Determine the interval where each function is continuous

Denominator 4 can be used to represent a distance less than 7/12 mile.

When working with fractions, the denominator represents the number of equal parts that make up a whole unit. In this case, the denominator of 4 means that the whole unit is divided into 4 equal parts. To represent a distance that is less than 7/12 mile, we need to find a fraction that has a smaller value. One way to do this is to simplify 7/12 by dividing both the numerator and denominator by their greatest common factor, which is 1. This gives us 7/12 = 21/36. To find a denominator of 4, we need to divide both the numerator and denominator of 21/36 by 9, which gives us 7/12 = 7/12 x 3/3 = 21/36 = 21/36 x 1/1 = 7/12 x 3/3 x 1/1 = 21/108. Now, we can see that 21/108 has a denominator of 4 (since 108 is divisible by 4), and its value is less than 7/12. Therefore, we can use denominator 4 to represent a distance less than 7/12 mile.

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At great axial distances x from a current-carrying loop, the magnetic field varies as:

b. x^-3

Here's the step-by-step explanation:

1. The magnetic field B at a point on the axis of a current-carrying loop can be given by the formula: B = (μ₀Ia²) / (2(a² + x²)^(3/2))

2. In this formula, μ₀ is the permeability of free space, I is current, a is the radius of the loop, and x is the axial distance from the loop.

3. When the axial distance x is much greater than the radius a (i.e., x >> a), the term a² in the denominator becomes negligible compared to x².

4. Therefore, the formula simplifies to: B ≈ (μ₀I) / (2x³)

As you can see, the magnetic field B varies as x^-3 at great axial distances from the current-carrying loop.

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The number of nodes in a perfect binary tree of height 5 is 63

Given data ,

Let the binary tree be represented as A

Now , the value of A is

A perfect binary tree of height 5 has 6 levels, including the root node.

The number of nodes in a perfect binary tree of height h can be calculated as n = 2^(h+1) - 1.

Therefore, for a perfect binary tree of height 5, the number of nodes is given by A = 2^(5+1) - 1

On simplifying the equation , we get

A = 63

Hence , the number of nodes is A = 63

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The taylor polynomial of degree n=4 for x near the point a=π for the function cos(x) is P4(x) = −1 + (x−π) − (x−π)^2/2 + (x−π)^3/6 − (x−π)^4/24.

To find the Taylor polynomial of degree 4 for cos(x) near the point a = π, we need to use the Taylor series formula:

cos(x) = ∑(−1)^k/k! (x−π)^k

where we have used the fact that cos(π) = −1 and the Maclaurin series for cos(x):

cos(x) = ∑(−1)^k/(2k)! x^2k.

We can then simplify the formula by only taking the first four terms of the series, as we want the Taylor polynomial of degree 4:

cos(x) ≈ ∑(−1)^k/k! (x−π)^k ≈ −1 + (x−π) − (x−π)^2/2 + (x−π)^3/6 − (x−π)^4/24.

Therefore, the Taylor polynomial of degree 4 for cos(x) near the point a=π is:

P4(x) = −1 + (x−π) − (x−π)^2/2 + (x−π)^3/6 − (x−π)^4/24.

Note that this polynomial is an approximation of cos(x) near x=π and becomes more accurate as x approaches π.

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It is recommended to consult with a licensed appraiser or real estate agent for guidance on how to accurately measure and report total area square footage.

According to national standards, a pool house may be included in total square footage if it meets certain criteria. Firstly, the pool house must have a permanent foundation and be heated and cooled to the same extent as the main house.

Additionally, the pool house must have finished living space that is connected to the main house by a covered walkway or enclosed breezeway. This living space may include a bathroom, bedroom, or kitchen.

However, it is important to note that some appraisers and real estate agents may not include the pool house in the total square footage if it is not considered "livable space" or if it is primarily used for storage or pool equipment. Ultimately, the decision to include the pool house in total square footage may vary depending on local real estate practices and regulations.

It is recommended to consult with a licensed appraiser or real estate agent for guidance on how to accurately measure and report total square footage.

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The unit price of the jump ropes is $0.90 per jump rope.

Unit price is the price at which a single quantity of a product is being sold. This can refer to the price per unit of measure, such as the price per pound, ounce, or pint.

To find the unit price of the jump ropes, we divide the total cost by the number of jump ropes in the pack.

Total cost of the 2-pack of jump ropes: $1.80

Number of jump ropes in the pack: 2

Unit price = Total cost / Number of items

Unit price = $1.80 / 2

Unit price = $0.90

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Given the rain barrel is shaped like a right circular cylinder. Therefore, volume of the rain barrel is approximately 3166.04 cubic inches.

To find the volume of the rain barrel, we can use the formula for the volume of a right circular cylinder, which is V = πr^2h. We are given the height and diameter of the rain barrel, so we can calculate its radius and then use the formula to find its volume.

Explanation:

The diameter of the rain barrel is 22 inches, so its radius is half of that, or 11 inches. The height of the rain barrel is 26 inches. We can now use the formula for the volume of a right circular cylinder to find the volume of the rain barrel:

V = πr^2h

V = π(11^2)(26)

V = π(121)(26)

V = 3166.04 cubic inches

Therefore, the volume of the rain barrel is approximately 3166.04 cubic inches.

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The forecast error for a specific year, such as 2002, we would compare the actual number of births that occurred to the forecasted number for that year. The difference between these two values is the forecast error.

Historical data is essential for forecasting future events such as animal births at a zoo.

Exponential smoothing, with a smoothing constant of 0.30 in this case, is a method that can be used to predict the number of births based on this data.

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We can conclude that the, diameter m GFH = 21° and m GZH = 116°.

In geometry, the diameter of a circle is any straight line segment that has an endpoint on the circle and travels through its center.

Since the diameter is the length of the line that runs tangent to the two points at either end of the circle. Thus, diameter of a circle is the distance between its two furthest points.

We are given that angle EGF is a right angle (90°) because EG is a diameter.

We have been given that m FG = 116°, we can calculate m FGE by subtracting it from 180°;

m FGE = 180 degrees - m FG = 180 degrees - 116 degrees

m FGE = 64 degrees

m GFH = 180° - m ZGF - m ZGFH = 180° - 90° - 69°

m GFH = 21°

Finally, we can find m GZH by subtracting m FGE from 180° (the sum of triangle angles):

180° m GZH - 180°

m FGE - 64° = 116°

Therefore, m GFH = 21° and m GZH = 116°.

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The surface area of the sphere is approximately 615.75 mm².

The surface area of a sphere with radius r is given by the formula:

SA = 4πr²

Substituting r = 7 mm and using π ≈ 3.14, we get:

SA = 4π(7)²

SA = 4π(49)

SA ≈ 615.75 mm²

Thus, Rounding to the nearest hundredth, the surface area of the sphere is approximately 615.75 mm².

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The point scored by the football team in the second half was 21 points.

The amount in Debbie's savings account is $ 834. 00.

The hourly pay rate for Harry is $ 5. 70 per hour.

The points scored by the football team can be found by the formula :

= Total scored - points in first half

= 45 - 24

= 21 points

The amount that Debbie has in her account would be :

= Amount Lonna has x 3

= 278 x 3

= $ 834.00

The pay rate per hour for Harry is:

= 171 / 3

= $ 5. 70 per hour

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A revenue function is a mathematical formula that represents the total income earned by a business or organization at different levels of output or sales. It is typically used in economics and finance.

(a) To find the revenue function, R(x), you need to multiply the number of memory sticks sold (x) by the price per unit (p). Using the given price function, p = 80000/x + 6. So,

R(x) = x * (80000/x + 6)

(b) To determine the cost function, C(x), use the financial department's estimates. The weekly fixed cost is 1500 dollars, and the additional cost per unit is 21 dollars. Therefore,

C(x) = 1500 + 21x

(c) To find the profit function, P(x), subtract the cost function from the revenue function:

P(x) = R(x) - C(x) = (x * (80000/x + 6)) - (1500 + 21x)

(d) To find the marginal profit when 20 memory sticks are produced and sold each week, you need to first find the derivative of the profit function with respect to x (P'(x)), and then evaluate it at x=20:

P'(x) = d/dx[(x * (80000/x + 6)) - (1500 + 21x)]

After calculating the derivative, evaluate P'(20) to find the marginal profit at 20 memory sticks produced and sold per week.

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The main difference between finitely repeated and infinitely repeated games is the number of times the game is played.

Players in a game that is finitely repeated have a limited amount of opportunities to observe and react to each other's actions. Depending on the number of rounds and their perceptions of the other player's actions, players can employ a range of strategies.

Players can see and react to each other's activities an endless number of times in an infinitely repeated game. Now that long-term collaboration and punishment are options, this alters the methods that players can employ.

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Tyler drank a total of 9 2/21 liters of water altogether.  

Tyler drank a fraction of 9 5/21 liters of water altogether. To find the total amount of water Tyler drank, we need to add the 2 3 liters of water he drank before jogging and the 3 7 liters of water he drank after jogging. To add these fractions, we need to find a common denominator, which is 21.

For the first fraction, we need to multiply both the numerator and denominator by 7 to get 14/21. For the second fraction, we need to multiply both the numerator and denominator by 3 to get 9/21. Now, we can add these two fractions:

14/21 + 9/21 = 23/21

This is an improper fraction, so we can simplify it to a mixed number by dividing the numerator by the denominator:

23 ÷ 21 = 1 remainder 2

So, Tyler drank a total of 9 2/21 liters of water altogether.

Drinking enough water is essential for staying hydrated and healthy, especially during exercise. It is recommended to drink at least 8 cups (64 ounces or 1.9 liters) of water per day, but this can vary depending on factors such as age, gender, weight, and physical activity level. Drinking enough water can help prevent dehydration, regulate body temperature, flush out toxins, and improve overall health and well-being.
 

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Only the second inequality and fourth inequality is true when x = -8, all other inequalities are false.

We have x = -8, now we will put this value in each inequality and check if it is true.

First inequality is 1/2x + 12 ≤ -4.

1/2 × -8 + 12 ≤ 4

-4 + 12 ≤ 4

8 ≤ 4. This is clearly false.

Second inequality is -2x + 9 ≥ 25

-2 × -8 + 9 ≥ 25

16 + 9 ≥ 25

25 ≥ 25. This statement is true.

Third inequality is 3x + 8 > -16

3 × -8 + 8 > -16

-24 + 8 > -16

-16 > -16. This is also not true because -16 = -16.

Fourth inequality is -1/8x - 2 > -3

- 1 / 8 × -8 - 2 >-3

1 - 2 > -3

-1 > -3. This is also true.

Fifth inequality is -x + 9 < 12

-(-8) + 9 <12

8 + 9 < 12

17 < 12. This statement is also false.

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The decision on whether to reject the null hypothesis would be we A. reject the null hypothesis.

Calculate the test statistic (z) using the formula:

z = (x - μ) / (σ / √n)

z = ( 0.47 - 0.5 ) / (0.06 / √76)

z = - 3.29

Determine the critical z - values for a two-tailed test at 1 % significance level (α = 0. 01 ).

z  -critical = ± 2. 576

Since our test statistic z = - 3. 29 is less than the lower critical z-value - 2. 576, we reject the null hypothesis (H₀).

We reject the null hypothesis because there is enough evidence at a 1% significance level to conclude that the cheeseburgers do not have exactly a half pound of meat.

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The error between the given expression is of 2 units.

The given expressions are

y = -2x+4  ....(i)

y = -2x+6  ...(ii)

To find the error

Subtract first expression from the second expression

As follows

⇒(-2x+6) - (-2x+4)

⇒-2x + 2x + 6 - 4

⇒ 2

Hence, Error = 2 unit.

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There are 846720 different values possible for the sum of xyx and yxy.

Let's denote the three digits of xyx as a, b, and c, such that xyx = 100a + 10b + c, and the three digits of yxy as d, e, and f, such that yxy = 100d + 10e + f. Note that x and y are distinct non-zero digits, so a, b, c, d, e, and f are all distinct non-zero digits.

The sum of xyx and yxy is (100a + 10b + c) + (100d + 10e + f), which simplifies to 100(a+d) + 20(b+e) + (c+f).

We want to find how many different values are possible for the sum. Since a, b, c, d, e, and f are all distinct non-zero digits, we can consider each of them separately.

For a given value of a, there are 9 choices for d (since d cannot be equal to a), and once we have chosen d, there are 8 choices for e (since e cannot be equal to either a or d). Similarly, there are 7 choices for f (since f cannot be equal to a, d, or e).

So, for a fixed value of a, the number of possible values of the sum is the number of possible values of (100(a+d) + 20(b+e) + (c+f)), which is simply the number of possible values of (20(b+e) + (c+f)), since 100(a+d) is fixed.

There are 8 choices for b (since b cannot be equal to a), and once we have chosen b, there are 7 choices for c (since c cannot be equal to either a or b). Similarly, there are 6 choices for e (since e cannot be equal to either a, d, or b), and 5 choices for f (since f cannot be equal to either a, d, e, or c).

Therefore, the total number of possible values of the sum is:

9 × 8 × 7 × 8 × 7 × 6 × 5 = 846720

Therefore, there are 846720 different values possible for the sum of xyx and yxy.

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The distinguishing feature of ecological economics is the concept of steady-state economies that neither grow nor shrink.

This concept recognizes the finite nature of natural resources and the need to balance economic activity with ecological limits.

Unlike traditional neoclassical economics, which assumes that growth is always desirable, ecological economics emphasizes the need for sustainable development that maintains ecological integrity while meeting human needs.

Additionally, ecological economics recognizes that environmental damage and benefits are external to the market system and must be internalized through policies like carbon taxes and cap-and-trade systems.

While cost-benefit analysis is an important tool in ecological economics, it is used within the framework of steady-state economics and sustainable development rather than as a justification for unlimited economic growth.

Finally, ecological economics rejects greenwashing as a means of increasing public acceptance of products, recognizing that true sustainability requires fundamental changes in economic and social systems.

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The answer to the question is that the grating has approximately 1.155 lines per centimeter.

The number of lines per centimeter for a grating can be calculated using the formula:
n = d * sinθ / λ

where n is the number of lines per centimeter, d is the spacing between the grating lines, θ is the angle at which a certain order appears, and λ is the wavelength of the incident light.

In this case, we are given that band 4 appears at an angle of 60 degrees. We can assume that this is the first order, meaning that n = 1.

We are not given the wavelength of the incident light, so we cannot solve for d directly.
However, we can use some trigonometry to find sinθ. Since we know that θ = 60 degrees, we can use the sin(60) = √3/2.

We can then rearrange the formula to solve for d:
d = n * λ / sinθ

Substituting n = 1 and sinθ = √3/2, we get:
d = λ / (√3/2)

We still cannot solve for d without knowing the wavelength of the incident light.

However, we can find the ratio of d to λ, which is equal to the number of lines per centimeter:

n = d / λ = 1 / (√3/2) ≈ 1.155 lines per centimeter

So the answer to the question is that the grating has approximately 1.155 lines per centimeter.

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A metric is a single, quantifiable type of data that can be utilized for various tasks in different domains. Metrics serve as essential tools for measuring progress, assessing performance, and making informed decisions.

By converting complex information into manageable and comparable units, metrics enable individuals and organizations to monitor their objectives and drive improvement.
In business, metrics are often employed to evaluate the success of marketing campaigns, employee performance, or financial growth. They assist in identifying trends, pinpointing areas that need attention, and guiding strategic planning. Key performance indicators (KPIs) are a common type of metric used to measure the effectiveness of business processes and track progress toward organizational goals.
In scientific research, metrics provide valuable insight into the progress and impact of studies. For example, citation metrics measure the influence of a published work within a given field. Metrics also facilitate the assessment of experimental results and the development of new theories, ultimately advancing scientific knowledge.
In education, metrics serve as essential tools for gauging student performance and the quality of educational programs. They provide data on student learning outcomes, helping educators refine their teaching methods and tailor instruction to the needs of individual learners.
Overall, a metric is a versatile, quantifiable data type that plays a vital role in assessing and improving various aspects of performance across multiple domains. By offering concrete, measurable information, metrics empower decision-makers to make informed choices and drive progress.

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As a result, the line integral ∫C F · dR depends on the path C connecting A(0, 0) to B(1, 1) and cannot be computed using just the endpoints.

To find a scalar potential function f, we need to check if the vector field F satisfies the condition of being conservative, which means that it must be the gradient of some scalar function.

Taking the partial derivative of F with respect to y, we get:

∂F/∂y = 1 + 2y

Taking the partial derivative of F with respect to x, we get:

∂F/∂x = -2x

Since these derivatives are not equal, F is not a conservative vector field.

Therefore, we cannot find a scalar potential function f for F.

As a result, the line integral ∫C F · dR depends on the path C connecting A(0, 0) to B(1, 1) and cannot be computed using just the endpoints.

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The line integral over any smooth path C connecting A(0, 0) to B(1, 1) is 2/3.

To find a scalar potential function f, we need to find a function whose gradient is equal to F.

Taking the partial derivative of F with respect to x, we get:

∂/∂x f(x, y) = y - x^2

Integrating with respect to x, we get:

f(x, y) = xy - (1/3) x^3 + g(y)

Taking the partial derivative of f with respect to y, we get:

∂/∂y f(x, y) = x + y^2

Comparing with the y-component of F, we see that g(y) = (1/3) y^3. Therefore, the potential function is:

f(x, y) = xy - (1/3) x^3 + (1/3) y^3

To evaluate the line integral over any smooth path C connecting A(0, 0) to B(1, 1), we can use the fundamental theorem of line integrals. This theorem states that the line integral of a conservative vector field F along any smooth path between two points A and B depends only on the values of the scalar potential function f at those two points. Therefore, we have:

∫C F · dR = f(B) - f(A)

Plugging in A(0, 0) and B(1, 1), we get:

∫C F · dR = f(1, 1) - f(0, 0)

= (1)(1) - (1/3)(1)^3 - (0)(0) + (1/3)(0)^3

= 2/3

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The number of non-fiction books sold on Friday will be two-and-a-half times the number of arts and crafts books sold on Friday. Therefore, option B is the correct answer.

We can see from the table that on Thursday, 7 sports and trivia books and 19 arts and crafts books were sold, for a difference of 12.

On Thursday, 13 non-fiction books and 19 arts and crafts books were sold. If we assume that the same ratio will hold on Friday, then we can predict that the number of non-fiction books sold will be (19/2)×2.5 = 23.75, which is not a whole number. Therefore, this prediction is not supported by the data.

Therefore, option B is the correct answer.

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The possible number of passengers on the flight over the past 5 days can range from 72 to 92 passengers

Since the mean number of passengers on a flight from Los Angeles to San Francisco is 82, we can assume that this is the average number of passengers over a period of time. To list the possible number of passengers on the flight over the past 5 days, we can use the range of values that are within one standard deviation of the mean.

Assuming a normal distribution of passengers on the flight, we can use the empirical rule to estimate the range of values within one standard deviation of the mean. The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean.

Therefore, the range of possible values for the number of passengers on the flight can be estimated as:

Mean - Standard deviation <= Number of passengers <= Mean + Standard deviation

Since we do not have the standard deviation of the data, we can use a conservative estimate of the standard deviation as 10% of the maximum capacity of the plane, which is 102 passengers. Therefore, the standard deviation can be estimated as 10.2 passengers.

Substituting these values into the formula, we get:

82 - 10.2 <= Number of passengers <= 82 + 10.2

Simplifying this inequality, we get:

71.8 <= Number of passengers <= 92.2

Therefore, the possible number of passengers on the flight over the past 5 days can range from 72 to 92 passengers, assuming a normal distribution of passengers and a standard deviation of 10.2 passengers.

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Silvia will save $330 in taxes by deducting her student loan interest from her taxable income.

We must first determine her taxable income.

Her total annual income is $50,577.

Next, we need to deduct her gross income from the interest she paid on her student loans:

$50,577 - $1,500 = $49,077

Her taxable income is $49077.

We need to know her tax rate in order to determine how much tax she will save. Assume that her tax rate is 22% based on the single filer tax brackets for 2021.

So the amount she will save in taxes is:

$1,500 x 0.22 = $330

Therefore, Silvia will save $330 in taxes by deducting her student loan interest from her taxable income.

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A conical tent made of canvas has a base that is 26 feet across and a slant height of 14 feet. The area of the canvas as 1102 square feet.

The radius of the circular base of the tent is half of the diameter, which is 26/2 = 13 feet.

Using the Pythagorean theorem, we can find the height of the cone as follows:

height = √(slant height^2 - radius^2)

height = √(14^2 - 13^2)

height ≈ 5 feet

The lateral area of the cone (excluding the base) is given by:

lateral area = πrs

lateral area = π(13)(14)

lateral area ≈ 571.47 square feet

The area of the circular base is:

base area = πr^2

base area = π(13)^2

base area ≈ 530.93 square feet

The total area of the canvas, including the floor, is the sum of the lateral area and the base area:

total area ≈ lateral area + base area

total area ≈ 571.47 + 530.93

total area ≈ 1102.40 square feet

Rounding this to the nearest whole unit, we get the area of the canvas as 1102 square feet.

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Rounding to the nearest hundredth using π = 3.14, the difference in volume is approximately 29.87 cm³.

The difference in the volume of the regulation soccer balls can be found using the formula for the volume of a sphere, which is V = (4/3)πr³, where r is the radius of the sphere.

The radius of the first soccer ball is (21.66/2) = 10.83 cm, and the radius of the second soccer ball is (22.29/2) = 11.145 cm.

Substituting the values, we get:

V₁ = (4/3)π(10.83)³ = 6576.67 cm³

V₂ = (4/3)π(11.145)³ = 7066.67 cm³

The difference in volume is:

V₂ - V₁ = (4/3)π(11.145)³ - (4/3)π(10.83)³

= (4/3)π[(11.145)³ - (10.83)³]

= (4/3)π(0.315)(3.007)(24.174)

= 9.52π

Therefore, the difference in the volume of the regulation soccer balls is approximately 29.87 cm³.

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In the least-squares regression line equation, the slope is defined as the change in the response variable to unit change in the explanatory variable. So, option (a) is right one.

A regression line is a linear line that best fits the data, but this does not mean that the fit is good. In some cases, there can still a lot of variability. The slope of least-squares regression line is change in the response variable by a unit change in the explanatory variable. Because a regression line will only pass through all the data points if r = ± 1, the slope is defined as the predicted change in the response variable by a unit change in the explanatory variable. This line is always passes through the means points of the explanatory and response variables respectively. So, the correct option is option(a).

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Answer: X = 18°

Step-by-step explanation:

Apply the cyclic quadrilaterals rule from circle theorems.

hence, opposite angles add up to 180

°

So ,

180

° - 144° = 36

2x = 36

x = 36/2

= 18°

Answer:

Quotient = 3x-5

Remainder = 3

Step-by-step explanation:

For any long division, I use the DMS principle, divide, multiply, subtract.

(x+6) | 3x² + 13 x - 27

First of all, divide the first part of the equation by x. (which is 3x²)

You get 3x.

Now take your 3x and multiply it by (x+6)

You get 3x²+18x. You need to subtract this from your equation.

(3x² + 13 x - 27) - (3x² + 18x) = -5x - 27

One round of DMS complete

Now,

(x+6) | - 5x - 27

We again divide the first part only by x again. You get -5

We multiply -5 by (x+6)

-5x-30.

Let's subtract it from the equation.

(-5x-27) - (-5x-30) = - 5x - 27 + 5x + 30 = 3

Second round of DMS complete.

We cannot divide 3 further by (x+6) so this is it.

The remainder is 3.

Now we take our answers, which is 3x and - 5 and add them to get 3x-5. This is our quotient.