Hind the quotient and remainder using synthetic division, (x^(4)-x^(3)+x^(2)-x+2)/(x-3)

Nicole has 8 nickels and 7 dimes.

A system of equations is simply two or more equations that share the same variables.

Let's use a system of equations to solve this problem:

Let x be the number of nickels, and y be the number of dimes.

From the problem, we know that:

x + y = 15 (equation 1) (the total number of coins is 15)

0.05x + 0.1y = 1.1 (equation 2) (the total value of the coins is $1.10)

To solve for x and y, we can use substitution or elimination. Let's use elimination:

Multiply equation 1 by 0.1:

0.1x + 0.1y = 1.5 (equation 3)

Subtract equation 3 from equation 2:

0.05x = 0.4

x = 8

Substitute x = 8 into equation 1:

8 + y = 15

y = 7

Therefore, Nicole has 8 nickels and 7 dimes.

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Solve for the first variable in one of the equations, then substitute the result into the other equation.

Answer:

Point Form:

(−8,−17)

Equation Form: x=−8,y=−17

Answer:

x=-4, y=5

Step-by-step explanation:

Because y=-3x-7 and y=x+9, so y=y, and -3x-7=x+9

-3x-7=x+9,

so 4x=-16

      x=-4

so we put x=-4 to y=-3x-7

      y=(-3)* (-4)-7

      y=12-7

       y=5

Zach still has to save 1/12 of the total amount after this month.

To find out how much of the total amount Zach still has to save after this month, we need to calculate how much he has already saved and how much he will save from this month's paycheck.

First, we know that Zach has already saved 2/3 of the money he will need for his trip. So, if we let x be the total amount he needs for the trip, we can write this as:
Already saved = 2/3 * x

Next, we know that Zach plans to put aside 3/4 of what he has left to save from this month's paycheck. Since he has already saved 2/3 of the total amount, he has 1/3 of the total amount left to save. So, the amount he will save from this month's paycheck is:
Amount from paycheck = 3/4 * (1/3 * x) = 1/4 * x

Now, we can add these two amounts to find out how much Zach has saved in total:
Total saved = Already saved + Amount from paycheck
Total saved = (2/3 * x) + (1/4 * x) = (8/12 * x) + (3/12 * x) = 11/12 * x

Finally, we can subtract this amount from the total amount he needs to find out how much he still has to save:
Amount still to save = Total amount - Total saved
Amount still to save = x - (11/12 * x) = (12/12 * x) - (11/12 * x) = 1/12 * x

So, Zach still has to save 1/12 of the total amount after this month.  

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In response to the query, we have that The two graphs are oriented and coordinates  have the same form in common. The moment where the two graphs come together (1, 2).

A coordinate system is a technique used in geometry to precisely locate points or other geometry objects on a manifold, also including Euclidean space, using one or more numbers or coordinates. A point or item on a two-dimensional sphere is located using a pair of numbers known as coordinates. The x and y positions are two numbers that describe a point's location on a 2D plane. a collection of integers that indicate exact locations. Typically, the figure has two numbers. The first number represents the front-to-back distance, while the second column represents the top-to-bottom distance. When there are 12 units below and 5 above, as in (12.5), this is an example.

f(1/x) = 2/(1/x) = 2x, where g(x) =

Hence, g(x) = f(1/x) is equivalent to f(x) = 2x.

The values of x and g(x) are as follows:

x= 1 2 3 4 5\sg(x)= 1/8 1/4 1/2 1 2

f(x) = 2x: (1, 2), (2, 4), (3, 6), (4, 8), (5, 10) (5, 10)

g(x) = f(1/x) = 2x: (1, 2), (1/2, 1), (1/3, 2/3), (1/4, 1/2), (1/5, 2/5)

The appropriate choices are thus:

A reflection of the graph of f(x) about the line y = x is the graph of g(x).

The two graphs are oriented and have the same form in common.

The moment where the two graphs come together (1, 2).

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The provide situation can be modeled as a linear equation, y = 50 + 20x, where y is total cost and x is number of person who will attend the party. The slope of line tells me that the additional cost per person attending the party is $20. Thus, write option is option (b).

Emmett wants to through a party at an escape room. There are two charges for

escape room, one is booking fee and other one is additional cost per person attending the party. Let the booking fee and additional cost per person who attended the party of escape room be 'a ' and 'b'. Also, consider total person who will attend the party are equal to 'x'. So, total cost = booking cost + additional cost per person × number of people's who will attend the party. Let total cost of escape room for x people be 'y'. Then,

y = a + bx --(1)

it is act as modeled linear relationship, see above graph, which represents it graphically. In the graph initial total cost is 50, when no additional cost occur, that is booking fee equals to $50 or a = $50. After that, number of people increase total cost also increases. When there are 10 people in party then total cost is $250 and the additional cost = $200 for 10 people. So, additional cost per person

= 200/10 = 20 , i.e., b = $20 and equation(1) is rewrite, y = 50 + 20x.

If we compare it with standard linear equation, y = mx + c

where m --> slope of line

c --> y-intercept

then slope of equation (1) is equals to 'b' and b = 20. In this situation, the slope of line tell me that except the booking fee, the additional cost per person who will attend the party is equals to $20. Hence, the required answer is option(b).

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Complete question:

Emmett is planning a party at an escape room. The escape room will charge Emmett a booking fee and an additional cost per person attending the party.

This situation can be modeled as a linear relationship, see above graph. What does the slope of the line tell you about the situation?

A. The booking fee is $50

B. After the booking fee the party will cost $20 per person

C. After the booking fee the party will cost $25 per person

D. Including the bookin fee it would cost emmett $250 total for 10 people to attend.

B. After the booking fee the party will cost $20 per person

Answer: The y-intercept is 30 and the coefficient is -5

Step-by-step explanation:

In the slope-intercept form (y = mx + b) b stands for your y-intercept, in this case, 30.

The coefficient is the number being multiplied by the variable, or in this case, the slope, or -5.

Answer:

Dude.....it's a CIRCLE!!

The distance between the two groups after the hikes is approximately 12.04 miles.

A right triangle's connection between its sides is described by the Pythagorean Theorem, a foundational idea in mathematics. It claims that the hypotenuse's square length, which is the side that faces the right angle, equals the sum of the squares of the lengths of the other two sides of any right triangle. The Pythagorean Theorem is frequently used to solve issues involving distance, velocity, and acceleration and has various applications in the disciplines of geometry, trigonometry, and physics.

The distance between the two groups serves as the hypotenuse of a right triangle, which is formed by the two pathways. The following formula can be used to determine the hypotenuse's length:

Distance = √((2+7)² + (5+3)²)

= √(9² + 8²)

= √(81 + 64)

= √(145)

≈ 12.04 miles

Therefore, the distance between the two groups after the hikes is approximately 12.04 miles.

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Answer:

2(−x 2−3)

Simplify the expression

−2x 2−6

Factor the expression

−2(x 2+3)

The rational expression is undefined when the denominator is equal to zero. This is because division by zero is not allowed in mathematics, and the result would be "undefined."

To find the values for which the rational expression is undefined, we need to set the denominator equal to zero and solve for the variable. In this case, the denominator is z:

z = 0

Therefore, the rational expression is undefined when z = 0.

So, the values for which the rational expression ((8x)/(z)) is undefined are:

z = 0

I hope this helps! Let me know if you have any further questions.

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Dina's older brother has a hot chocolate that is more chocolaty, as his recipe has a higher proportion of cocoa.

The more chocolaty recipe is defined obtain the proportion of cocoa for each recipe.

The proportion of cocoa for each recipe is obtained dividing the amount of cocoa used by the sum of the amounts of cocoa and milk used.

Hence the proportion for Dina's older brother is calculated as follows:

2/(2 + 3) = 2/5 = 0.4 = 40% cocoa.

The proportion for Dina's is obtained as follows:

3/(3 + 5) = 3/8 = 0.375 = 37.5% cocoa.

40 > 37.5, hence Dina's older brother has a hot chocolate that is more chocolaty, as his recipe has a higher proportion of cocoa.

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Answer:

300

Step-by-step explanation:

x(100%-7%)=279

        0.93x=279

                x=300

The set builder nοtatiοn wοuld be {(x, y) | x = 1/125, 1/25, 1/5, 1, 5, 25, 125 and y = -3, -2, -1, 0, 1, 2, 3}

A set builder nοtatiοn is a cοncise way οf representing a set οf elements based οn a rule οr cοnditiοn. In set builder nοtatiοn, we use braces { } tο enclοse the set οf elements, and a vertical bar | οr a cοlοn : tο separate the cοnditiοn frοm the elements. The general fοrmat οf a set builder nοtatiοn is:

{elements | cοnditiοn}

Using this nοtatiοn, we can represent the set οf οrdered pairs (x, y) given in the questiοn as:

{(x, y) | x = 1/125, 1/25, 1/5, 1, 5, 25, 125 and y = -3, -2, -1, 0, 1, 2, 3}

This nοtatiοn reads as "the set οf all οrdered pairs (x, y) such that x takes οn the values 1/125, 1/25, 1/5, 1, 5, 25, and 125, and y takes οn the values -3, -2, -1, 0, 1, 2, and 3."

Hence, the set builder nοtatiοn wοuld be {(x, y) | x = 1/125, 1/25, 1/5, 1, 5, 25, 125 and y = -3, -2, -1, 0, 1, 2, 3}

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One cheeseburgers have 550 Cal and on smalls Fries have 320 Cal.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

Two cheeseburgers and one small order of fries contain a total of 1420 calories.

Three cheeseburgers and two small orders of fries contain a total of 2290 calories.

let the number of Calories in one cheeseburgers  be x and in one fries be y.

So, the system of Equation is:

2x + y = 1420

3x + 2y= 2290

solving the above two equation

x= 550 and y= 320

So, one cheeseburgers have 550 Cal and on smalls Fries have 320 Cal.

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Answer:

To find the name of the restaurant, we need to find the intersection point of the two streets.

The equation of Second Street is y = (-4/3)x + 4/3.

The equation of Washington Boulevard is y = (3/7)x - 3/7.

To find the intersection point, we can set the two equations equal to each other:

(-4/3)x + 4/3 = (3/7)x - 3/7

Multiplying both sides by 21 (the least common multiple of 3 and 7) to eliminate fractions, we get:

-28x + 28 = 9x - 9

Bringing all the x terms to one side and all the constants to the other side, we get:

-37x = -37

Dividing both sides by -37, we get:

x = 1

Substituting this value of x into either equation, we can find y:

y = (-4/3)(1) + 4/3 = 0

Therefore, the intersection point of Second Street and Washington Boulevard is (1,0), which means the restaurant is located at that point.

Step-by-step explanation:

Answer: I'd say A

Step-by-step explanation:

-1+1=0

The value of the variable x for the similar triangles is equal to 3

The triangles PTQ and STR are similar, this implies that the length ST of the smaller triangle is similar to the length PT of the larger triangle

and the length RT of the smaller triangle is similar to the length QT of the larger triangle

so;

5/(2x + 9) = 7/(3x + 12)

5(3x + 12) = 7(2x + 9) {cross multiplication}

15x + 60 = 14x + 63

15x - 14x = 63 - 60 {collect like terms}

x = 3.

Therefore, the value of the variable x for the similar triangles is equal to 3

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Ezra should increase the stage's dimensions by the following factor in order to make the picture half the size of the original blueprint:

50%.

The modified blueprint will have a perimeter that is half that of the original layout.

When the scale factor is multiplied by the coordinates of an image's vertices, the result is a dilatation, which alters the side lengths of a figure.

The scale factor of the dilation is supplied as follows because, for this issue, we want the updated stage's dimensions to be half those of the original stage.

k = 1/2.

Given that both the perimeter and the side lengths have the same unit, 50%, or half of the original stage's perimeter, will be represented by the updated stage's perimeter.

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The probability that Ann's score is above 600 is 1 - 0.6628 = 0.3372.

(a) The shape of the sampling distribution of the sample means will be approximately normal. This is because of the Central Limit Theorem, which states that the sampling distribution of the sample means will be approximately normal if the sample size is large enough, regardless of the shape of the population distribution. In this case, the sample size of 35 is large enough for the Central Limit Theorem to apply.

(b) The mean of the sampling distribution of the sample means will be equal to the mean of the population, which is 550. The standard deviation of the sampling distribution of the sample means will be equal to the standard deviation of the population divided by the square root of the sample size, which is 120/√35 ≈ 20.28.

(c) To find the probability that the average score of the 35 selected students is above 600, we need to find the z-score for 600 using the mean and standard deviation of the sampling distribution of the sample means. The z-score is (600 - 550)/20.28 ≈ 2.47. Using a z-table, we can find the probability that the z-score is less than 2.47, which is 0.9932. Therefore, the probability that the average score of the 35 selected students is above 600 is 1 - 0.9932 = 0.0068.

(d) To find the probability that Ann's score is above 600, we need to find the z-score for 600 using the mean and standard deviation of the population. The z-score is (600 - 550)/120 ≈ 0.42. Using a z-table, we can find the probability that the z-score is less than 0.42, which is 0.6628.

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Answer:

Q1    Value of  angle B = 50°

Q2   BC =  1.3

Q3    AB = 2.57

Step-by-step explanation:

For Part 1
The three angles of  a triangle must add up to 180°

Therefore m∠B + 30 + 100 = 180

m∠B + 130 = 180

m∠B = 180 - 30 = 50°

part 2
The law of sines states that, in  a triangle, the ratio of each side to the sine of the angle opposite that side must be the same for all sides

We have side AC opposite ∠B

AC = 2 and we found that m∠B = 50° from part 1

The side BC is opposite ∠A which is 30°

Therefore, applying the law of sines
[tex]\dfrac{{AC}}{\sin 50^\circ} = \dfrac{{BC}}{\sin 30^\circ}\\\\\\\dfrac{2}{\sin 50^\circ} = \dfrac{{BC}}{\sin 30^\circ}\\\\[/tex]

Multiplying both sides by sin 30° we get
[tex]\dfrac{2}{\sin 50^\circ} \times \sin 30^\circ =BC\\\\or\\\\BC = \dfrac{2}{\sin 50^\circ} \times \sin 30^\circ[/tex]

Using a calculator to compute the right side we get
BC = 1.30540 ≈ 1.3

part 3

Similarly
[tex]\dfrac{AB}{\sin 100} = \dfrac{2}{\sin50}\\\\AB = \sin 100 \times \dfrac{2}{\sin 50} = 2.57115 \approx 2.57[/tex]

The expression are classified as follows

1. Decay

2. Growth

3. Growth

3.  Decay

4. Growth

5. Decay

6. Decay

Exponential function refers to functions of the form

f(x) = a(b)^x

where

the starting = a

the base = b

the exponents = x

The base is what determines exponential growth or decay

when the base is greater than 1 we have exponential growth

when the base is 0 < x < 1 we have exponential decay

comparing the base of the functions shows that

1. y = 500(0.30)  decay

2. y = 500(1.70) growth

3. y = 0.3(500) growth

3. y=500(0.30) - 6  decay

4.y = 0.3(1.7)¹-2 growth

5. y = 500(0.30)+2 decay

6. y = 500(0.30)-6  decay

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Answer:

4.2 represents the initial height of the ball.

Step-by-step explanation:

h(x) = -16x² + 24x + 4.2

At x = 0, h(0) = 4.2

4.2 represents the initial height of the ball.

In a linear equation, There is one more test this quarter 39.

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                    Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables. Equations with variables of power 1 are referred to as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

lowest average score = 80% * 50  = 40

 suppose the final exam score she must get at least x.

   ( x + 45 + 38 + 36) = 4 ≥ 40

                                x ≥ 39

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The odds against the next car leaving the lot being black = option(A) is correct option.

The odds against getting three tails when you flip three coins = option(B) is the correct option.

What is probability?

The potential of any random event's result is referred to as probability. To determine the likelihood that any event will occur is the definition of this phrase.

There 34 cars out of which 20 are green, 8 are gold and 6 are black.

So, The odds against the next car leaving the lot being black

    = no. of black cars / no. of non black cars

    = 6/28

    =3/14 (A)

If three coins are flipped together,

the total number of outcomes will be = 2³ = 8

no. of outcomes getting three tails = 1

The odds against getting three tails when you flip three coins

= no. of outcomes getting three tails/ no. of outcomes not getting three tails

= 1/7 (option B)

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Yes, the binomial 2x+3 is a factor of the given polynomial P(x)=2x^(3)+3x^(2)-2x-3. We can verify this by using synthetic division.

First, we need to find the zero of the binomial, which is -3/2. Then, we can use synthetic division to divide the polynomial by the binomial.

Here are the steps for synthetic division:

1. Write the coefficients of the polynomial in a row: 2  3  -2  -3
2. Write the zero of the binomial to the left of the row: -3/2 | 2  3  -2  -3
3. Bring down the first coefficient: -3/2 | 2  3  -2  -3
               0  -3   6  -6
               ---------------
               2  0   4   -9
4. Multiply the zero by the first coefficient and write the result below the second coefficient: -3/2 * 2 = -3
5. Add the second coefficient and the result: 3 + (-3) = 0
6. Repeat steps 4 and 5 for the remaining coefficients: -3/2 * 0 = 0, -2 + 0 = -2; -3/2 * -2 = 3, -3 + 3 = 0
7. The final row of numbers represents the coefficients of the quotient: 2x^(2) + 0x + 4, with a remainder of -9.

Since the remainder is not zero, the binomial is not a factor of the polynomial. However, if we divide the polynomial by the binomial using long division, we get the following:

2x^(2) + 4 - (9/(2x+3))

This means that the polynomial can be written as: P(x) = (2x+3)(2x^(2) + 4) - 9

So, the binomial 2x+3 is a factor of the polynomial P(x)=2x^(3)+3x^(2)-2x-3.

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The solution of the exponential equation is x = 0.35

An exponential equation is an equation that contains exponents.

Since we have the exponential equation

4ˣ = 8ˣ - 1

We proceed to solve as follows

4ˣ = 8ˣ - 1

(2²)ˣ = (2⁴)ˣ - 1

(2ˣ)² = (2ˣ)⁴ - 1

Let 2ˣ = y

So, we have that

y² = y⁴ - 1

Re- arranging, we have that

y⁴ - y² - 1 = 0

Also, let y² = p. So, we have that

p² - p - 1 = 0

Now, we find p using the quadratic formula.

[tex]p = \frac{-b +/-\sqrt{b^{2} - 4ac} }{2a}[/tex]

where a = 1 b = -1 and c = -1

So, [tex]p = \frac{-(-1) +/-\sqrt{(-1)^{2} - 4(1)(-1)} }{2(1)}\\= \frac{1 +/-\sqrt{1 + 4} }{2}\\= \frac{1 +/-\sqrt{5} }{2}\\p = \frac{1 + 2.24 }{2} or p = \frac{1 - 2.24 }{2}\\p = \frac{3.24 }{2} or p = \frac{-1.24 }{2}\\p = 1.62 or p = -0.62[/tex]

We ignore the negative value.

So, p = 1.62

y² = 1.62

y = ±√1.62

y = ±1.273

Since y = 2ˣ, we have that

2ˣ = ±1.273

We ignore the negative value since the value of y cannot be negative.

So, 2ˣ = 1.273

Taking natural logarithm of both sides, we have that

㏑2ˣ = ㏑1.273

x㏑2 = ㏑1.273

x = ㏑1.273/㏑2

= 0.2412/0.693

= 0.348

≅ 0.35

So, x = 0.35

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When a point is reflected across the x-axis, its y-coordinate changes sign while its x-coordinate remains the same.

Therefore, to find the coordinates of the reflected point D, we simply change the sign of the y-coordinate of point D.

Given that the coordinates of point D are (3/4,3), its reflected point D' across the x-axis will have the coordinates (3/4, -3).

Therefore, the reflected point D' is located at the point (3/4, -3).

Coordinates are pairs of numbers used to identify the position of a point in a two-dimensional plane. In a Cartesian coordinate system, the two numbers represent the x-coordinate (the horizontal position) and the y-coordinate (the vertical position) of the point.

For example, the point (3, 5) has an x-coordinate of 3 and a y-coordinate of 5. This means that the point is located 3 units to the right of the origin (the point where the x- and y-axes intersect) and 5 units above the x-axis. Similarly, the point (-2, -1) has an x-coordinate of -2 and a y-coordinate of -1, which means that it is located 2 units to the left of the origin and 1 unit below the x-axis.

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a. The standard error of the mean would be 9.49.

b. The probability that the sample mean will be less than $125 is 0.1539.

c. The probability that the sample mean will be more than $145 is 0.1401.

d. The probability that the sample mean will be between $120 and $160 is 0.9356.

e. The symmetrical interval that includes 95% of the sample means is ($116.11, $153.31).

a. The standard error of the mean is calculated using the formula:

SE = σ / √n

where σ is the standard deviation and n is the sample size.

In this case, σ = $30 and n = 10. So, the standard error of the mean is:

SE = 30 / √10

SE = 9.49

b. To find the probability that the sample mean will be less than $125, we need to calculate the z-score:

z = (x - μ) / SE

where x is the sample mean, μ is the population mean, and SE is the standard error of the mean.

In this case, x = $125, μ = $134.71, and SE = 9.49. So, the z-score is:

z = (125 - 134.71) / 9.49

z = -1.02

Using a z-table, we find that the probability of getting a z-score less than -1.02 is 0.1539. So, the probability that the sample mean will be less than $125 is 0.1539.

c. To find the probability that the sample mean will be more than $145, we need to calculate the z-score:

z = (x - μ) / SE

where x is the sample mean, μ is the population mean, and SE is the standard error of the mean.

In this case, x = $145, μ = $134.71, and SE = 9.49. So, the z-score is:

z = (145 - 134.71) / 9.49

z = 1.08

Using a z-table, we find that the probability of getting a z-score less than 1.08 is 0.8599. So, the probability that the sample mean will be more than $145 is 1 - 0.8599 = 0.1401.

d. To find the probability that the sample mean will be between $120 and $160, we need to calculate the z-scores for both values:

z1 = (x1 - μ) / SE

z2 = (x2 - μ) / SE

where x1 and x2 are the sample means, μ is the population mean, and SE is the standard error of the mean.

In this case, x1 = $120, x2 = $160, μ = $134.71, and SE = 9.49. So, the z-scores are:

z1 = (120 - 134.71) / 9.49

z1 = -1.55

z2 = (160 - 134.71) / 9.49

z2 = 2.67

Using a z-table, we find that the probability of getting a z-score less than -1.55 is 0.0606 and the probability of getting a z-score less than 2.67 is 0.9962. So, the probability that the sample mean will be between $120 and $160 is 0.9962 - 0.0606 = 0.9356.

e. To find the symmetrical interval that includes 95% of the sample means, we need to use the formula:

x = μ ± z*SE

where x is the sample mean, μ is the population mean, z is the z-score, and SE is the standard error of the mean.

In this case, μ = $134.71, z = 1.96 (for a 95% confidence interval), and SE = 9.49. So, the symmetrical interval is:

x = 134.71 ± 1.96*9.49

x = 134.71 ± 18.6

x = (116.11, 153.31)

So, the symmetrical interval that includes 95% of the sample means is ($116.11, $153.31).

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The slope of the linear equation is; 0.5

The y-intercept of the Linear Equation is: 4.75

Yes the equation is proportional

The equation we are given is expressed as;

y = ¹/₂x + 4³/₄

When x = 0,

y = ¹/₂(0) + 4³/₄

y = 4.75

When x = 1,

y = ¹/₂(1) + 4³/₄

y = 5.25

When x = 2,

y = ¹/₂(2) + 4³/₄

y = 5.75

When x = 3,

y = ¹/₂(3) + 4³/₄

y = 6.25

If we go on and on till x = 10, we will see the constant difference in y for every change in x is 0.5

Thus, slope = 0.5

y-intercept = 4.75

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Answer:

$0.45

Step-by-step explanation: