what is 8 x 1 ????????????

Answer:

15.

The trick to working out 5 divided by 1/3 is similar to the method we use to work out dividing a fraction by a whole number.All we need to do here is multiply the whole number by the numerator and make that number the new numerator. The old numerator then becomes the new denominator.

So, the answer to the question "what is 5 divided by 1/3?" is
15!

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Answer:

0.03

Step-by-step explanation:

So we can tell that our population parameter is p-hat = 0.637, which is 0.03 apart from the upper and lower bounds of the confidence interval. This means that 0.03 is our margin of error, or how much room there is for error.

The correct answer to the given question is:

Null hypothesis: There is no difference in the satisfaction levels of patients between the downtown and uptown offices.

Alternative hypothesis: There is a difference in the satisfaction levels of patients between the downtown and uptown offices.

The null hypothesis is an assumption that is tested in order to determine its validity.

In this case, the null hypothesis is that there is no difference in the satisfaction levels of patients between the downtown and uptown offices.

This is an educated guess because Amanda has no way of knowing if there is a difference until she collects the survey results and analyzes them.

The alternative hypothesis is the opposite of the null hypothesis. In this case, the alternative hypothesis is that there is a difference in the satisfaction levels of patients between the downtown and uptown offices.

This is based on the survey results where the downtown office staff had 84% of clients reporting satisfaction with services, while the uptown office staff had 76% of clients reporting satisfaction with services.

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Null hypothesis: There is no difference in the satisfaction levels of patients between the downtown and uptown offices.

Alternative hypothesis: There is a difference in the satisfaction levels of patients between the downtown and uptown offices.

The null hypothesis is an assumption that is tested in order to determine its validity.

In this case, the null hypothesis is that there is no difference in the satisfaction levels of patients between the downtown and uptown offices.

This is an educated guess because Amanda has no way of knowing if there is a difference until she collects the 600 survey results and analyzes them.

The alternative hypothesis is the opposite of the null hypothesis.

In this case, the alternative hypothesis is that there is a difference in the satisfaction levels of patients between the downtown and uptown offices.

This is based on the survey results where the downtown office staff had 84% of clients reporting satisfaction with services, while the uptown office staff had 76% of clients reporting satisfaction with services.

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Total number of steps taken by an average man in a year while walking with 7192 steps a day is equals to 2,625,080 steps/year.

Number of steps taken by average man in a day is equals to  7192

Then the total number of steps he takes in a year is equals to,

Calculate it by multiplying the average number of steps per day by the number of days in a year.

There are different ways to define a year,

But assuming a regular calendar year of 365 days, the calculation would be,

Total number of days in a year = 365 days

Total number of steps in a year

= 7192 steps/day x 365 days/year

= 2,625,080 steps/year

Therefore, on average the man would walk about 2,625,080 steps in a year.

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The given question is incomplete, I answer the question in general according to my knowledge:

If a man walks with random steps and the average man takes 7192 steps a day about how many steps does the average man take in a year?

Method to list all non-isomorphic trees on n vertices is to add edges to a single vertex tree. Using A, B, C, D, E, we list 5 non-isomorphic trees on 6 vertices.

A by-hand method to give a list of all non-isomorphic trees on exactly n vertices is to start with a tree on n vertices and then generate all possible trees by adding edges between vertices that are not already connected.

For example, to find all non-isomorphic trees on 4 vertices, we can start with a single vertex and then add edges to form a tree with 2 vertices, then add edges to form a tree with 3 vertices, and finally add edges to form a tree with 4 vertices. We can then check each tree for isomorphism by comparing their adjacency matrices.

Using the method from (a), we can find all non-isomorphic trees on exactly six vertices by starting with a single vertex and adding edges until we have a tree on six vertices.

To ensure that we generate all possible trees, we can use the following five vertices: A, B, C, D, E. We can then generate all trees by adding edges between vertices that are not already connected, making sure to avoid creating cycles. After generating all trees, we can check for isomorphism by comparing their adjacency matrices.

The resulting list of non-isomorphic trees on six vertices, in alphabetical order, is shown. The tree 1 and tree 2 are the same. Also, trees 3, 4, and 5 are not isomorphic to each other or to trees 1 and 2.

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Answer:

A≈1.83m²

Step-by-step explanation:

Using the formulas

A=πr2C=2πr

Solving forAA=C24π=4.824·π≈1.83346m²

Answer:

1.83 (approximate)

Step-by-step explanation:

First, we need to find the radius of the circle.

Using the formula:

[tex]C=2 \pi r[/tex]

We have to reorganize terms:

[tex]r=\frac{C}{2\pi}[/tex]

[tex]\frac{4.8}{2 \times \pi}[/tex]

r ≈ 0.76

Now we have the radius and the circumference in order to find the area.

Use the formulas:

[tex]A= \pi r^2[/tex]

[tex]C=2 \pi r[/tex]

A = C^2/4[tex]\pi[/tex]  = 4.8^2 / 4 * pi = 1.83

Answer:

Loan term of the truck in years:

The loan term of the truck is 60 months. To convert this to years, we divide by 12 since there are 12 months in a year. Therefore, the loan term of the truck in years is:

60 months ÷ 12 months/year = 5 years

Monthly repayments including insurance:

The monthly instalment of the truck is R22,301.67 and the insurance cover is R500. Therefore, the total monthly repayment including insurance is:

R22,301.67 + R500 = R22,801.67

Total amount owed to the financer excluding insurance in June 2022:

Half of the monthly instalments have been made towards the truck in June 2022, which means that 5 months' worth of payments have been made. Therefore, the total amount owed to the financer excluding insurance in June 2022 is:

(R22,301.67 × 55) - R500 = R1,223,592.35

Significance of including insurance cover:

Including an insurance cover when purchasing the water tanker is important because it provides protection for the contractor against unforeseen events such as accidents, theft, or damage to the vehicle. In the event of an accident or theft, the insurance cover can help cover the costs of repairs or replacement of the vehicle, which can be a significant expense for the contractor. Additionally, having insurance can provide peace of mind for the contractor, knowing that they are covered in case of unexpected events.

Tanker driver's daily pay during weekdays:

The driver works for 5½ days per week and 9 hours per day on weekdays, which amounts to:

5.5 days/week × 9 hours/day = 49.5 hours/week

The rate per hour is R92.50, so the driver's pay per day during weekdays is:

49.5 hours/week × R92.50/hour ÷ 5 days/week = R909.75/day

Answer:

shelf area = 19018.33 square inches

Step-by-step explanation:

shelf area calculation:

Ignacio makes a display shelf from 4 wooden boards. All angles formed by the boards are right angles. Ignacio plans to stain all faces of the shelf, except the back face, which will be against the wall. What is the total area Ignacio will stain? Show your work. 512 7 in. 1 2 32 in. 30 in. 72 17 56 9 in. 4 in. 512 24 A 329-30.7 72

To find the total area that Ignacio will stain, we first need to determine the area of each face that will be stained.

Let's label the boards as follows:

Board 1: 512 in x 7 in

Board 2: 12 in x 32 in

Board 3: 30 in x 72 in

Board 4: 17 56/9 in x 4 in

For each board, we need to find the total area of all the faces that will be stained.

Board 1 has two faces that will be stained: the top face and the two side faces. The area of the top face is 512 in x 7 in = 3584 in^2. The area of each side face is 512 in x 12 in = 6144 in^2. So the total area of all three faces is 3584 in^2 + 2 x 6144 in^2 = 15872 in^2.

Board 2 has three faces that will be stained: the top face and the two side faces. The area of the top face is 12 in x 32 in = 384 in^2. The area of each side face is 12 in x 4 in = 48 in^2. So the total area of all three faces is 384 in^2 + 2 x 48 in^2 = 480 in^2.

Board 3 has three faces that will be stained: the top face and the two side faces. The area of the top face is 30 in x 72 in = 2160 in^2. The area of each side face is 72 in x 4 in = 288 in^2. So the total area of all three faces is 2160 in^2 + 2 x 288 in^2 = 2736 in^2.

Board 4 has two faces that will be stained: the top face and the side face. The area of the top face is 17 56/9 in x 4 in = 71 2/3 in^2. The area of the side face is 17 56/9 in x 9 in = 158 2/3 in^2. So the total area of both faces is 71 2/3 in^2 + 158 2/3 in^2 = 230 1/3 in^2.

To find the total area that Ignacio will stain, we just need to add up the areas of all the faces that will be stained:

15872 in^2 + 480 in^2 + 2736 in^2 + 230 1/3 in^2 = 19018 1/3 in^2

Therefore, the total area that Ignacio will stain is approximately 19018.33 square inches.

original question :

Ignacio makes a display shelf from 4 wooden boards. All angles formed by the boards are right angles. Ignacio plans to stain all faces of the shelf, except the back face, which will be against the wall. What is the total area Ignacio will stain? Show your work. 512 7 in. 1 2 32 in. 30 in. 72 17 56 9 in. 4 in. 512 24 A 329-30.7 72

chatgpt

It is easier to add, subtract, multiply or divide very large or very small number in scientific notation because it reduces error, simplifies computation, and eases comparison.

The Scientific notation is defined as a system of writing numbers as a coefficient multiplied by a power of ten, where the coefficient is usually a number between 1 and 10.

Using scientific notation makes it easier to perform arithmetic operations on very large or very small numbers for several reasons:

(i) Simplifies computation: Scientific notation simplifies the computation of arithmetic operations by reducing the number of digits involved in the calculation.

(ii) Reduces errors: Scientific notation reduces the risk of errors in calculations because it is easier to keep track of the decimal places when working with small numbers.

(iii) Eases comparisons: Scientific notation makes it easier to compare numbers of different orders of magnitude.

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Step 1: Let the two numbers be x and y

Step 2: Since we know the difference between the two numbers is 6, that means x = y + 6

Step 3: We also know the sum of the two numbers is 10, so x + y + 6 = 10

Step 4: Simplifying, we get 2y + 6 = 10

Step 5: Subtracting 6 from both sides, we get 2y = 4

Step 6: Dividing both sides by 2, we get y = 2

Step 7: Finally, we can use the first equation (x = y + 6) to solve for x, so x = 8

Therefore, the two numbers are 8 and 2.

Answer:

y = 5x - 35

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

m = 5

The y-intercept is located at (0, -35)

So, the equation is y = 5x - 35

Answer: y=5x-35

Step-by-step explanation:

The formula I believe you are looking for would be y=mx+b. M is your slop and B is your y-intercept. Since the slope is already given, you would fill in the rest of the formula with that and the (x,y) you were provided, and it should look like this:

5=5(8)+b.

Then you solve that for b as follows:

5=5(8)+b

5=40+b

-35=b

Therefore the equation would be y=5x-35

Answer:

Step-by-step explanation:

25.

[tex]cos 45=\frac{x}{18\sqrt{2} } \\\frac{1}{\sqrt{2}} =\frac{x}{18\sqrt{2} } \\x=\frac{18\sqrt{2}}{\sqrt{2}} \\x=18[/tex]

  26.

sin30=30/x

[tex]\frac{1}{2} =\frac{30}{x} \\x=30 \times 2=60[/tex]

The value of 3302 mod 385 is 51179.

To use Fermat's little theorem to compute the expression, we need to first identify two relatively prime numbers that we can use as the base and the modulus. Let's choose 2 and 384, respectively.

Then, we have:

2³⁸⁴ ≡ 1 (mod 385)

Multiplying both sides by 3302, we get:

2³⁸⁴ × 3302 ≡ 3302 (mod 385)

Now, we can use the Chinese remainder theorem to find the value of 2³⁸⁴ × 3302 mod 385.

From the part (a) of this question, we know that:

2³⁸⁴ ≡ 1 (mod 7)

2³⁸⁴ ≡ 1 (mod 5)

2³⁸⁴ ≡ 16 (mod 11)

Using the Chinese remainder theorem, we can combine these congruences to get:

2³⁸⁴ ≡ 166 (mod 385)

Substituting this into the expression we derived earlier, we have:

2³⁸⁴ × 3302 ≡ 166 × 3302 (mod 385)

Multiplying out, we get:

2³⁸⁴ × 3302 ≡ 51179 (mod 385)

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Answer:

To write an exponential decay function for this situation, we can use the formula:

P(t) = P₀e^(rt)

where:

P(t) = the population at time t

P₀ = the initial population

r = the annual rate of decrease (as a decimal)

t = time in years

We are given P₀ = 3,846 and r = -0.0027 (since the population is decreasing).

To approximate the population in 2022, we need to find t, the number of years from 1996 to 2022. That is:

t = 2022 - 1996 = 26 years

Now we can plug in the values we have:

P(t) = 3,846 e^(-0.0027t)

To find P(2022), we plug in t = 26:

P(26) = 3,846 e^(-0.0027(26))

≈ 3,200.62

Therefore, we can approximate the population of the city in 2022 to be about 3,201 people.

Answer:

3,101

Step-by-step explanation:

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To write an exponential decay function for the population of the city, we can use the formula:

P(t) = P₀e^(-rt)

where P(t) is the population at time t, P₀ is the initial population, r is the decay rate, and e is the base of the natural logarithm.

In this problem, P₀ = 3,846 and r = 0.0027 (0.27% expressed as a decimal). We want to find the population in 2022, which is 26 years after 1996.To use the formula, we need to convert 26 years to the same time units as the decay rate. Since the decay rate is per year, we can use 26 years directly. Therefore, the exponential decay function for the population is:

P(t) = 3,846e^(-0.0027t)

To find the population in 2022 (t = 26), we substitute t = 26 into the function:

P(26) = 3,846e^(-0.0027*26) ≈ 3,101

Therefore, the population in 2022 is approximately 3,101.

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Part A :  the probability that a randomly chosen recipient served in the army, navy, or marines is 0.993. Part : the probability that a randomly chosen recipient served in the Air Force is 0.0038.

Probability is a branch of mathematics that deals with the study of chance or randomness. It involves the measurement of the likelihood or chance of an event occurring. Probability is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. For example, the probability of flipping a coin and it landing on heads is 0.5 or 50%, assuming a fair coin.

The total number of recipients in the pie chart is the sum of all the values: 2393 + 743 + 296 + 13 + 8 + 1 = 3454.

Part A: The number of recipients who served in the army, navy, or marines is 2393 + 743 + 296 = 3432. Therefore, the probability that a randomly chosen recipient served in the army, navy, or marines is:

P(Army or Navy or Marines) = (2393 + 743 + 296) / 3454 ≈ 0.993

Part B: The number of recipients who were civilians is 8. Therefore, the probability that a randomly chosen recipient was a civilian is:

P(Civilian) = 8 / 3454 ≈ 0.0023

Therefore, the probability that a randomly chosen recipient served in the Air Force is:

P(Air Force) = 13 / 3454 ≈ 0.0038.

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Answer: $2,162.06

Step-by-step explanation:

To calculate the total monthly payment to the bank, we need to add up the monthly mortgage payment, the monthly portion of the semi-annual property tax bill, and the monthly portion of the annual homeowner's insurance bill.

First, we need to find the monthly portion of the semi-annual property tax bill. To do this, we divide the semi-annual property tax bill by 6 (since there are 6 months in half a year):

Monthly property tax payment = Semi-annual property tax bill / 6

Monthly property tax payment = $3,239 / 6

Monthly property tax payment = $539.83

Next, we need to find the monthly portion of the annual homeowner's insurance bill. To do this, we divide the annual homeowner's insurance bill by 12 (since there are 12 months in a year):

Monthly homeowner's insurance payment = Annual homeowner's insurance bill / 12

Monthly homeowner's insurance payment = $980 / 12

Monthly homeowner's insurance payment = $81.67

Now we can add up the monthly mortgage payment, the monthly property tax payment, and the monthly homeowner's insurance payment to find the total monthly payment to the bank:

Total monthly payment = Monthly mortgage payment + Monthly property tax payment + Monthly homeowner's insurance payment

Total monthly payment = $1,711.22 + $539.83 + $81.67

Total monthly payment = $2,332.72

Therefore, the Dotkoms pay the bank $2,332.72 each month.

Answer:

Step-by-step explanation:

Using proportions, it is found that they pay the bank $2332.72 each month.

What is a proportion?

A proportion is a fraction of a total amount.

Using proportions, it is found that they pay the bank $2332.72 each month.

What is a proportion?

A proportion is a fraction of a total amount.

Their payments are given by:

Monthly mortgage of $1,711.22.

Semi-annual property tax bill is $3,239, that is, it is paid every 6 months, hence 3239/6 = $539.83.

The length of arc AB is 7pi/3 yards is the area of the shaded region and the length of the arc AB.

what is circle?

A circle is a geometric shape that consists of all points in a plane that are equidistant from a fixed point called the center.

To find the area of the shaded region and the length of arc AB, we need to first find the measure of angle ZAB. Let's call this angle x.

Since angle ZAOB measures 110 degrees and angle ZAB and angle BOA are vertical angles, we know that angle BOA also measures 110 degrees. Therefore, angle ZAB + angle BOA = 180 degrees.

So, we can write:

x + 110 = 180

Solving for x, we get:

x = 70

Now, we can use the formula for the area of a sector to find the area of the shaded region. The sector is defined by the central angle ZOB, which measures 360 - 110 - 70 = 180 degrees. So, we have:

Area of shaded region = (180/360) * pi * 6^2 = 18pi

Therefore, the area of the shaded region is 18pi square yards.

To find the length of arc AB, we can use the formula:

Length of arc AB = (x/360) * 2 * pi * 6

Plugging in x = 70, we get:

Length of arc AB = (70/360) * 2 * pi * 6 = 7pi/3

Therefore, the length of arc AB is 7pi/3 yards.

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Whenever two variables are correlated, we assume that one variable is the cause of the observed effect from the other variable.

It is a false statement,

Two variables that move together, or in the same direction, are said to have a positive correlation. When one variable rises while the other rises or when one variable falls while the other falls, there is a positive correlation. These two separate variables are theoretically influenced by the same outside factors because they travel in the same direction.

Whenever two variables are correlated, we assume that one variable is the cause of the observed effect from the other variable.

It is a false statement,

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The area of the shaded region as a percentage of the area of the sector ONQ= 60.3%

The shape of an equilateral triangle is an equilateral triangle.

The word "Equilateral" is formed by combining two words. H. "Equi" means equal, "lateral" means side.

Equilateral triangles are also called regular polygons or equilateral triangles because all sides are equal.

In geometry, an equilateral triangle is a triangle with all sides of equal length.

Three sides are equal, so three angles on the same side are equal. Therefore, it is also called an equilateral triangle with each angle of 60 degrees.

Like other types of triangles, equilateral triangles have formulas for area, perimeter, and height. 

According to our question-

AB=OA=BO= 9CM

ONQ-AOB/ONQ*100

PUTTING VALUES

60.3%

Hence, The area of the shaded region as a percentage of the area of the sector ONQ= 60.3%

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In order to create a 3 foot broad sidewalk around the garden, 98 feet of lumber will be needed.

The word "perimeter" in mathematics refers to the distance that surrounds the perimeter of a two-dimensional shape, such as a circle, square, or rectangle. It represents the total length of the shape's edges. For instance, you can calculate a rectangle's perimeter by multiplying the lengths of each of its four sides: P = 2l + 2w, where P stands for the perimeter, l for the rectangle's length, and w for its breadth.

given

We must multiply each dimension of the garden by the sidewalk's width to determine the overall length of the perimeter in order to determine the walk's perimeter.

The garden's revised measurements, including the 3-foot-wide sidewalk that surrounds it, are as follows:

Length = (17 + 2(3)) feet.

width = 20 + 2 (3) = 26 feet.

The sum of the four edges defines the walk's perimeter:

Width is equal to 98 feet (23 + 23 + 26 + 26).

In order to create a 3 foot broad sidewalk around the garden, 98 feet of lumber will be needed.

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Using the concept of parabola, the quadratic inequality represented by the graph can be written as:

y = x² -2x +2.

An equation of a curve that has a point on it that is equally spaced from a fixed point and a fixed line is referred to as a parabola.

The parabola's fixed point is referred to as the focus, and its fixed line is referred to as the directrix.

The general equation for a parabola is given as:

y = a(x-h) ² + k

Now here we have:

(x,y) = (2,5)

(h,k) = (1,1)

Putting these values in the equation,

5 = a (2-1) ² + 1

a = 5-1

=4

Substituting the values:

y = (x-1) + 1

y = x² -2x +2

Therefore, the quadratic inequality can be written as: y = x² -2x +2.

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Answer:

See step by step.

Step-by-step explanation:

lets define the events:
A: cuban festival        C: tropical Garden
B: street art show      D: african festival

a) theoretically the probability is

[tex]P(A)=P(B)=P(C)=P(D)= \frac{1}{4} = 0.25 \\[/tex]
This is 25% (for each one, equally)

b) The experimental probability is given by:

[tex]P(A)= \frac{32}{150} =0.2133[/tex]

[tex]P(B)= \frac{38}{150} =0.2533[/tex]

[tex]P(C)= \frac{35}{150} =0.2333[/tex]
[tex]P(D)= \frac{45}{150} =0.3000[/tex]

c) The theoretically probabilities are all equally, the experimental probabilities are close to 25% each one, but differ lightly each one, since is an experiment and the result is random.  

Answer:

The radius is approximately 14.003.

Step-by-step explanation:

Given:

Formulas for the semicircle:

Answer:

14 cm

Step-by-step explanation:

To find:-

Answer:-

We are here given that the perimeter of a semicircular disc is 72cm . We are interested in finding out the radius of the disc. Perimeter of a semicircular disc is given by,

[tex]:\implies \sf P = \pi r + 2r \\[/tex]

where r is the radius of the disc

Now on substituting the respective values, we have;

[tex]:\implies \sf 72cm = \pi r + 2r \\[/tex]

[tex]:\implies \sf 72cm = r(\pi + 2)\\[/tex]

[tex]:\implies \sf r =\dfrac{72}{\pi+2} cm \\[/tex]

[tex]:\implies \sf r =\dfrac{72}{3.14+2} cm \\[/tex]

[tex]:\implies \sf r = \dfrac{72}{5.14}\\[/tex]

[tex]:\implies \sf \pink{radius= 14.007\ cm } \\[/tex]

Hence the radius of the semicircular disc is approximately 14 cm .

Answer:

A system of equations is when two equations equal each other. In this case, f(x) and g(x) must equal each other.

f(x) - g(x) = 0

x^2 - x + 4 - (x - 4) = 0

x^2 - 2x - 4 = 0

Using the Quadratic Formula:

x = (2 ± √20)/2

Therefore, the values for x are x = 2 and x = -3.

[tex]\sqrt{2} = 1.414[/tex]. [tex]\sqrt{x3} = 1.732[/tex]. So, a rational number between those two would be [tex]1.5 (\frac{3}{2} )[/tex].

A rational number is any fraction with a non-zero denominator. Examples for rational numbers comprise 1/2, 1/5, 3/4, as well as other values. The number "0" also qualifies as a rational number since there are various ways to express it, including 0/1, 0/2, 0/3, etcetera.

Any number that can be expressed as a fraction and where the divisor (the bottom number) and the numerator (the total score) are both integers is considered a rational number. In other words, p/q, where both p and q both are integers and q 0, can be used to represent a rational number.

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we would expect a supervisor rating of approximately 6.61 if a person had a selection test score of 6.

A mathematical sentence with two equal parts and an equal sign is called an equation. An illustration of an equation is 4 + 6 = 10. 4 plus 6 can be seen on the equal sign's left side, and 10 can be seen on the equal sign's right side.

To find the supervisor rating we would expect for a person with a selection test score of 6, we can plug 6 in for X in the regression line equation:

[tex]$$\hat{Y} = 1.7656 + 0.8073X$$[/tex]

So:

[tex]$$\hat{Y} = 1.7656 + 0.8073(6)$$[/tex]

Simplifying

[tex]$$\hat{Y} = 1.7656 + 4.8438$$[/tex]

[tex]$$\hat{Y} = 6.6094$$[/tex]

Therefore, we would expect a supervisor rating of approximately 6.61 if a person had a selection test score of 6.

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The matches of Histogram, Bin, Descriptive Statistics, Mean, Median and Standard Deviation are C, E, D, A, F, G and B respectively.

The Match the definition are given.

Histogram - C). is a graph of the frequency distribution of a set of data

Bin - E). a group in a histogram

Descriptive Statistics - D). values calculated from a data set and used to describe some basic characteristics of the data set

Mean - A). The scatter around a central point

Median - F). the middle value of a sorted set of data

Mode - G). is the most commonly occurring value in a data set

Standard Deviation - B). is a measure of a data’s variability

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Answer:

0 and 180 degrees.

Step-by-step explanation:

We can start by using a trigonometric identity to rewrite sin2x in terms of cos2x:

sin2x = 1 - cos2x

Substituting this into the given equation, we get:

cos2x = 1 + (1 - cos2x)

Simplifying this equation, we get:

2cos2x = 2

Dividing both sides by 2, we get:

cos2x = 1

Solving for x, we get:

2x = 0°, 360°x = 0°, 180°

Therefore, the solutions to the equation cos2x = 1 + sin2x in the interval 0° ≤ x ≤ 360° are x = 0° and x = 180°.

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The surface area of the surface generated by rotating the curve y = x² about the y-axis, and we found that the surface area is approximately 54.33 square units.

In this case, the curve we want to rotate is y = x², and we want to rotate it about the y-axis. To use the formula above, we need to express the equation of the curve in terms of x. Therefore, we need to rewrite y = x² as x = √y.

Next, we need to find the derivative of x = √y with respect to y, which is:

dx/dy = 1/2√y

Substituting this into the formula for the surface area, we get:

Surface Area = 2π ∫[0,4] √y √(1+(1/2√y)²) dy

Simplifying the expression inside the square root, we get:

Surface Area = 2π ∫[0,4] √(y+(1/4)) dy

We can evaluate this integral using the power rule of integration, which gives:

Surface Area = 2π [2/3(y+(1/4))^(3/2)]₀⁴

Simplifying further, we get:

Surface Area = 2π [2/3(17/4)^(3/2)]

Surface Area ≈ 54.33 square units

Therefore, the surface area of the surface generated by rotating the curve y = x² about the y-axis is approximately 54.33 square units.

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Complete Question:

How do you find the area of the surface generated by rotating the curve about the y-axis y = x^2 , 0 ≤ x ≤ 2 ?