which is equivalent to (3/125)*?
​125 1/3×
125 1/3X
125 3x
125 [1/3]×

From the line plot, we can see that the statement "Team C has the most people" is true, since the count of 7 appears three times, which is more than any other count.

A line plot is a type of graph used to display data along a number line. It is also known as a dot plot or a strip plot. In a line plot, each data point is represented by a dot or a symbol above the number line, indicating the frequency or count of the data point. The dots or symbols are aligned horizontally along the number line, making it easy to see the distribution of the data.

Here,

To create a line plot from the data, we need to count the number of people on each team and mark a dot on the number line for each count. The number line should start at the smallest count (which is 3 in this case) and end at the largest count (which is 8).

Looking at the data, we can see that there are two counts of 3, three counts of 5, three counts of 7, one count of 6, and one count of 8.

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Complete question:

Answer: $68.90 for each quarter

Step-by-step explanation:

To calculate the state and federal unemployment taxes that Paula will pay, we need to first calculate the taxable wages for each employee and then apply the appropriate tax rates.

For Quarter 1:

Employee 1's taxable wages: $410

Employee 2's taxable wages: $650

Total taxable wages: $410 + $650 = $1,060

State unemployment tax:

Taxable wage base for the state: $9,000 per employee per year

Taxable wage base for two employees in a quarter: $18,000

Tax rate: 5.9%

State unemployment tax: $1,060 x 5.9% = $62.54

Federal unemployment tax:

Taxable wage base for the federal government: $7,000 per employee per year

Taxable wage base for two employees in a quarter: $14,000

Tax rate: 0.6%

Federal unemployment tax: $1,060 x 0.6% = $6.36

Therefore, Paula will pay a total of $62.54 + $6.36 = $68.90 in state and federal unemployment taxes for Quarter 1.

For Quarter 2:

Employee 1's taxable wages: $410

Employee 2's taxable wages: $650

Total taxable wages: $410 + $650 = $1,060

State unemployment tax:

Taxable wage base for the state: $9,000 per employee per year

Taxable wage base for two employees in a quarter: $18,000

Tax rate: 5.9%

State unemployment tax: $1,060 x 5.9% = $62.54

Federal unemployment tax:

Taxable wage base for the federal government: $7,000 per employee per year

Taxable wage base for two employees in a quarter: $14,000

Tax rate: 0.6%

Federal unemployment tax: $1,060 x 0.6% = $6.36

Therefore, Paula will pay a total of $62.54 + $6.36 = $68.90 in state and federal unemployment taxes for Quarter 2 as well.

After answering the presented question, we can conclude that As a result, the player may expect to lose a total of $5,580, with an average loss of $5.58 per game.

Probability is a measure of how likely an event is to occur. It is represented by a number between 0 and 1, with 0 representing a rare event and 1 representing an inescapable event. Switching a fair coin and coin flips has a chance of 0.5 or 50% because there are two equally likely outcomes. (Heads or tails). Probabilistic theory is an area of mathematics that studies random events rather than their attributes. It is applied in many fields, including statistics, economics, science, and engineering.

(38/39) x $315 - (1/39) x $9 = $306.54

The player will lose the remaining 620 games, totaling $9 x 620 = $5,580.

As a result, the player may expect to lose a total of $5,580, with an average loss of $5.58 per game.

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The correct option is "IQR, because Desert Landing is skewed."

Skewness is a term used in statistics to describe the degree of asymmetry of a probability distribution or a dataset. If a dataset is skewed, it means that the distribution of the data is not symmetrical.

To determine the location with the most consistent temperature between the two sets of data, we should use the IQR (Interquartile Range) measure of variability. This is because the IQR is a measure of the spread of the middle 50% of the data, which is a good measure of variability for skewed data such as the data from Desert Landing.

In the case of Flower Town, the range measure of variability could also be used, but it is generally less robust than the IQR and can be affected by extreme values. The symmetry of the histograms does not necessarily determine which measure of variability should be used.

Therefore, the correct option is "IQR, because Desert Landing is skewed." This is because the IQR is a robust measure of variability that is less affected by extreme values and is suitable for skewed data. The data from Desert Landing is skewed, while the data from Flower Town is not specified as being skewed or symmetric, so the IQR is the best choice for both sets of data.

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Answer:

(what is the question).

We can say that the domain of f(x) is (-∞, ∞), which means that any real number can be plugged into the function and it will produce a valid output.

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.

The function f(x) = -x³ + 6x² -10x + 4 is a polynomial function, which means that it is defined for all real values of x. Unlike some functions (such as square roots or logarithms) which have restrictions on their input values, a polynomial function can take any real number as an input.

To express the domain of the function in interval notation, we use the notation (-∞, ∞), which represents all real numbers. The symbol ∞ stands for infinity, and the open interval notation with the parentheses indicates that there are no specific endpoints for the domain; it extends indefinitely in both directions along the real number line.

Therefore, we can say that the domain of f(x) is (-∞, ∞), which means that any real number can be plugged into the function and it will produce a valid output.

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Hello!
In this question, we are solving "Composite Functions".

We were given the functions:

We will use these to solve each part of the question.

Solve:

Whenever we see "(f ∘ g)", it means f of g. Another way to write (f ∘ g) is f(g(x)) where g(x) is plugged into the "x" variable of our f(x) function.

(a)

(f ∘ g)(x) = f(g(x))

Plug the g(x) function into the "x" variable of the f(x) function.

(f ∘ g)(x) = 4x + 5 − 2

Simplify.

(f ∘ g)(x) = 4x + 3

(b)

For question "b", it is the same concept but reversed. Plug the f(x) function into the "x" variable of the g(x) function.

(g ∘ f)(x) = g(f(x)) = 4(x − 2) + 5

Simplify.

(g ∘ f)(x) = g(f(x)) = 4(x − 2) + 5

Use the distributive property to distribute the 4.

(g ∘ f)(x) = g(f(x)) = 4x − 8 + 5

(g ∘ f)(x) = g(f(x)) = 4x − 3

(c)

For problem "c", we will plug our number into the "x" variable after compositing the functions. Use the same concept described above.

(f ∘ g)(5) = f(g(x))

(f ∘ g)(5) = 4x + 5 − 2

Simplify.

(f ∘ g)(5) = 4x + 3

Plug in "5" to "x" and simplify.

(f ∘ g)(5) = 4(5) + 3

(f ∘ g)(5) = 23

(d)

Problem "d" is just like problem "c", but the functions are reversed

(g ∘ f)(5) = g(f(x)) = 4(x − 2) + 5

Use the distributive property to distribute the 4.

(g ∘ f)(5) = g(f(x)) = 4x − 8 + 5

(g ∘ f)(5) = g(f(x)) = 4x − 3

Plug in "5" to "x" and simplify.

(g ∘ f)(5)  = 4(5) − 3

(g ∘ f)(5) = 17

Answer:

(a) 4x + 3

(b) 4x − 3

(c) 23

(d) 17

The cost is minimized when all the remaining requirements are met by Food B, we can purchase 36 units of Food A and 30 units of Food B to meet all the requirements and minimize the cost.

What is probability?

Probability is a measure of the likelihood of an event occurring.

Let x be the number of units of Food A and y be the number of units of Food B.

The objective is to minimize the cost, which is given by the expression 2x + y.

The constraints are:

50x + 20y >= 2400 (minimum vitamin requirement)

30x + 20y >= 1800 (minimum mineral requirement)

10x + 40y >= 1200 (minimum calorie requirement)

x >= 0, y >= 0 (non-negative quantities)

We can use linear programming to solve this problem. The standard form is:

Minimize: 2x + y

Subject to:

50x + 20y >= 2400

30x + 20y >= 1800

10x + 40y >= 1200

x >= 0, y >= 0

Therefore, the minimum cost is $30, and it is achieved by purchasing 30 units of Food B.

To find the number of units of Food A, we can use one of the constraints, for example:

50x + 20y >= 2400

50x + 20(30) >= 2400

50x >= 1800

x >= 36

So, we need at least 36 units of Food A to meet the vitamin requirement.

Since the cost is minimized when all the remaining requirements are met by Food B, we can purchase 36 units of Food A and 30 units of Food B to meet all the requirements and minimize the cost.

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The right number that fits the sequence 54 , 48 , 18 , 24 , 6 , 12 , is 2

To find the next number in the sequence, we need to look for a pattern in the given numbers.

So, we have

54 .. 18 .. 6 ... division by 3

48 .. 24 .. 12 .. division by 2

So, it seems that we are dividing each pair of consecutive numbers by a certain factor.

So, the factor by which we are dividing each pair of consecutive numbers seems to be decreasing by a factor of 3 each time, for the first sequence.

Therefore, the next number in the sequence should be:

6/3 = 2

So, the complete sequence is:

54, 48, 18, 24, 6, 12, 2

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The earning οf an emplοyee is 8.25 dοllars per hοur and hοurly pay rate is 8.25 dοllars.

Pay rate refers tο the amοunt οf mοney an emplοyer pays an emplοyee fοr their wοrk, typically expressed as an hοurly, weekly, οr mοnthly rate. It is the rate at which an emplοyee is cοmpensated fοr their time and labοr, and can be influenced by a number οf factοrs, such as jοb skills, experience, industry, lοcatiοn, and market demand fοr the particular rοle. The pay rate may alsο be subject tο deductiοns fοr taxes, insurance, and οther benefits prοvided by the emplοyer.

Here,

The hοurly pay rate οf the emplοyee can be calculated by dividing the earnings by the number οf hοurs wοrked. Using the given equatiοn, we can rewrite it as E = 8.25h

Dividing bοth sides by h,

E/h = 8.25

Therefοre, the hοurly pay rate οf the emplοyee is 8.25 dοllars per hοur.

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Answer:

[tex]\frac{-2(24t-25)}{45}[/tex]

Step-by-step explanation:

In order to find the expression, you have to read the problem from left to right and make sure to follow PEMDAS.

[tex](-\frac{7}{5} * t+\frac{2}{9})-(-\frac{1}{3}*t+\frac{4}{3})\\\\\\(-\frac{7t}{5} +\frac{2}{9} )-(-\frac{t}{3} +\frac{4}{3} )\\\\(-\frac{7t}{5} *\frac{9}{9} +\frac{2}{9} *\frac{5}{5} )-(\frac{-t+4}{3} )\\\\(-\frac{63t}{45} +\frac{10}{45} )+\frac{t-4}{3} \\\\\frac{-63t+10}{45} +\frac{t-4}{3} \\\\\frac{-63t+10}{45}*\frac{3}{3}+\frac{t-4}{3}*\frac{45}{45}\\\\\frac{3(-63t+10)}{135}+\frac{45(t-4)}{135}\\\\\frac{-189t+30}{135}+\frac{45t-180}{135}\\\\\frac{-189t+30+45t-180}{135}[/tex]

Once you have it combined into only one expression, simplify the expression.

[tex]\frac{-189t+30+45t-180}{135}\\\\\frac{-144t-150}{135}\\\\\frac{-6(24t-25)}{135}\\\\\frac{-2(24t-25)}{45}\\\\[/tex]

The slope of a line is a measure of its steepness and is defined as the ratio of the change in y (vertical distance) to the change in x (horizontal distance) between any two points on the line.

We can use the point-slope form of the equation of a line to find the equation of the line that passes through the point (2, 5) and has a slope of -3. The point-slope form of the equation of a line is:

y - y1 = m(x - x1)

where m is the slope of the line, and (x1, y1) is a point on the line.

Substituting the values we have, we get:

y - 5 = -3(x - 2)

Simplifying this equation, we get:

y - 5 = -3x + 6

Adding 5 to both sides, we get:

y = -3x + 11

Therefore, the equation of the line that passes through the point (2, 5) and has a slope of -3 is y = -3x + 11.

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If the quadratic function f(x) has roots of 3 and 7, and it passes through the point (1, 12), the correct vertex form of the equation is f(x) = (x-5)² - 4. So, correct option is D.

Since the quadratic function has roots of 3 and 7, we can write it in factored form as f(x) = a(x-3)(x-7), where "a" is a constant.

To find the value of "a", we can use the point (1,12) that the function passes through.

Substituting x=1 and y=12 in the equation f(x) = a(x-3)(x-7), we get:

12 = a(1-3)(1-7)

12 = a(-2)(-6)

12 = 12a

a = 1

Therefore, the equation of the function in factored form is f(x) = (x-3)(x-7), and we can expand it to vertex form by completing the square:

f(x) = (x-5)² - 4

The vertex of the parabola is at (5, -4) and the coefficient of the squared term is positive, which means the parabola opens upwards.

So, correct option is D.

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The article's development of key ideas by showcasing the importance of precise criteria and the ever-evolving nature of scientific knowledge. The IAU's decision on Pluto serves as a prime example of these concepts in action.

Lines 25 and 26 contribute to the development of key ideas in the article "Ideas That Work... And Those That Don't" from "When is a Planet Not a Planet? The Story of Pluto" by providing a comparison between two concepts in the field of astronomy. These lines state, "The IAU's decision to demote Pluto was based on a more nuanced understanding of what constitutes a planet."

Firstly, this statement demonstrates that the International Astronomical Union (IAU) relies on a "nuanced understanding" to define what a planet is, emphasizing the importance of precise criteria in categorizing celestial bodies. The IAU's decision-making process shows that some ideas work, such as establishing clear guidelines for planetary classification, while others don't, like the previous, less accurate definition of a planet.

Secondly, by mentioning the demotion of Pluto, the article highlights the impact of changing scientific perspectives on our understanding of the solar system. This reinforces the idea that knowledge is constantly evolving, and sometimes previously accepted ideas must be reconsidered or discarded to make way for more accurate information.

In summary, lines 25 and 26 contribute to the article's development of key ideas by showcasing the importance of precise criteria and the ever-evolving nature of scientific knowledge. The IAU's decision on Pluto serves as a prime example of these concepts in action.

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to have $110,000 available for the child's education in 12 years at 6% interest, we need to set aside $54,661.55 now. This assumes that the interest rate remains constant over the 12-year period, which may not be the case in reality.

To calculate how much money must be set aside at 6% interest, we can use the future value formula:

[tex]FV = PV*(1 + r)^n[/tex]

where:
FV = future value (the amount of money we want to have in the future)
PV = present value (the amount of money we need to set aside now)
r = interest rate per period (in this case, per year)
n = number of periods (in this case, the number of years until the child starts college)

In this case, we want to have $110,000 in the future, the child is 6 years old now, and we assume that the child will start college at age 18, which is 12 years from now. The interest rate is given as 6% per year.

So, we can plug in the values into the formula:

[tex]110,000 = PV*(1 + 0.06)^12[/tex]

Simplifying the right-hand side of the equation:

110,000 = PV x 2.012594

Dividing both sides by 2.012594:

PV = 110,000 / 2.012594

PV = $54,661.55 (rounded to the nearest cent)

Therefore, to have $110,000 available for the child's education in 12 years at 6% interest, we need to set aside $54,661.55 now. This assumes that the interest rate remains constant over the 12-year period, which may not be the case in reality.
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The lines are tangent,     The value of x is -7.

To determine the value of x, we need to apply the Tangent-Chord Angle Theorem, which states that the measure of an angle formed by a chord and a tangent line is equal to half the difference of the intercepted arc measures.

In this case, let's call the angle WZY as angle A and angle WYX as angle B. According to the given information, angle A = 2x + 9, and angle B = 3x + 6.

Since , we can apply the Tangent-Chord Angle Theorem:

Angle A = (1/2) * (arc WY - arc ZX)

Substitute the given angle measures:

2x + 9 = (1/2) * ((3x + 6) - (2x + 9))

Multiply both sides by 2 to get rid of the fraction:

4x + 18 = (3x + 6) - (2x + 9)

Simplify the equation:

4x + 18 = x - 3

Subtract x from both sides:

3x + 18 = -3

Subtract 18 from both sides:

3x = -21

Divide both sides by 3:

x = -7

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1. The each ingredient amount will be multiplied by 2.5.

2. [tex]\frac{5}{8}[/tex] ounces of yeast will make 30 rolls.

3.  [tex]\frac{5}{6}[/tex] cup of powdered milk will make 30 rolls.

4. 15 tablespoons of butter will make 30 rolls.

5.  [tex]\frac{15}{2}[/tex] more cups of bread flour than whole wheat flour will be needed to make 30 rolls.

6. The original recipe has been multiplied by 1.5 and this will serve 18 people.

The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

All real numbers are thought to be sufficiently described by the four basic operations, also referred to as "arithmetic operations." Quotient, product, sum, and difference are the mathematical operations that come after division, multiplication, addition, and subtraction.

1. We are given that the given recipe makes 12 rolls.

In order to make 30 rolls, the amount will be multiplied by 2.5 because when 12 is multiplied by 2.5, we get 30.

2. For 12 rolls, [tex]\frac{1}{4}[/tex] ounce package is required.

So, using the multiplication operation, we get

⇒ [tex]\frac{1}{4}[/tex] * 2.5 = [tex]\frac{5}{8}[/tex]

So,  [tex]\frac{5}{8}[/tex] ounces of yeast will make 30 rolls.

3. For 12 rolls, [tex]\frac{1}{3}[/tex] cup of powdered milk is required.

So, using the multiplication operation, we get

⇒ [tex]\frac{1}{3}[/tex] * 2.5 = [tex]\frac{5}{6}[/tex]

So,  [tex]\frac{5}{6}[/tex] cup of powdered milk will make 30 rolls.

4. For 12 rolls, 6 tablespoons of butter is required.

So, using the multiplication operation, we get

⇒ 6 * 2.5 = 15

So,  15 tablespoons of butter will make 30 rolls.

5. 4. For 12 rolls, 3 [tex]\frac{1}{2}[/tex] cups of bread flour is required.

So, using the multiplication operation, we get

⇒ 3 [tex]\frac{1}{2}[/tex]  * 2.5 = [tex]\frac{35}{4}[/tex]

Similarly, for 12 rolls,  [tex]\frac{1}{2}[/tex] cup of wheat flour is required.

So, using the multiplication operation, we get

⇒ [tex]\frac{1}{2}[/tex]  * 2.5 = [tex]\frac{5}{4}[/tex]

Now, using the subtraction operation, we get

⇒ [tex]\frac{35}{4}[/tex] -  [tex]\frac{5}{4}[/tex] =  [tex]\frac{30}{4}[/tex]

⇒ [tex]\frac{35}{4}[/tex] -  [tex]\frac{5}{4}[/tex] = [tex]\frac{15}{2}[/tex]

So, [tex]\frac{15}{2}[/tex] more cups of bread flour than whole wheat flour will be needed to make 30 rolls.

6. Let the number be x.

So,

⇒ [tex]\frac{1}{4}[/tex]x = [tex]\frac{3}{8}[/tex]

⇒ x = 1.5

So, the original recipe has been multiplied by 1.5.

Now,

⇒ Number of people served = 12 * 1.5

⇒ Number of people served = 18

So, this will serve 18 people.

Hence, the required solutions have been obtained.

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The complete question has been attached below.

Question:

Recipe:

Cinnamon Rolls

makes 12 rolls

1/4 -ounce package active dry yeast

1 cup warm water

3 tablespoons molasses

1/3 cup powdered milk

1 egg yolk,beaten

1 egg white, beaten

3 1/2 cups bread flour, divided

6 tablespoons butter, melted and divided

1 teaspoon salt

1/2 cup whole wheat flour

1/3 cup raisins

1 large egg, beaten

​​​​​​​1/4 cup water

You will use the recipe above to answer the following questions:

Respond to the questions about the second recipe:

1. This recipe makes 12 rolls, but you need to make 30 rolls. What number will you need to multiply the amount of each ingredient by to adjust the recipe? How did you determine this number?

2. How many ounces of yeast will you need to make 30 rolls?

3. How many cups of powdered milk will you need to make 30 rolls?

4. How many tablespoons of butter will you need to make 30 rolls?

5. How many more cups of bread flour than whole wheat flour will you need to make 30 rolls?

6. notice that the original recipe calls for 1/4 cup of water. If you put in 3/8 cup of water, by how much have you multiplied the original recipe? How many people will your new recipe serve?

Answer:

8n³ + 3n² + 8

Step-by-step explanation:

To find the sum of the two polynomials, we can simply add the corresponding terms:

(-5n^4 + 8) + (5n^4 + 8n³ + 3n²)

Combine the like terms:

(-5n^4 + 5n^4) + 8n³ + 3n² + 8

The -5n^4 and 5n^4 terms cancel each other out:

0 + 8n³ + 3n² + 8

So the expanded polynomial in standard form is:

8n³ + 3n² + 8

It would take approximately 1.3 hours for 12 rice to reach the velocity of Earth's orbit at a velocity of 23 km/hr.

The velocity of Earth's orbit around the sun is approximately 30 km/s.

To determine how many hours it takes for 12 rice to reach this velocity, you must use the formula for calculating average speed:

Average Speed = Total Distance/Total Time

Since the total distance is the speed of Earth's orbit (30 km/s) and the total time is the amount of time it takes to reach this speed, then you can solve for the total time by rearranging the equation:

Total Time = Total Distance/Average Speed

Therefore, the total time it takes to reach the velocity of Earth's orbit is:

Total Time = 30 km/s/23 km/hr

= 1.3 hours

It would take approximately 1.3 hours for 12 rice to reach the velocity of Earth's orbit at a velocity of 23 km/hr.

This is because the average speed of the rice is equal to the total distance (30 km/s) divided by the total time (1.3 hours).

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The percentage change in attendance after rounding off the answer to the nearest hundredth of a percent will be approximately 48.78%.

To find the percentage change in attendance, we need to calculate the difference between the new and old attendance, divide by the old attendance, and then multiply by 100 to express the result as a percentage:

percentage change = (new attendance - old attendance) ÷ old attendance × 100%

Plugging in the given values, we get:

percentage change = (61 - 41) ÷ 41 × 100%

percentage change = 20 ÷ 41 × 100%

percentage change = 0.4878 × 100%

percentage change ≈ 48.78%

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To find when the apple will reach the ground, we need to find when the height of the apple is zero.

So, we set h(t) = 0 and solve for t:

-5t^2 + 3t + 1 = 0

Using the quadratic formula, we get:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = -5, b = 3, and c = 1.

Plugging in the values, we get:

t = (-3 ± sqrt(3^2 - 4(-5)(1))) / 2(-5)

Simplifying:

t = (-3 ± sqrt(49)) / (-10)

t = (-3 ± 7) / (-10)

t = 0.4 or t = 1.3

We discard the negative value since time cannot be negative, so the apple will reach the ground after approximately 1.3 seconds.

Therefore, it will take 1.3 seconds for the apple to reach the ground.

Hope this helped (:

the correct answer is A. LM is translated 5 units left and 2 units down to L'M', which is then obtained by reflecting LM over the x-axis. The coordinates of L'M' are 5X, -B, 5+8, -4, 7, -2, 11, 13, 15, -M, -M.

How to transform line?

To transform the LM to L'M', we need to apply a combination of translation and reflection operations. The given coordinates of LM are 10X, B, 10-8, 6, -2, 4, -6, -8, -10, M, M, and we need to translate them 5 units left and 2 units down.

Translation is a geometric transformation that moves each point of an object a fixed distance in a specified direction. To translate the LM 5 units left and 2 units down, we subtract 5 from the x-coordinates and 2 from the y-coordinates. The new coordinates become 5X, B, 5-8, 4, -7, 2, -11, -13, -15, M, M.

Next, we need to reflect the translated LM over an axis to obtain L'M'. A reflection is a transformation that flips a shape over a line or plane. We have two choices here, reflecting over the x-axis or the y-axis.

If we reflect the translated LM over the x-axis, the y-coordinates will change sign. The new coordinates become 5X, -B, 5+8, -4, 7, -2, 11, 13, 15, -M, -M. This is L'M', obtained by reflecting LM over the x-axis.

On the other hand, if we reflect the translated LM over the y-axis, the x-coordinates will change sign. The new coordinates become -5X, B, -5+8, 4, 7, 2, 11, 13, 15, M, M. This is not L'M', as it is reflected over the wrong axis.

Therefore, the correct answer is A. LM is translated 5 units left and 2 units down to L'M', which is then obtained by reflecting LM over the x-axis. The coordinates of L'M' are 5X, -B, 5+8, -4, 7, -2, 11, 13, 15, -M, -M.

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Answer:

Let's start with the distance each person covers in 3 minutes.

Beth's distance = 70 meters/minute x 3 minutes = 210 meters

Liz's distance = 60 meters/minute x 3 minutes = 180 meters

Now we can find their distance apart after 3 minutes.

Distance apart = Initial distance - distance covered

Distance apart = 520 meters - (210 meters + 180 meters)

Distance apart = 520 meters - 390 meters

Distance apart = 130 meters

Therefore, Beth and Liz are 130 meters away from each other after walking towards each other for 3 minutes.

(A) Two critical number are x = 1, x = -2

(b) Function is increasing on the intervals (-∞,-2) and (1,∞), and decreasing on the interval (-2,1).

(c) f'(x) changes sign at x = -2, this critical number corresponds to a relative maximum and  f'(x) also changes sign at x = 1, this critical number  a relative minimum.

(d) The graph is given in the attachment.

The derivative of function is the rate of change of a function at a specific point.

(a) find the values of x where the derivative of the function equals zero or does not exist.

[tex]$f'(X)=6X^{2}+6X-12=6(X^{2}+X-2)$[/tex]

Setting f'(x) equal to zero gives:

[tex]$6(X^{2}+X-2)=0$[/tex]

[tex]$X^{2}+X-2=0$[/tex]

(X + 2)(X - 1) = 0

So the critical numbers are x = -2 and x = 1.

(b)  we need to examine the sign of the derivative on either side of the critical numbers.

For x < -2, f'(x) is positive, indicating that the function is increasing.

For -2 < x < 1, f'(x) is negative, indicating that the function is decreasing.

For x > 1, f'(x) is positive, indicating that the function is increasing.

Therefore, the function is increasing on the intervals (-∞,-2) and (1,∞), and decreasing on the interval (-2,1).

(c)  identify all relative extrema, we examine the sign of the derivative in the neighborhood of each critical number. Since f'(x) changes sign at x = -2, this critical number corresponds to a relative maximum. Since f'(x) also changes sign at x = 1, this critical number  a relative minimum.

(d) The graph is given in the attachment.

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Step-by-step explanation:

The factored form of the polynomial is:

f(x) = (x-2)³(x+4)³(x+6)⁶

From this, we can see that the zeros of the function are: x = 2 (with multiplicity 3), x = -4 (with multiplicity 3), and x = -6 (with multiplicity 6).

Therefore, the correct answer is:

-6 with multiplicity 6; x = -4 with multiplicity 3; x = 2 with multiplicity 3.

Answer:

the answer is the 3rd one and I am 100% sure