why do deviation scores have to be squared before calculating the standard deviation?

Answers

Answer 1

To answer your question: deviation scores have to be squared before calculating the standard deviation to eliminate negative values, ensure consistency, and emphasize larger deviations.

Here's a step-by-step explanation:

1. Deviation scores represent the differences between each data point and the mean. These differences can be both positive and negative.

2. If we simply add up the deviation scores without squaring them, the sum may equal zero due to the negative and positive values canceling each other out, which does not accurately represent the variation in the data.

3. By squaring the deviation scores, we eliminate negative values, ensuring that all deviations are considered in the calculation.

4. Squaring also emphasizes larger deviations, making the standard deviation more sensitive to extreme values or outliers.

5. After squaring the deviation scores and calculating their mean (i.e., the variance), we take the square root to obtain the standard deviation, which is measured in the same unit as the original data. This allows for easier interpretation and comparison of the data.

In summary, squaring deviation scores before calculating the standard deviation helps eliminate negative values, ensures consistency, and emphasizes larger deviations, resulting in a more accurate representation of the data's variation.

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Related Questions

Activity 2 Compute for the following: Speed; Distance, Time
1. A car travels 300 kom in 6 hours. What is the average speed of the car in km/h?
Given
Solution
Car travels in 300 kom ( distance) in
6 hours (time)

2. What is the speed of a jet plane that flies 8100 kon in 9 hours? (in km/h)
Given:
Solution
Jet travels 8100 km (distance)in
9 hours (time)

3. A storm is moving toward your house at a speed of 20 kmh.it is now 60 km away from your house. In what time will the storm reach your house?
Given:
Solution:
Storm moving at 20 km/h (speed) at
60 kun (distance)
Speed and direction
In describing the motion of an object, we do not just describe​

Answers

Answer:

1. 50 km/h

2. 900 km/h

3. 3 hours

Step-by-step explanation:

1. The distance the car travels, Δd = 300 km

For how long it takes the car to travel the 300 km, Δt = 6 hours

Average speed, \(\overline v\), is given as follows;

\(\overline v = \dfrac{The \ total \ distance }{The \ total \ time \ it \ takes } = \dfrac{\Delta d}{\Delta t}\),

\(\therefore \overline v = \dfrac{300 \ km}{6 \ hours} = 50 \ \dfrac{km}{hour}\)

The average speed of the car, \(\overline v\) = 50 km/h

2. The distance the jet plane flies, d = 8,100 km

For how long it takes the the jet plane to fly the 8,100 km, t = 9 hours

The speed, v, of the jet plane is given as follows;

\(Speed, \ v = \dfrac{Distance}{Time} = \dfrac{d}{t}\)

\(v = \dfrac{The\ distance \ the \ jet plane \ flies}{The \ time \ it \ takes \ to \ fly \ thart \ distance } = \dfrac{d}{t }\),

\(\therefore v = \dfrac{8,100 \ km}{9 \ hours} = 900 \ \dfrac{km}{hour}\)

The speed of the jet plane, v = 900 km/h

3. The speed of the storm, v = 20 km/h

The distance of the storm from the house, d = 60 km

The time 't' it takes the storm to reach the house is given as follows;

\(Speed, \ v = \dfrac{d}{t}\)

Therefore, by cross multiplying and dividing by 'v', we get;

\(Time, \ t = \dfrac{d}{v}\)

Plugging in the known values of 'd' and 'v' gives;

\(Time, \ t = \dfrac{60 \ km}{20 \ \dfrac{km}{hour} } = 3 \ km \times \dfrac{hour}{km} = 3 \ hours\)

The time it takes the storm to reach the house, t = 3 hours.

for f(x) = -4x -9 evaluate for f(6.5)​

Answers

Answer:

f=−35

Step-by-step explanation:

1.Simplify  4×6.5 to 26

f=-26-9

2. Simplify  -26-9 to  -35

f=-35

How do we most easily graph an equation in the form Ax+By=C? Explain, and use at least one example.

Answers

Answer:

Rewrite the equation into the form of "y=mx+b", plot the y-intercept (b), plot the x-intercept(s), then create the line.

Step-by-step explanation:

Example: 2x + y = 1

Rewrite: y = -2x + 1Plot y-intercept: (0, 1)Plot x-intercept: (1/2, 0)Draw line

I hope this helps!

How do we most easily graph an equation in the form Ax+By=C? Explain, and use at least one example.

which number does NOT represent an integer? A. 3 B. 20.1 C. -10 D. 20/4​

Answers

Answer:

the number doesnt represent an integer?

A. 3

B. 20.1

C. -10

D. 20/4​

Step-by-step explanation:

We call integers the “counting numbers, their negatives and zero”. I.e.  0,1,−1,2,−2,3,−3,.. etc.

non-integers means “everything except integers”. Which is not well-defined (i.e. nonsense). Why? Because nobody said what “everything” is.

Therefore, when somebody says “non-integer” he has to specify how he defines “everything”. In this case, our “everything” is probably “real numbers”.

Real numbers have an interesting definition concerning an abstract mathematical object called “field” . Let’s forget about that and let’s focus on a high school definition: Real numbers are probably all the numbers you know. They are those represented by a decimal and their negatives e.g.  345.232…  and  −243.13242240…  where there are “infinitely many” digits at the end. Note that  2.5  is also a real number. Integers are too. Basically, real numbers are the numbers used to measure distances and their negatives.

To summirize, your answer is the following:

“non integers” means everything except the integers, where everything is defined however we want. The most common definition of everything in this case is “real numbers” and therefore the most common interpretation of “non integers” is “reals which are not integers”.

Examples of “reals which are not integers”:  1.5,2.88,1.3333…  etc

the answer is

B. 20.1

In a hypothesis test, what should we conclude if the data would be very unusual if the original assumption about our parameter were correct?

Answers

We conclude  the hypothesis test as Alternative Hypothesis if the data would be very unusual if the original assumption about our parameter were correct.

A hypothesis in statistics is a claim or supposition on the properties of one or more variables in one or more populations. There are two hypothesis to choose between because a statement might either be true or wrong.The null hypothesis is the assertion that we (or someone else) consider to be true. Our hypothesis test will come to one of two conclusions: "reject H0" or "do not reject H0." Remember that until data provide evidence to the contrary, we always proceed under the null hypothesis.If the null hypothesis is incorrect, the alternative hypothesis must be true. The hypothesis test can be different in one of three ways: greater than, smaller than, or just different (not equal). As a result, there will always be an inequality requirement in the notation for H.

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Help.. brainlist?

1) Which occupational group is projected to have the biggest employment growth?

What skill level is required for this occupational group?

Help.. brainlist?1) Which occupational group is projected to have the biggest employment growth? What

Answers

Answer:

Bachelor degree or higher qualification

OCCUPATIONAL GROUP: LABOURERS OR PROFESSIONAL
SKILL LEVEL: ADVANCED DIPLOMA OR DIPLOMA

It should be Skill Level 2 because, all those who work or with have profession need to have and provide those needed achievements.

Please Correct me if I’m wrong.

1. what does the regular expression ^comp141\.\*zip$ mean? that is, can you write an english sentence to describe it?

Answers

It is well known that utilizing a zip file is an effective method for reducing the size of the document in order to boost storage and sharing possibilities. A zipped folder may be smoothly moved from one place (the transmitter) to another without any issues (the receiver).

What is Zip and how does it work?

One or more files can be compressed and stored together in one location using the ubiquitous ZIP file format. As a result, the file size is less and is simpler to transport and store. After transmission, the receiver can use the files in their original form by unzipping (or extracting) her ZIP file.

In order to increase storage and sharing capabilities by lowering the size of the document, it is known that using a zip file is an efficient technique. A zipped folder may be effortlessly transferred without any hiccups from one location (the transmitter) to another (the receiver). Using Zip files has the primary benefit of reducing the time required to send large files over email. The document's quality is unaffected and is identical to the original file when it is compressed using a zip file.

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what is the unit rate if 12 inches of snow in 3 hours?​

Answers

Answer:

4 inches of snow has fallen rate per hour

therefore the unit rate is 4 inches of snow per hr.

Step-by-step explanation:

Answer:

4 per hour

Step-by-step explanation:

because 12 divided by 3 is 4

A traditional unit of length in japan is the ken (1 ken = 1.97 m). what are the ratios of (a) square kens to square meters?

Answers

The ratios of (a) square kens to square meters (1 ken² / 1 m²) is 3.88

For given question,

We have been given the unit conversion.

A traditional unit of length in Japan is the ken

and 1 ken = 1.97 m

We need to find the the ratios of (a) square kens to square meters.

First we find the square of 1 ken.

⇒ 1 square kens = (1.97 m)²

⇒ 1 square kens = 1.97 × 1.97 m²

⇒ 1 square kens = 3.88 m²                  ..................(1)

And 1 square meters = 1 m²                 ................. (2)

Now we take the ratio of square kens to square meters.

From (1) and (2),

⇒ 1 ken² / 1 m² = 3.88 / 1

⇒ 1 ken² / 1 m² = 3.88

Therefore, the ratios of (a) square kens to square meters (1 ken² / 1 m²) is 3.88

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Lydia worked the following hours last Tuesday and Wednesday.

Tuesday: 7AM to 11:30AM and 12PM to 3:45PM
Wednesday: 7AM to 11:45AM and 12:30PM to 4PM

If Lydia is paid $18 an hour, how much did she make in total for Tuesday and Wednesday?

Answers

Answer:

$297

Step-by-step explanation:

Tuesday:

7AM to 11:30AM (4.5 hours)

12PM to 3:45PM (3.75 hours)

Wednesday:

7AM to 11:45AM (4.75 hours)

12:30PM to 4PM (3.5 hours)

Then add:

4.5+3.75+4.75+3.5= 16.5

Then multiply:

16.5*18= 297

Hope this helps!

The total time of Tuesday and Wednesday will be 16.5 hours. Then the total amount that she made on Tuesday and Wednesday is $297.

What is the linear system?

A Linear system is a system in which the degree of the variable in the equation is one. It may contain one, two, or more than two variables.

Lydia worked the following hours last Tuesday and Wednesday.

Tuesday: 7AM to 11:30AM and 12PM to 3:45PM

Wednesday: 7AM to 11:45AM and 12:30PM to 4PM

If Lydia is paid $18 an hour. Then the equation will be

y = 18x

where x is the number of hours.

Then the number of hours on Tuesday will be

→ 11:30 AM - 7:00 AM = 4:30

→ 3:45 PM - 12:00 PM = 3:45

The total hours will be

→ 5:30 + 3:45 = 8:15

Then the number of hours on Wednesday will be

→ 11:45 AM - 7:00 AM = 4:45

→ 4:00 PM - 12:30 PM = 3:30

The total hours will be

→ 4:45 + 3:30 = 8:15

Then the total time of Tuesday and Wednesday will be

→ 8:15 + 8:15 = 16:30

The time can be written as

→ 16:30 = 16.5 hours

Then the total amount she made will be given as

y = 18 × 16.5

y = $297

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Which of the following is
closest in value to 8074?
8,138
8.128
8.145
8.009

Which of the following isclosest in value to 8074?8,1388.1288.1458.009

Answers

Answer:

8.145

Step-by-step explanation:

Answer:

8128

Step-by-step explanation:

this is because this is the one that when you subject from 8074, it is the lowest number

90 is 75% of what number

Answers

Answer:

120

Step-by-step explanation:

90 x 100 ÷ 75 = 120

Is the following statement a good definition? Why?

An obtuse angle is an angle whose measure is greater than 90°.

Is the following statement a good definition? Why?An obtuse angle is an angle whose measure is greater

Answers

Answer:

yes it is all obtuse angles have measures greater than 90°.

Answer:

Yes it is. All obtuse angles have measures greater than 90.

Step-by-step explanation:

Reasoning It has to be Higher then 90 degrees to be called obtuse. This rule applys for all.

Exercise 1-9 (Algo) Using the accounting equation LO A1 Determine the missing amount from each of the separate situations given below.

Answers

The accounting equation is a fundamental principle in accounting that states that assets equal liabilities plus equity. Using this equation, missing amounts can be determined in various situations.

The accounting equation is expressed as Assets = Liabilities + Equity. It serves as the foundation for double-entry bookkeeping and ensures that the financial statements are balanced. By rearranging the equation, missing amounts can be determined.

For example, if the assets and equity are given, the missing amount of liabilities can be calculated by subtracting equity from assets. Conversely, if the liabilities and equity are known, the missing amount of assets can be calculated by adding liabilities to equity.

Similarly, if the assets and liabilities are provided, the missing amount of equity can be calculated by subtracting liabilities from assets. Alternatively, if the assets and equity are known, the missing amount of liabilities can be calculated by subtracting equity from assets.

In each situation, the missing amount can be determined by applying the accounting equation and rearranging it to solve for the missing variable. This equation provides a framework for ensuring that all financial transactions are properly recorded and that the financial statements accurately reflect the financial position of a business.

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The sum of three lengths of a fence ranges from 45 to 60 inches. Two side lengths are 9 and 18 inches. If the length of the third side is x inches, write and solve a compound inequality to show the possible lengths of the third side.
a. 9 ≤ x ≤ 18
b. 18 ≤ x ≤ 33
c. 36 ≤ x ≤ 42
d. 45 ≤ x ≤ 60

Answers

The solution to a compound inequality to show the possible lengths of the third side is 18 ≤ x ≤ 33.

What is Compound Inequality?

Two or more inequalities are connected together with or or and to form a compound inequality (also known as a combined inequality).

A value must only make one aspect of an inequality true in order to be the solution to an or inequality. An inequality's solution must make both portions true in order to be effective.

According to the given values;

given parameters are

Range = 45 to 60 inches

Sides = 9, 18 and x

The sum of the three sides is

sum=9 + 18 + x

Evaluate

Sum = 27 + x

Recall that

Range = 47 to 60 inches

So, we have

45 ≤ 27 + x ≤  60

Subtract through by 27

So, we have

18 ≤ x ≤ 33

Hence, the solution to a compound inequality to show the possible lengths of the third side is 18 ≤ x ≤ 33.

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UL.Z This question is designed to be answered with a calculator. A midpoint approximation of the area under the curve f(x) = 2x(x - 4)(x - 8) over the interval [0, 4) with 4 subintervals is 0 111. 0 120 O 132. O 160.

Answers

The midpoint approximation of the area under the curve f(x) = 2x(x - 4)(x - 8) over the interval [0, 4) with 4 subintervals is 0. To approximate the area under the curve using a midpoint approximation, we divide the interval [0, 4) into four subintervals of equal width.

The width of each subinterval is (4 - 0) / 4 = 1.

Now, we need to evaluate the function at the midpoint of each subinterval and multiply it by the width of the subinterval.

The midpoints of the subintervals are: 0.5, 1.5, 2.5, and 3.5.

Evaluating the function at these midpoints, we get:

f(0.5) = 2 * 0.5 * (0.5 - 4) * (0.5 - 8) = 6

f(1.5) = 2 * 1.5 * (1.5 - 4) * (1.5 - 8) = -54

f(2.5) = 2 * 2.5 * (2.5 - 4) * (2.5 - 8) = 54

f(3.5) = 2 * 3.5 * (3.5 - 4) * (3.5 - 8) = -6

Now, we calculate the sum of these values and multiply it by the width of the subinterval:

Area ≈ (6 + (-54) + 54 + (-6)) * 1 = 0.

Therefore, the midpoint approximation of the area under the curve f(x) = 2x(x - 4)(x - 8) over the interval [0, 4) with 4 subintervals is 0.

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A newspaper conducted a statewide survey concerning the 2008 race for state senator. The newspaper took a random sample (assume it is a SRS) of 1200 registered voters and found that 620 would vote for the Republican candidate. Let p represent the proportion of registered voters in the state that would vote for the Republican candidate. A 90% confidence interval for p is:_________.

A. (0.503, 0.531)

B. (0.495, 0.539)

C. (0.493, 0.541)

D. (0.489, 0.545)

E. (0.268, 0.766)

Answers

Therefore, the 90% confidence interval for p is approximately (0.4959, 0.5375) that is option B.

To calculate the 90% confidence interval for the proportion (p) of registered voters who would vote for the Republican candidate, we can use the formula:

Confidence interval = p ± z * √((p(1-p))/n)

where:

p is the sample proportion (620/1200 = 0.5167),

z is the z-score corresponding to the desired confidence level (90% corresponds to a z-score of approximately 1.645),

n is the sample size (1200).

Substituting the values into the formula:

Confidence interval = 0.5167 ± 1.645 * √((0.5167*(1-0.5167))/1200)

Calculating the confidence interval:

Confidence interval ≈ 0.5167 ± 1.645 * √(0.2497/1200)

Confidence interval ≈ 0.5167 ± 1.645 * 0.0126

Confidence interval ≈ 0.5167 ± 0.0208

Confidence interval ≈ (0.4959, 0.5375)

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Suppose that a fourth order differential equation has a solution y = - 3e^3x x sin(x). Find the initial conditions that this solution satisfies.

Answers

The initial conditions that the solution y = -3e^(3x)sin(x) satisfies are:

y(0) = 0, y'(0) = 3, y''(0) = -6, y'''(0) = 0, and y''''(0) = 18.

To find the initial conditions that the solution y = -3e^(3x)sin(x) satisfies, we need to differentiate it four times and then evaluate it at the initial time t = 0.

y = -3e^(3x)sin(x)

y' = -3e^(3x)sin(x) + 3e^(3x)cos(x)

y'' = -6e^(3x)cos(x)

y''' = 18e^(3x)sin(x)

y'''' = 18e^(3x)cos(x)

Now, evaluating these expressions at t = 0:

y(0) = -3e^(3*0)sin(0) = 0

y'(0) = -3e^(30)sin(0) + 3e^(30)cos(0) = 3

y''(0) = -6e^(3*0)cos(0) = -6

y'''(0) = 18e^(3*0)sin(0) = 0

y''''(0) = 18e^(3*0)cos(0) = 18

Therefore, the initial conditions that the solution y = -3e^(3x)sin(x) satisfies are:

y(0) = 0, y'(0) = 3, y''(0) = -6, y'''(0) = 0, and y''''(0) = 18.

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Solve the following system of equations.
5 a-b=17
3 a+2 b=5

Answers

Answer:

a = 3 , b = - 2

Step-by-step explanation:

5a + b = 17 → (1)

3a + 2b = 5 → (2)

multiplying (1) by 2 and adding to (2) will eliminate b

10a + 2b = 34 → (3)

add (2) and (3) term by term to eliminate b

13a + 0 = 39

13a = 39 ( divide both sides by 13 )

a = 3

substitute a = 3 into either of the 2 equations and solve for b

substituting into (1)

5(3) + b = 17

15 - b = 17 ( subtract 15 from both sides )

- b = 2 ( multiply both sides by - 1 )

b = - 2

then solution is a = 3 , b = - 2

Your homework grades are 92, 87, 74, 96, 83, 88, 91, 82, and 85. What score on your next homework will make the median and the mode equal? List the scores in order from least to greatest. You can use the strategy Systematic Guess and Check to help you find the solution.

Answers

Answer:Next  homework  Score should be  87.

Step-by-step explanation:

STEP 1

Listing the scores in ascending order ie from least to greatest gives

74, 82, 83, 85, 87, 88, 91,92, 96

STEP 2

we find the median--- The median score is the score  that is in the middle of the set of results when arranged from lowest to highest as we have done above , which give  us 87 as the median.

74, 82, 83, 85,   87,    88, 91,92, 96

Also Mode is the score with the highest frequency , ie the score which appears more than other numbers. For now we have no mode as all the numbers occurred once

Therefore in next exam, I should get a score of 87,  making my scores to become

74, 82, 83, 85, 87, 87, 88, 91,92, 96

Now my mode becomes 87 ( highest frequency of  occurrence ) and

My median becomes (87+ 87)/2 = 87  ( with the addition of a new score the total score become ten, which is now even, so we divide the addtion of the two middle numbers by 2 )

8. Write the equation of a horizontal line that passes through the point (-2, 2).
x = -2
y = 2
X = 2
y = -2

Answers

Answer:

its y=2

Step-by-step explanation:

An ice cream shop is testing some new flavors of ice cream. They invent 252525 new flavors for customers to try, and throw out the 131313 least popular flavors. The shop makes 808080 cartons of each of the remaining flavors. It takes 222 cartons to get one liter of ice cream. How much total ice cream is in the cartons?

Answers

Answer: total 480 liters of ice creams is in the cartons

Step-by-step explanation: cuz yea

PLEASE CAN SOMEONE TELL ME IF IM SUPPOSE TO USE TAN OR SIN TO SOLVE 11b?

PLEASE CAN SOMEONE TELL ME IF IM SUPPOSE TO USE TAN OR SIN TO SOLVE 11b?

Answers

Answer:

Either or, it doesn’t make a difference here. If you used the tangent formula, you would need to use AC to find BC but if you used the sine formula you would need to use AB to find BC.

No, You use GCD take all measurements do GCD

What are not exponential functions?.

Answers

It is not of exponential order for h(t) = et2. F is of exponential order and has order. F is piecewise continuous. No function's Laplace transform is possible. g(t) = eat, where t [0,.

Non-exponential form:

To represent a scientifically notated number as a non-exponential quantity: • Remove the exponential component of the number by moving the decimal point the same number of places as the exponent's value. The decimal is moved to the right by the same number if the exponent is positive.

Here are a few instances of functions other than exponential ones. y = 3 1 x as a result. n = 0 3 p as a result. Because y = ( 4) x. Since b = 1, y = 6 0 x.

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3. Diego works at a carwash and makes a $11 per hour. If he works 9 hours of overtime in a week, how much overtime would he earn?
Consider that overtime is paid at time-and-a-half of his regular rate.
- $158.40
-$130.00
- $150.40
-$148.50

Answers

The amount of overtime that Diego would earn would be D. $ 148. 50.

How to find the overtime ?

Diego's regular rate is $11 per hour. When he works overtime, he is paid time-and-a-half of his regular rate, which means he earns an additional 50% of his regular rate for each hour of overtime.

So his overtime rate is:

= 11 + ( 11 x 0.5)

= 11 +  5.50

= $16.50 per hour

If Diego works 9 hours of overtime in a week, he would earn:

= 9 x 16.50

= $ 148. 50

In conclusion, if Diego was to work 9 hours of overtime, he would be paid $ 148. 50 for his efforts.

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Which of the following decimals is equal to 3 1/40 ? Select TWO that apply

A 3.0025
B 3.025
С 3. overline 0025
D 3.overline 025
E 3.0025 overline0
F. 3.025overline0​

Answers

Answers:  Choice B, Choice F

=======================================================

Explanation:

Use long division or your calculator to find that 1/40 = 0.025

Add on 3 to get 3 & 1/40 = 3 + 1/40 = 3 + 0.025 = 3.025 which is choice B

For some reason, your teacher has decided to add on a string of 0's onto this to form \(3.025\overline{0}\) that horizontal line over the '0' means the 0's go on forever. So, \(3.025 = 3.025\overline{0} = 3.025000000...\), but we can always chop off the trailing zeros to make life simpler. I'm not sure why your teacher has decided to use this fairly strange format.

The decimal value is equivalent to 3.025.

What are decimal numbers?A decimal is a number that consists of a whole and a fractional part. Decimal numbers lie between integers and represent numerical value for quantities that are whole plus some part of a whole.

We have the following fraction -

3 \(\frac{1}{40\\}\)

We can write the mixed fraction as -

3 \(\frac{1}{40\\}\)

(40 x 3 + 1)/40

121/40

3.025

Therefore, the decimal value is equivalent to 3.025.

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The radius of circle a is 6 in. the radius of circle b is 5 in. greater than the radius of circle a. the radius of circle c is 5 in. greater than the radius of circle b. the radius of circle d is 2 in. less than the radius of circle c. what is the area of each​ circle? how many times greater than the area of circle a is the area of circle d​?

Answers

Answer:

Step-by-step explanation:

The radius of circle a is 6 in. the radius of circle b is 5 in. greater than the radius of circle a.

how do you represent an union set on a Venn diagram?? ​

Answers

I have attached an image that will hopefully answer your question.

Hope this helps! Please make me the brainliest, it’s not necessary but appreciated, I put a lot of effort and research into my answers. Have a good day, stay safe and stay healthy, and have a happy New Years!!!
how do you represent an union set on a Venn diagram??

The union set combines the elements that are in different sets to form a new set

Assume that we have the following sets

Set ASet B

The union of both sets is the collection of the elements of both sets (without repetition)

Assume that, we have:

A = {2,4,5}

B = {1,2,7}

The union of both sets would be:

A u B = {1,2,4,5,7}

See attachment for the graphical representation of the union set

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how do you represent an union set on a Venn diagram??



Write an inequality for which the solution is the set of all real numbers.

Answers

Any inequality of the form "x ≥ x" or "x ≤ x" represents a solution set of all real numbers. Inequality "x ≥ x" means that any value of x that is greater than or equal to itself satisfies the inequality.

Since every real number is equal to itself, the solution set is all real numbers. Similarly, "x ≤ x" indicates that any value of x that is less than or equal to itself satisfies the inequality, resulting in the solution set of all real numbers. This is always true, regardless of the value of x, since any number less than 1 is positive. Therefore, the solution set for x is all real numbers.

The inequality "x ≥ x" or "x ≤ x" represents the set of all real numbers as its solution, as any real number is greater than or equal to itself, and any real number is also less than or equal to itself. Therefore, the solution set for x is all real numbers.

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[11 ma Given the points P(2.1.3). Q(1.1.1).R(2.3.4) and S(0,1,2) a) Find the equation of the plane that contains the points P. Q, and R. Write your answer in the linear form. ax + by + cx+d=0 b) Find

Answers

The distance between the point S and the plane is 10 / √14.

To find the equation of the plane that contains the points P(2,1,3), Q(1,1,1), and R(2,3,4), we can use the method of cross products.

Step 1: Find two vectors on the plane.

We can choose two vectors from the given points. Let's take vectors PQ and PR.

Vector PQ = Q - P = (1-2, 1-1, 1-3) = (-1, 0, -2)

Vector PR = R - P = (2-2, 3-1, 4-3) = (0, 2, 1)

Step 2: Find the cross product of the two vectors.

To find the normal vector to the plane, we take the cross product of vectors PQ and PR.

Normal vector N = PQ × PR

N = (-1, 0, -2) × (0, 2, 1)

= (-2, 1, 0)

Step 3: Write the equation of the plane in the form ax + by + cz + d = 0.

Since the normal vector to the plane is (-2, 1, 0), we have the coefficients of x, y, and z. To find d, we substitute the coordinates of one of the given points (P, Q, or R) into the equation.

Using point P(2,1,3):

-2(2) + 1(1) + 0(3) + d = 0

-4 + 1 + d = 0

-3 + d = 0

d = 3

Therefore, the equation of the plane that contains the points P, Q, and R is:

-2x + y + 3z + 3 = 0

b) To find the distance between the point S(0,1,2) and the plane, we can use the formula for the distance between a point and a plane.

The formula for the distance between a point (x0, y0, z0) and a plane ax + by + cz + d = 0 is:

Distance = |ax0 + by0 + cz0 + d| / √(a^2 + b^2 + c^2)

Plugging in the values from the given point S(0,1,2) and the equation of the plane -2x + y + 3z + 3 = 0, we have:

Distance = |-2(0) + 1(1) + 3(2) + 3| / √((-2)^2 + 1^2 + 3^2)

= |1 + 6 + 3| / √(4 + 1 + 9)

= |10| / √14

= 10 / √14

Therefore, the distance between the point S and the plane is 10 / √14.

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