How to program the quadratic formula into a ti-84 plus.

The calculated value of c in the expression x² + 10x + c is 25

From the question, we have the following parameters that can be used in our computation:

x² + 10x + c

We understand that

The expression is a perfect square trinomial

So, we do the following:

Divide the coefficient of x by 2

k = 10/2

k = 5

Next, we square the result to get c

k² = 25

This means that

c = k² = 25

This means that the value of c is 25

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Question

The algebra tiles represent the perfect square trinomial x2 + 10x + c. What is the value of c?

The measure of ∠Q is 62 degrees.

Given that, QRST is a parallelogram.

Angle Q has measure (4x + 10) degrees, angle R is (9x + 1) degrees, angle S is (5x − 3) degrees.

We have to find the measure of angle Q.

In parallelogram opposite angles are equal, and adjacent angles are supplementary.

Therefore, we can say that,

Angle T = Angle R

= 9x + 1°

Angle Q = Angle S

= 5x - 3°

Also,

Angle Q + Angle R

= 180°(4x + 10) + (9x + 1) = 180°

Solving the above equation,

4x + 10 + 9x + 1

= 18013x + 11

= 18013x

= 180 - 11

= 169x

= 169/13

Therefore, x = 13

Now, we can calculate the measure of angle Q, Angle Q = 5x - 3°= 5 × 13 - 3°= 65 - 3°= 62°

Hence, the measure of ∠Q is 62 degrees.

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Given, Aika is building a square garden. She places a garden post at (3.5, 3.5)

To find: The location of the corner that reflects (3.5, 3.5) across the y-axis.

We know that the y-axis is the vertical line through the point (0,0) and it divides the plane into two parts: left and right. When we reflect a point across the y-axis, the x-coordinate changes sign. For example, the reflection of (2,3) is (-2,3).Therefore, the reflection of (3.5, 3.5) across the y-axis is (-3.5, 3.5)

Since Aika is building a square garden, the corner opposite to (3.5, 3.5) will have coordinates (-3.5, -3.5).

Hence, the location of the corner that reflects (3.5, 3.5) across the y-axis is (-3.5, -3.5).

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The quote emphasizes the importance of acting in a way that brings honor and respect, ensuring that one's actions align with societal expectations and moral values. It calls upon individuals to consider the impact of their behavior on their loved ones and their community.

The quote reflects a call for personal responsibility and ethical conduct. By urging individuals to "be men," it does not limit the message to a specific gender but rather encourages everyone to exhibit qualities associated with strength, integrity, and honor. The subsequent line highlights the significance of acting in a manner that preserves one's dignity and integrity, emphasizing the importance of living in a way that would not bring shame or disgrace upon oneself when facing others.

The second paragraph of the quote urges individuals to contemplate the repercussions of their actions on their loved ones and their broader social connections. It prompts individuals to think of their children, spouse, parents, and home, whether they are still alive or have passed away. This appeal to familial and personal relationships emphasizes the interconnectedness of human lives and the responsibility to act in a way that protects and uplifts one's family and community. The quote serves as a reminder that our actions have consequences beyond ourselves and that we should consider the potential impact on those who are closest to us. It encourages individuals to cultivate a sense of empathy, accountability, and social consciousness in their decision-making process.

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The expression that can be used to determine the total distance in miles Natalie jogged over these 3 nights is:

The model for finding the total distanceNatalie went on a jog 3 nights in a row and jogged the same distance each night.

The model for finding the total distance that Natalie jogged during the 3 nights is shown below:

Model for finding the total distance where each column represents one mile, and the shaded parts of each column represent the fraction of a mile that Natalie jogged each night.

From the model, we can find the total distance in miles Natalie jogged by counting the number of shaded parts in each column and then adding them together.

The number of shaded parts in each column represents the fraction of a mile that Natalie jogged each night.

Therefore, the expression that can be used to determine the total distance in miles Natalie jogged over these 3 nights is:

[tex]$$3 \cdot 1 + \frac{1}{2} + \frac{3}{4}$$ $$= 3 + \frac{2}{4} + \frac{3}{4}$$$$= 3 + \frac{5}{4}$$$$= \frac{12}{4} + \frac{5}{4}$$$$= \frac{17}{4}$$$$= \boxed{4\frac{1}{4}}\ miles$$[/tex]

Therefore, Natalie jogged a total distance of 4 and 1/4 miles over these three nights.

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Alexander Hamilton was a staunch supporter of the Federalist Party and, in particular, a strong supporter of a powerful central government. He opposed Thomas Jefferson's philosophy of a strict interpretation of the Constitution and advocated for the creation of a strong economy, and the promotion of manufacturing and industry.

Their visions differed. Hamilton had a vision for America that was far more centralized and industrialized than that of Jefferson. He wanted a strong national government that would be able to support a thriving economy by promoting industry, commerce, and manufacturing, while Jefferson favored a limited federal government that would be unable to interfere in the lives of individual citizens.In Hamilton's view, the United States needed to establish itself as a world power, and he believed that this could be accomplished through a strong military and a powerful economy.

He saw the United States as a great commercial and manufacturing nation, and he believed that it could only achieve this status by embracing industrialization and creating a national bank that would provide the capital necessary to finance economic growth. Jefferson, on the other hand, believed that the federal government should have only limited powers and that these powers should be strictly defined by the Constitution. He believed that the states should have more power than the federal government and that the country should be primarily agrarian.

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The system of equations that could be used to solve the problem is:

1. p + c = 10   (equation representing the total weight of the mixture)

2. 4.00p + 6.50c = 4.75(10)   (equation representing the cost of the mixture)

Let's break down the given information and use it to set up the system of equations.

1. Total weight equation:

The grocer wants to make a 10-pound mixture of peanuts and cashews. Since we are given that p represents the pounds of peanuts and c represents the pounds of cashews, we can write the equation:

p + c = 10

2. Cost equation:

The grocer wants to sell the mixture for $4.75 per pound. The cost of the peanuts is $4.00 per pound and the cost of cashews is $6.50 per pound. To calculate the total cost, we multiply the cost per pound by the weight of each component (peanuts and cashews) and sum them up. This can be expressed as:

4.00p + 6.50c = 4.75(10)

By setting up this system of equations, we can solve for the values of p and c, which represent the pounds of peanuts and cashews, respectively, that the grocer should use in order to make the 10-pound mixture.

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The growth of Manchester during the Industrial Revolution gave rise to several issues that had significant social, economic, and environmental implications.

As a result, reactions to these issues emerged and evolved over the course of the nineteenth century.

One major issue raised by the growth of Manchester was poor working and living conditions for the working class. Rapid industrialization led to the establishment of large factories and mills, which attracted workers from rural areas. These workers often faced long working hours, low wages, and hazardous working conditions. They lived in overcrowded and unsanitary slums, lacking proper housing, sanitation, and access to basic amenities. These harsh conditions resulted in widespread poverty, disease, and social unrest.

In response to these issues, various movements and reforms emerged throughout the nineteenth century. The labor movement gained momentum as workers organized themselves to demand better working conditions, higher wages, and shorter hours. The formation of trade unions aimed to protect workers' rights and negotiate with employers. Additionally, reformers such as Robert Owen and the Chartists advocated for social and political reforms to address the plight of the working class.

Another issue that arose with the growth of Manchester was environmental degradation. The rapid expansion of industries led to pollution of air and water sources. Factories emitted smoke and pollutants, contributing to air pollution and poor air quality. Rivers and streams became contaminated with industrial waste and sewage, leading to water pollution and health hazards.

As awareness of these environmental issues grew, there were efforts to address them. The establishment of legislation and regulations aimed to control pollution and improve public health. For example, the Alkali Act of 1863 imposed restrictions on the emission of harmful gases from factories. These measures, although limited, marked the beginning of environmental consciousness and attempts to mitigate the negative impact of industrialization.

Furthermore, the growth of Manchester highlighted class divisions and inequalities. The wealthy factory owners and industrialists thrived while the working class suffered. This socioeconomic divide led to social tensions and movements advocating for greater equality and social reforms.

Throughout the nineteenth century, the issues raised by the growth of Manchester prompted a gradual transformation in society. Reactions to these issues ranged from grassroots movements to legislative reforms. Although progress was often gradual and incremental, the recognition of the hardships faced by the working class and the need for improved working conditions, social reforms, and environmental conservation laid the groundwork for future advancements in labor rights, social equality, and environmental protection.

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Each pizza pie costs $8.75. The cost of each bottle of soda is represented by "y" dollars.

Let's assume the cost of each pizza pie is represented by "x" dollars, and the cost of each bottle of soda is represented by "y" dollars.

Based on the given information, we can set up the following system of equations:

Equation 1: 5x + 3y = 50.50 (First trip cost without tax)

Equation 2: 2x + 6y = 31.00 (Second trip cost without tax)

To solve this system of equations, we can use the method of elimination.

Multiply Equation 1 by 2 and Equation 2 by 5 to create coefficients of "x" that will cancel each other out:

2(5x + 3y) = 2(50.50)

5(2x + 6y) = 5(31.00)

Simplifying:

10x + 6y = 101

10x + 30y = 155

Now, subtract Equation 1 from Equation 2:

(10x + 30y) - (10x + 6y) = 155 - 101

24y = 54

y = 54/24

y = 2.25

Now substitute the value of "y" back into Equation 1 to solve for "x":

5x + 3(2.25) = 50.50

5x + 6.75 = 50.50

5x = 43.75

x = 43.75/5

x = 8.75

Therefore, each pizza pie costs $8.75.

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The value of mZWXY is 103.5° , mWVX = (13x + 9)° and mZWXZ = (5x + 36)°

To find: mZWXY

Here, mWVX and mZWXY are two angles that are formed when the two parallel lines are intersected by a transversal. Therefore,mWVX = mZWXY (alternate interior angles)

and mZWXZ is the third angle that is formed and the sum of the three angles is equal to 180°.

So,mWVX + mZWXY + mZWXZ = 180°

Therefore,mZWXY + mZWXY + (5x + 36)°

= 180°13x + 9 + (5x + 36)°

= 180°18x + 45

= 180°18x

= 180° - 45x

= 135/18x = 7.5

Therefore, mZWXY = mWVX= (13 × 7.5 + 9)°= 103.5°

Hence, the value of mZWXY is 103.5°.

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Therefore, the diagonals PS and ET are equal, and they bisect each other. This parallelogram is a rectangle and a rhombus because its diagonals are equal and bisect each other.

The diagonals to determine whether a parallelogram with vertices P(2,4), E(4,8), S(12,4), and T(10,0) is a rectangle, rhombus, or square is described below:

Given coordinates of vertices are

P(2, 4), E(4, 8), S(12, 4), and T(10, 0).

Therefore, PS and ET are the diagonals of the parallelogram.

Here, the midpoint of the diagonal is

M = [ (x₁ + x₂)/2, (y₁ + y₂)/2 ].

The midpoint of PS is the midpoint of the line segment PS with coordinates (7,4).

The midpoint of ET is the midpoint of the line segment ET with coordinates (7,4).

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In parallelogram ABCD, what is the relationship between angle a° and angle b° is: a° = b°.

Correct answers of given question are given below:

1. The correct relationship between angle a° and angle b° in parallelogram ABCD is: a° = b°. In a parallelogram, opposite angles are congruent, meaning they have the same measure. Therefore, angle a° and angle b° have equal measures.

2. The area of rectangle FGHK can be calculated by multiplying the length and width. However, the width is not given in the information provided. Therefore, it is not possible to determine the area of the rectangle based on the given information. The correct answer cannot be determined.

3.In rectangle FGHK, FC = CH = 8.5 cm. Since FC and CH are equal, they represent the width of the rectangle. The length of the rectangle is not provided in the information. Therefore, it is not possible to determine the length of GK based on the given information. The correct answer cannot be determined.

4. The correct relationship between angle e and angle g in parallelogram EFGH is: e° = g°. In a parallelogram, opposite angles are congruent, meaning they have the same measure. Therefore, angle e° and angle g° have equal measures.

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The limit of the expression (x^2+4x-43.61)/(x^2-0.2x-23.03) as x approaches 4.9 is equal to a specific value. The limit of the given expression as x approaches 4.9 is approximately 1.51.

To find the limit as x approaches 4.9, we substitute 4.9 into the expression and evaluate it. Plugging in 4.9 for x, we get ((4.9)^2 + 4(4.9) - 43.61)/((4.9)^2 - 0.2(4.9) - 23.03). Simplifying this expression gives us (24.01 + 19.6 - 43.61)/(24.01 - 0.98 - 23.03), which further simplifies to 0/0.

When we encounter an indeterminate form like 0/0, we can apply mathematical techniques to evaluate the limit. One approach is to use L'Hôpital's Rule, which states that if the limit of the ratio of two functions f(x)/g(x) is of the form 0/0 or ∞/∞ as x approaches a certain value, then the limit of the ratio is equal to the limit of the derivative of f(x) divided by the derivative of g(x) as x approaches the same value.

In this case, we can differentiate the numerator and denominator separately and apply L'Hôpital's Rule. After differentiating, we obtain the new expression (2x + 4)/(2x - 0.2). Evaluating this expression at x = 4.9 gives us (2(4.9) + 4)/(2(4.9) - 0.2), which simplifies to 14.8/9.8. Therefore, the limit of the given expression as x approaches 4.9 is approximately 1.51.

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The actual width of the rectangular swimming pool is given as 28 feet. Jason's drawing is made to a scale of 1 inch : 4 feet.

To find out how many inches wide Jason's drawing is, we need to divide the actual width of the pool by the scaling factor (4 feet).

28 feet / 4 feet = 7

Therefore, Jason's drawing of the pool is 7 inches wide.

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The expression "x - 12" can be modeled in a real world situation where you are trying to find the difference between a number x and 12. Here is an example:Suppose you have a jar containing x marbles.

You give away 12 marbles to your friend. The number of marbles you have left in the jar can be modeled by the expression "x - 12". In this situation, x represents the original number of marbles in the jar, and 12 represents the number of marbles given away to your friend. The expression "x - 12" calculates the number of marbles you have left after giving away 12.

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The pattern to the number series 5, 12, 26, ____, 110, 222 is 50. There are different methods to solve the pattern. For example, you can find the difference between consecutive numbers and check if it follows a pattern.

The pattern to the number series 5, 12, 26, ____, 110, 222 is 50.

How?

There are different methods to solve the pattern. For example, you can find the difference between consecutive numbers and check if it follows a pattern. Here, I am using this method to explain the answer. To start, we will find the difference between consecutive numbers.

5 to 12 = 7 (12 - 5 = 7)

12 to 26 = 14 (26 - 12 = 14)

26 to ____ = ?

____ to 110 = 84 (110 - ____ = 84)

110 to 222 = 112 (222 - 110 = 112)

Now, we will find the difference between the second difference.

7 to 14 = 7 (14 - 7 = 7)

14 to ____ = ?

____ to 84 = 70 (84 - ____ = 70)

84 to 112 = 28 (112 - 84 = 28)

Since we are given 5, 12, 26, ____, 110, 222

fill in the patterning blank, we need to find the blank space. So, let's work on that.

7 to 14 = 7 (14 - 7 = 7)

14 to ____ = ?

7 + 7 = 14

____ = 28 (14 + 14 = 28)

28 to 84 = 56 (84 - 28 = 56)

84 to 112 = 28 (112 - 84 = 28)

28 + 28 = 56

Hence, the blank in the patterning is 50. This number series has a pattern to solve. If you learn how to solve such patterns, you can easily find the blank in any patterning series. There are different methods to find the answer, as explained above, but the one I used is the most common one. Here, we found the difference between consecutive numbers and checked if it follows a pattern. The pattern we found is that the second difference is constant. The second difference is the difference between the first difference of the consecutive numbers. When we calculated the second difference, we found that the blank in the patterning series is 50. It means that the difference between 26 and the blank is 28, and the difference between the blank and 110 is 84.

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The midpoint of the line segment connecting two points can be found by averaging their corresponding coordinates. Therefore, to obtain the midpoint of line BC, we add the coordinates of B and C and divide by 2.

Midpoint of line BC is given by:

\[\left(\frac{-1-3}{2},\frac{1-1}{2},\frac{2+0}{2}\right)=(-2,0,1)\]

The length of line AD is found by using the distance formula, which is given as:

\[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]

Thus, we need to find the coordinates of point D and A to determine the length of AD. The coordinates of D are the average of the coordinates of B and C.

\[\left(\frac{-1-3}{2},\frac{1-1}{2},\frac{2+0}{2}\right)=(-2,0,1)\]The coordinates of A are (1,2,-1).

The distance between A and D is found by substituting these values into the distance formula:

\[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\] \[d=\sqrt{(1-(-2))^2+(2-0)^2+(-1-1)^2}\] \[d=\sqrt{(3)^2+(2)^2+(-2)^2}\] \[d=\sqrt{9+4+4}\] \[d=\sqrt{17}\]

Thus, the distance between points A and D is sqrt(17).Therefore, the midpoint of line BC is (-2,0,1) and the distance between points A and D is sqrt(17).

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Therefore, the size of a computer monitor whose diagonal measures d = 5/6 inches is 5/6 inches.

Most computer monitors range from 19 to 34 inches, measured diagonally from corner to corner. The average user will be happy with 22-24" screens. This range provides enough screen real estate for general productivity tasks and even light multitasking without overcrowding your desktop.

Full HD (1080p) stays fairly sharp on monitors of this size. As they're not too big, you can easily place them in a setup with 2 screens. A 24-inch monitor is also suitab

The size of a computer monitor is given as the length of the screens diagonal d in inches.

The equation d=5/6.

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Each section of the garden will be 3.4 feet long.

To find the length of each section, we can divide the total length of the garden by the number of sections.

Length of the garden: 40.8 feet

Number of sections: 12

Length of each section = Total length of the garden / Number of sections

Length of each section = 40.8 feet / 12

Length of each section = 3.4 feet

Therefore, each section of the garden will be 3.4 feet long.

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Ned will have travelled 2/9 mile after 1 hour

From the question, we have the following parameters that can be used in our computation:

Ned walked 1/9 of a mile in one 1/2

using the above as a guide, we have the following:

Rate = (1/9)/(1/2)

Evaluate the the quotient

Rate = 2/9

This means that he will have travelled 2/9 mile after 1 hour

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none of the provided approximations for the probabilities are accurate.To determine which statement is true, we need to calculate the probabilities of selecting each color marker.

Total number of markers = 20 green + 44 orange + 30 blue = 94 markers.

P(blue) = Number of blue markers / Total number of markers = 30 / 94 ≈ 0.319.

P(green) = Number of green markers / Total number of markers = 20 / 94 ≈ 0.213.

P(orange) = Number of orange markers / Total number of markers = 44 / 94 ≈ 0.468.

Based on the calculations, none of the given statements are true. The actual probabilities are approximately:

P(blue) ≈ 0.319,
P(green) ≈ 0.213,
P(orange) ≈ 0.468.

Therefore, none of the provided approximations for the probabilities are accurate.

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The measure of segment QS can be determined by adding the lengths of QR and RS. In this case, since QR is 10 units long and RS is 7 units long, the measure of QS would be 17 units.

To find the measure of segment QS, we need to add the lengths of QR and RS. Given that QR is 10 units long and RS is 7 units long, we can calculate the measure of QS by adding these two lengths together. Therefore, QS = QR + RS = 10 + 7 = 17. Hence, the measure of segment QS is 17 units. By adding the lengths of the two segments that make up QS, we obtain the total length of the segment itself.

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The difference between the radii of the two pastry stamps represents the thickness of the donut. In this case, the thickness would be (1.5 - 0.5) inches, which is 1 inch.

Petra's method of using pastry stamps with different radii to create donuts with specific thickness and distinct ring shape. A radius of 1.5 inches for outer circle, another with radius of 0.5 inches for donut hole.

Petra is using two different pastry stamps to make donuts, one with a radius of 1.5 inches for the outer circle and another with a radius of 0.5 inches for the donut hole.

Outer Circle: The pastry stamp with a radius of 1.5 inches is used to stamp out the outer circle of the donut. The outer circle represents the main body of the donut.

Donut Hole: The pastry stamp with a radius of 0.5 inches is used to stamp out the donut hole in the center of the donut. The donut hole is the circular space left in the middle of the donut.

Difference: The difference between the radii of the two pastry stamps represents the thickness of the donut. In this case, the thickness would be (1.5 - 0.5) inches, which is 1 inch.

Overall Shape: The combination of the outer circle and the donut hole creates the characteristic ring shape of a donut, with the thickness determined by the difference in radii.

Therefore, Petra's method of using pastry stamps with different radii allows her to create donuts with a specific thickness and a distinct ring shape.

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The total cost of fencing to David, based on the dimensions of the fence would be $ 996.

Since the area of a rectangle is length times width, the other side must be:

240 sq ft / 12 ft

= 20 ft

For the side along the street (12 feet), using the more expensive fencing ($18 per linear foot):

Cost = 12 ft x $18/ft

= $216

For the other three sides (two sides of 20 feet and one of 12 feet), using the less expensive fencing ($15 per linear foot):

Cost = (20 ft + 20 ft + 12 ft) x $15/ft

= 52 ft x $15/ft

= $780

So, the total cost for the fencing will be:

Total cost = Cost of expensive fencing + Cost of less expensive fencing

Total cost = $216 + $780

= $996

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Full question is:

David wants to install a fence fence around his garden. The garden is rectangular and has an area of 240 square feet. One side of the garden is along the street and is 12 feet long. For this side of the garden, David will use fencing that costs 18$ per linear foot. For the other three sides of the garden, David will use fencing that costs 15$ per linear foot.

The number line can be used to determine the sum of the location of point T and -1 1/2. By visually representing the positions of these points on the number line, we can determine their sum by adding their locations.

To determine the sum of the location of point T and -1 1/2, we can plot the point T on the number line and then locate -1 1/2 on the number line. We can then add the locations of these points on the number line to find their sum.

For example, if point T is located at 3 on the number line and -1 1/2 is located at -1.5, we can add 3 and -1.5 to find their sum, which would be 3 + (-1.5) = 1.5.

By using the number line, we can visually represent the locations of the points and perform addition to find their sum accurately.

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The sample of 10 prize winners from the 5000 tickets in the raffle is an example of simple random sampling without replacement.

Simple random sampling involves randomly selecting individuals from a population in such a way that each individual has an equal chance of being chosen. In this case, the 5000 tickets represent the population, and the business uses a random number generator to select 10 tickets as the prize winners.

The "without replacement" aspect means that once a ticket is selected as a winner, it is removed from the pool of available tickets for subsequent selections. This ensures that each ticket can only win the prize once, maintaining the randomness and fairness of the selection process.

Simple random sampling without replacement is a common sampling scheme used in various contexts, including raffles, surveys, and statisticalresearch. It provides a representative subset of the population, allowing for generalizations and conclusions to be drawn about the larger population based on the characteristics of the sample.

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`The measure or dimensions of the shorter sides will be 2.65 cm, given that the longer sides measure 40. 5 centimeters and the shorter sides measure 34. 5 centimeters and the scale-factor used for reduction is 13.

Given that,

The longer sides of the rectangular measure 40.5 cm.

The shorter sides of the rectangular measure 34.5 cm.

Scale factor = 13.

The scale factor is the ratio of the length of a side of one figure to the length of the corresponding side of the second figure.

To find the measure of the shorter sides of a rectangle, multiply the length of the shorter sides by the scale factor.Abby decides to reduce the scale drawing of her living room by a scale factor of 13.

Multiply 34.5 cm by 1/13 to find the length of the shorter sides in the reduced scale drawing as follows;`

34.5*1/13=2.65

The measure of the shorter sides will be 2.65 cm.

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1/6 is the fraction of Marjorie's ribbons that are green. Marjorie has a total of 6 ribbons and spools of thread for art, consisting of 3 pink ribbons, 1 green ribbon, and 2 blue spools of thread.

1. To determine the fraction of Marjorie's ribbons that are green, we divide the number of green ribbons by the total number of ribbons.

2. To find the fraction of Marjorie's ribbons that are green, we need to calculate the ratio of green ribbons to the total number of ribbons. The total number of ribbons is the sum of all the ribbons and spools of thread, which is 3 pink ribbons + 1 green ribbon + 2 blue spools of thread, equaling 6 ribbons in total.

3. Since Marjorie has only 1 green ribbon, we can say that the fraction of her ribbons that are green is 1 out of the total 6 ribbons. This can be expressed as 1/6. Therefore, 1/6 is the fraction of Marjorie's ribbons that are green.

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In this scenario, Tony is the one who comes out behind, and by $14. Tony's loss of $14 is smaller than Dave's loss of $20, making Tony the one who comes out behind by a smaller amount.

Tony initially purchases a hat for $14 and gives Dave a $20 bill. Since Dave doesn't have any change, he goes to Laura at Watson's Hardware store and exchanges the $20 bill for twenty $1 bills. Dave gives Tony $6 in change, which means Tony effectively paid $8 for the hat ($14 - $6).

However, it is later discovered that the $20 bill given to Laura by Dave was counterfeit. As a result, Dave compensates Laura by giving her a genuine $20 bill from the store's till. This means that Dave essentially lost $20 in the process.

So, in total, Tony ends up $14 behind because he paid $8 for the hat and received $6 in change, while Dave ends up $20 behind due to the counterfeit $20 bill. Tony's loss of $14 is smaller than Dave's loss of $20, making Tony the one who comes out behind by a smaller amount.

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