB: (-1 - (-1))/(-1 - (-7)) = 0/6 = 0

Slope of BC: (-9 - (-1))/(-7 - (-1)) = -8/(-6) = 4/3

Step 3: Find the perpendicular bisectors of two sides

We can now find the equations of the perpendicular bisectors of the two sides AB and BC. Since the slope of the perpendicular bisector is the negative reciprocal of the slope of the side, we have:

Equation of perpendicular bisector of AB:

y - (-1) = (1/0)[x - (-4)]

x = -4

Equation of perpendicular bisector of BC:

y - (-5) = (-3/4)[x - (-4)]

y + 5 = (-3/4)x - 3

y = (-3/4)x - 8

Step 4: Find the intersection of perpendicular bisectors

We now find the point of intersection of the two perpendicular bisectors. Solving for x and y from the two equations, we get:

(-4, -8)

Therefore, the circumcenter of the triangle with vertices (-7, -1), (-1, -1), and (-7, -9) is (-4, -8).

The function f(x) will be after shifting is g(x)= (x – 1)² + 1

The function f is used to move the graph of a function f(x) to the right by h units (x-h). This is because when we substitute x with x-h in f(x), we obtain f(x-h), meaning that we are substituting x with x+h in the initial function f. (x). Hence, if we wish to move the graph of f(x) = (x + 2)² + 1 three units to the right, we may do it by using the function f(x-3) to move the graph three units to the right.

g(x)=(x-3+2)²+1

g(x)= (x – 1)² + 1

The complete question is

Consider the function f(x) = (x + 2)2 + 1. Which of the following functions shifts the graph of f(x) to the right three units?

g(x) = (x + 5)2 + 1

g(x) = (x + 2)2 + 3

g(x) = (x – 1)2 + 1

g(x) = (x + 2)2 – 2

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Answer:

Step-by-step explanation:

Reflecting a point across the x-axis negates the y-coordinate, so the point (-5, 7) reflected across the x-axis becomes (-5, -7).

Reflecting a point across the y-axis negates the x-coordinate, so the point (-5, -7) reflected across the y-axis becomes (5, -7).

Therefore, the third point is (5, -7).

Based on the given information, Yelena jumped the highest with a jump of 74 inches (6.17 feet).

Measurement is the process of assigning a numerical value to a physical quantity, such as length, mass, time, temperature, or volume. It is a fundamental aspect of science, engineering, and everyday life. In order to measure something, we need a unit of measurement, which is a standard reference quantity that is used to express the measurement. For example, meters or feet are commonly used units for length, while grams or pounds are used for mass.

To compare the high jumps of the four athletes, we need to convert all the measurements to the same unit.

Anna: 2.1 yards = 6.3 feet

Javier: 72 inches = 6 feet

Charles: 5 feet 11 inches = 71 inches = 5.92 feet

Yelena: 74 inches = 6.17 feet

So, Yelena jumped the highest with a jump of 74 inches (6.17 feet).

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Answer:

that's the slope of the y intercept

Answer:

27 million dollars

Step-by-step explanation:

Let x be the total amount of the charity's donated funds. We know that the donation of 15.2 million represents 56% of x.

We can set up the following equation to solve for x:

15.2 million = 0.56x

Dividing both sides by 0.56, we get:

x = 15.2 million / 0.56 = 27.14 million

Therefore, the total amount of the charity's donated funds is approximately 27 million dollars.

Answer:

27 million

Step-by-step explanation:

create the equation

[tex]\frac{15200000}{x} =\frac{56}{100}[/tex], where x is equivalent to the total amount of donation funds

solve the equation by cross multiplying

[tex]x=2714285 \frac{5}{7}[/tex],

this can be simplified to 27 million

Answer:

D

Step-by-step explanation:

-6 is constant because it has a degree of 0

3x is linear because it has a degree of 1

[tex]4x^{2}[/tex] is quadratic because it has a degree of 2

So the answer is D

The expected value for the amount of money made selling all of the candy in one bag is $15, and the standard deviation is approximately $24.33.

The standard deviation is a measurement of how widely apart a set of numbers or statistics are from their mean.

The expected value for the amount of money made selling all of the candy in one bag can be found by;

Expected value = mean number of candy bars per bag x price per candy bar

Expected value = 12 x $1.25 = $15

Formula for the standard deviation of a product of random variables:

[tex]SD (XY) = \sqrt{((SD(X)^2)(E(Y^2)) + (SD(Y)^2)(E(X^2)) + 2(Cov(X,Y))(E(X))(E(Y)))}[/tex]

where X and Y are random variables, SD is the standard deviation, and Cov is the covariance.

X is the number of candy bars in a bag, which has a mean of 12 and a standard deviation of 2. Y is the price per candy bar, which is a constant $1.25. So we have:

E(Y²) = $1.25² = $1.5625

E(X²) = (SD(X)²) + (E(X)²) = 2² + 12² = 148

Cov (X,Y) = 0 (because X and Y are independent)

Using these values, we can calculate the standard deviation for the amount of money made selling all of the candy in one bag:

[tex]SD = sqrt((2^{2} )(148) + (0)(12)(1.25)^{2} + 2(0)(2)(12)(1.25))[/tex]

SD = √(592)

SD ≈ $24.33

Therefore, the expected value for the amount of money made selling all of the candy in one bag is $15, and the standard deviation is approximately $24.33.

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Buzz earned $65 - $55 = $10 more than Woody for his pink bear.

How to calculate earnings?

Woody sold his bear for $50 + 10% of $50 = $55.

Buzz sold his bear for $50 + 30% of $50 = $65.

Therefore, Buzz earned $65 - $55 = $10 more than Woody for his pink bear.

To determine how much more money Buzz earned than Woody for their pink bears, we first needed to calculate the selling price for each of them.

We know that both of them bought their bears for $50. Woody sold his bear for 10% more, which means his selling price was $50 + 10% of $50 = $55. Buzz sold his bear for 30% more, which means his selling price was $50 + 30% of $50 = $65. Therefore, Buzz earned $65 - $55 = $10 more than Woody for his pink bear.

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Answer:

To calculate the monthly payment for each option, we can use the loan formula:

Payment = (P * r) / (1 - (1 + r)^(-n))

where P is the principal amount, r is the monthly interest rate, and n is the total number of payments.

For Option A, the principal amount is $60,000, the interest rate is 4% per year, and the loan term is 10 years. We first need to convert the annual interest rate to a monthly interest rate:

r = 4% / 12 = 0.00333333 (rounded to 8 decimal places)

n = 10 years * 12 months/year = 120 months

Using the loan formula, we get:

Payment = (60000 * 0.00333333) / (1 - (1 + 0.00333333)^(-120)) = $630.55

Therefore, the monthly payment for Option A is $630.55.

For Option B, the principal amount is also $60,000, the interest rate is 3% per year, and the loan term is 20 years. We convert the annual interest rate to a monthly interest rate:

r = 3% / 12 = 0.0025 (rounded to 4 decimal places)

n = 20 years * 12 months/year = 240 months

Using the loan formula, we get:

Payment = (60000 * 0.0025) / (1 - (1 + 0.0025)^(-240)) = $342.61

Therefore, the monthly payment for Option B is $342.61.

To calculate the total amount paid over the life of the loan for each option, we simply multiply the monthly payment by the total number of payments:

For Option A, the total amount paid = $630.55 * 120 months = $75,665.92

For Option B, the total amount paid = $342.61 * 240 months = $82,226.40

To calculate the total interest paid over the life of the loan for each option, we subtract the principal amount from the total amount paid:

For Option A, the total interest paid = $75,665.92 - $60,000 = $15,665.92

For Option B, the total interest paid = $82,226.40 - $60,000 = $22,226.40

Therefore, Option A has a lower monthly payment and total amount paid over the life of the loan, but Option B has a longer loan term and a lower interest rate, resulting in a higher total interest paid over the life of the loan

Answer:

Certainly, here's an illustration:

       ----------------------

       |                    |

 2cm   |                    |

       |                    |  12cm

       |                    |

       ----------------------

           <---- ¾ of 12cm --->

             (9cm - 2cm)

               = 7cm

In this illustration, the rectangle is represented by a box with a length of 12cm and a width of 7cm. The width is calculated by taking ¾ of the length (which is 9cm) and subtracting 2cm from it. The dimensions of the rectangle are shown inside the box, with the length indicated by the vertical line and the width indicated by the horizontal line.

The fourth root of -81 is not a real number is 3√i.

A number that can be represented in the form a + bi is an imaginary number. In this case, a and b are real numbers, while I is the imaginary unit, which is equal to the square root of -1. In mathematics, imaginary numbers are used to expand on the real number system and to describe values that cannot be stated using real numbers.

Complex numbers, which are numbers of the type a + bi, where a and b are real numbers, are one of the principal applications for imaginary numbers. Several mathematical and scientific disciplines, such as electrical engineering, signal processing, and quantum physics, require complex numbers.

The value of -81 can be written as i²(3)⁴.

Taking the fourth root of the number we get 3√.

Hence, the fourth root of -81 is not a real number is 3√i.

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The solution to the problem is that the number "x" must be less than or equal to -3.9 in order for the sum of "a number times 10 and 20" to be at most -19.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can contain variables, constants, and operators (such as addition, subtraction, multiplication, and division) that are used to represent quantities and their relationships.

Let's use algebra to solve this problem.

Let's call the number we're trying to find "x".

The sum of "a number times 10 and 20" can be written as "10x + 20".

So, we can translate the statement "the sum of a number times 10 and 20 is at most -19" into an equation:

10x + 20 ≤ -19

Now we can solve for x:

10x + 20 ≤ -19

Subtract 20 from both sides:

10x ≤ -39

Divide both sides by 10:

x ≤ -3.9

Therefore, the solution to the problem is that the number "x" must be less than or equal to -3.9 in order for the sum of "a number times 10 and 20" to be at most -19.

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After answering the presented question, we can conclude that  area of Circle M, [tex]C = 2\pi [(17/\pi )^0.5] km\\[/tex]

A circle appears to be a two-dimensional component that is defined as the collection of all places in a jet that are equidistant from the hub. A circle is typically depicted with a capital "O" for the centre and a lower portion "r" for the radius, which represents the distance from the origin to any point on the circle. The formula 2r gives the girth (the distance from the centre of the circle), where (pi) is a proportionality constant about equal to 3.14159. The formula r2 computes the circumference of a circle, which relates to the amount of space inside the circle.

area of Circle M,

[tex]289 = \PI(x + 6)^2\\289/\PI = (x + 6)^2\\\sqrt(289/\pi ) = x + 6\\(17/\pi )^0.5 - 6 = x\\x = (17/\pi )^0.5 - 6 km\\C = \pi (2x + 12) km\\C = 2\pi (x + 6) km\\[/tex]

[tex]C = 2\pi [(17/\pi )^0.5] km\\[/tex]

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Answer:

B is greater than A by 1/4 of A

Step-by-step explanation:

Let's use the information given in the problem to write expressions for the values of a and b:

a = b - (1/5)b = (4/5)b

b is greater than a by the difference:

b - a = b - (4/5)b = (1/5)b

To express this difference as a fraction of a, we divide by a:

(b - a)/a = ((1/5)b)/((4/5)b) = 1/4

Therefore, b is greater than a by 1/4 of a.

About 25% of students ride the bus for less than 13 minutes.

About 75% of students ride the bus for less than 27 minutes.

Median is a measure of central tendency that represents the middle value in a dataset when the values are arranged in order of magnitude.

In this case, the median is 21 minutes. This means that half of the students ride the bus for less than 21 minutes and half of the students ride the bus for more than 21 minutes.

The lower quartile (Q1) is the value that separates the lowest 25% of the data from the other 75%. In this case, the lower quartile is 13 minutes. This means that 25% of the students ride the bus for less than 13 minutes and 75% ride the bus for more than 13 minutes.

The upper quartile (Q3) is the value that separates the highest 25% of the data from the other 75%. In this case, the upper quartile is 27 minutes. This means that 75% of the students ride the bus for less than 27 minutes and 25% ride the bus for more than 27 minutes.

So, to answer the statement about the data set:

About 25% of students ride the bus for less than 13 minutes. (this is because the lower quartile separates the lowest 25% of the data from the other 75%)

About 75% of students ride the bus for less than 27 minutes. (this is because the upper quartile separates the highest 25% of the data from the other 75%)

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Quotient ➔ 2x² - 7x + 18

Remainder ➔ - 37

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SolutioN ::

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[tex]\begin{gathered} \\ \\ \qquad{\rule{120pt}{7pt}} \\ \\ \end{gathered}[/tex]

VerificatioN ::

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[tex]\begin{gathered} \\ \\ \; \; :\longmapsto \; \sf {x(2x^{2} - 7x + 18) + 2(2x^{2} - 7x + 18) + ( - 37)} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \\ \; \; :\longmapsto \; \sf {2x^{3} - 7x^{2} + 18x + 4x^{2} - 14x + 36 - 37} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \\ \; \; :\longmapsto \; \sf {2x^{3} - 7x^{2} + 4x^{2} + 18x - 14x - 1} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \\ \; \; :\longmapsto \; \sf \pink{2x^{3} - 3x^{2} + 4x - 1} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \\ \\ \; \; :\longmapsto \; \sf \pink{2x^{3} - 3x^{2} + 4x - 1} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \\ \; \; :\longmapsto \; \sf {LHS = RHS} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \\ \; \; :\longmapsto \; \underline{\boxed{\sf{Verified}}} \; \pmb{\red{\bigstar}} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \\ {\underline{\rule{150pt}{10pt}}} \end{gathered}[/tex]

[tex]\blue{\huge {\mathrm{DIVIDING \; POLYNOMIALS}}}[/tex]

[tex]\\[/tex]

[tex]{===========================================}[/tex]

[tex]{\underline{\huge \mathbb{Q} {\large \mathrm {UESTION : }}}}[/tex]

[tex]{===========================================}[/tex]

[tex] {\underline{\huge \mathbb{A} {\large \mathrm {NSWER : }}}} [/tex]

[tex]{===========================================}[/tex]

[tex]{\underline{\huge \mathbb{S} {\large \mathrm {OLUTION : }}}}[/tex]

[tex]\qquad\begin{aligned}\sf \blue{2x^2 - 7x + 18} \:\:\:\:\:\:\:\:\:\: &\\\sf x + 2 \: \: ) \overline{2x^3 - 3x^2+4x-1} \quad&\\\sf \: \: \underline{ - 2x^3 - 4x^2 \qquad\qquad \: \: \: \: } \\\sf - 7x^2 + 4x \qquad \: \: \: \\\sf \:\:\: \underline{ - 7x^2 + 14x \qquad \: } \\ \sf 18x - 1 \: \: \: \\ \underline{ \sf{ - 18x - 36 \: }} \\ \sf \red{ - 37} \end{aligned}[/tex]

[tex]{===========================================}[/tex]

[tex]- \large\sf\copyright \: \large\tt{AriesLaveau}\large\qquad\qquad\qquad\qquad\qquad\qquad\tt 04/02/2023[/tex]

The domain of fog is: Domain of fog = {x ∈ R | 3 ≤ g(x) ≤ 9}

The range of fog is:  Range of fog = {1, 2, 5}

The domain of a function is the set of all possible input values (usually denoted by x) for which the function is defined.

The range of a function is the set of all possible output values (usually denoted by y) that the function can produce for its corresponding inputs in the domain.

(a) Domain of fog:

The domain of fog is the set of all inputs for which the composition is defined. Since g is defined for all values in its domain, and f is defined for all values in the range of g, the domain of fog is the set of all values in the domain of g for which g(x) is in the domain of f.

(b) Range of fog:

The range of fog is the set of all possible outputs of the composition. Since g is defined for all values in its domain, and f is defined for all values in the range of g, the range of fog is the set of all possible outputs of f when its input is an output of g.

Range of fog = {f(g(x)) | x ∈ R, 3 ≤ g(x) ≤ 9}

To determine the values in the range of fog, we need to evaluate f(g(x)) for each x in the domain of fog. We can do this by first determining the outputs of g for each value in its domain, and then evaluating f at those outputs.

The outputs of g for x = 4, 5, 7 are:

g(4) = 6

g(5) = 8

g(7) = 9

Since f is defined for all values in the range of g, we can evaluate f at each of these outputs to get:

f(g(4)) = f(6) = 5

f(g(5)) = f(8) = 2

f(g(7)) = f(9) = 1

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