This composite figure is made up of three simpler shapes. What is the area of

The probability of x being less than 79 is 0.2119.

Given, mean `μ = 83` and standard deviation `σ = 5`.

We need to find the indicated probability `P(x < 79)`.

Using the z-score formula we can find the probability as follows: `z = (x-μ)/σ`Here, `x = 79`, `μ = 83` and `σ = 5`. `z = (79-83)/5 = -0.8`

We can look up the probability corresponding to z-score `-0.8` in the standard normal distribution table, which gives us `0.2119`.

Hence, the indicated probability `P(x < 79) = 0.2119`.Answer: `0.2119`

The explanation is well described in the above text containing 82 words.

Therefore, the solution in 150 words are obtained by adding context to the solution as shown below:

The given fandom variable `x` is normally distributed with mean `μ = 83` and standard deviation `σ = 5`. We need to find the indicated probability `P(x < 79)`.

Using the z-score formula `z = (x-μ)/σ`, we have `x = 79`, `μ = 83` and `σ = 5`.

Substituting these values into the formula gives us `z = (79-83)/5 = -0.8`.

We can then look up the probability corresponding to z-score `-0.8` in the standard normal distribution table, which gives us `0.2119`.Hence, the indicated probability `P(x < 79) = 0.2119`.

Therefore, the probability of x being less than 79 is 0.2119.

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The required time taken by the diver to reach the water is 1.31 seconds (approx.).

Given data: Height of the diving board= 5 m The angle of the diving board with the vertical=22° Initial velocity of the diver=5 m/s.

To find: The time to the water for the diver Solution: Let's consider the motion of the diver in the y-direction Initial velocity of the diver in the y-direction, uy = usinθ = 5 sin 22° = 1.83 m/s

Acceleration due to gravity, g = 9.8 m/s²Let's use the formula of motion in the y-directions = ut + 1/2 gt²where, s = displacement of the diver in the y-direction u = initial velocity in the y-direction t = time taken by the diver in the air Putting all the values in the above equation, we get5 = 1.83t + 1/2 × 9.8 × t²

Simplifying the above equation, we get 4.9 t² + 1.83 t - 5 = 0

On solving the above quadratic equation using the quadratic formula, we get t = [ -b ± √(b² - 4ac) ] / 2awhere, a = 4.9, b = 1.83, c = -5

Putting all the values in the above equation, we get t = [ -1.83 ± √(1.83² - 4 × 4.9 × -5) ] / 2 × 4.9

On solving the above equation, we get t = 1.31 s (approx.) or t = -0.78 s We know that the time taken by the diver cannot be negative, therefore the time taken by the diver to reach the water is 1.31 seconds (approx.).

Hence, the required time taken by the diver to reach the water is 1.31 seconds (approx.).

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The trained runner needs to exercise at a speed of approximately 2313.6 meters per minute to maintain a 12 MET level during the inclement New England winter weather. This is equivalent to about 8.3 miles per hour or 13.4 kilometers per hour.

To determine the speed at which the trained runner needs to exercise to maintain a 12 MET level, we can use the following formula:

METs = VO2/kg/min * 3.5

where VO2 is the rate of oxygen consumption during exercise, expressed in milliliters per kilogram of body weight per minute.

For a 68 kg runner exercising at a 12 MET level, we have:

12 = VO2/68 * 3.5

Solving for VO2, we get:

VO2 = 12 * 68 / 3.5 = 234.86 ml/kg/min

Next, we can use the following formula to convert VO2 to speed:

VO2 = (0.1 * speed) + 3.5

where speed is expressed in meters per minute.

Solving for speed, we get:

speed = (VO2 - 3.5) / 0.1 = (234.86 - 3.5) / 0.1 = 2313.6 meters per minute

Therefore, the trained runner needs to exercise at a speed of approximately 2313.6 meters per minute to maintain a 12 MET level during the inclement New England winter weather. This is equivalent to about 8.3 miles per hour or 13.4 kilometers per hour.

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The vector V that makes an angle of 40 degrees with W and which is of the same length as W and is counterclockwise to W is given by V = -7.92i - 9.63j.

The given vector is W = -10i + 7j.I + J is a unit vector that makes an angle of 45 degrees with the positive direction of x-axis.

A vector that makes an angle of 40 degrees with W can be obtained by rotating the vector W counterclockwise by 5 degrees.

Using the rotation matrix, the vector V can be obtained as follows: V = R(θ)Wwhere R(θ) is the rotation matrix and θ is the angle of rotation.

The counterclockwise rotation matrix is given as:R(θ) = [cos θ  -sin θ][sin θ  cos θ]

Substituting the values of θ = 5 degrees, x = -10 and y = 7, we get:

R(5°) = [0.9962  -0.0872][0.0872  0.9962]V = [0.9962  -0.0872][0.0872  0.9962][-10][7]= [-7.920  -9.634]

Hence, the vector V that makes an angle of 40 degrees with W and which is of the same length as W and is counterclockwise to W is given by V = -7.92i - 9.63j.

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The sample space has 22 outcomes, represented by Y, W, and G. The first ball can be any color, and if yellow, the process stops. If white, the second ball must also be white to reach the desired outcome. The total number of possibilities is 22. To verify, calculate the sum of the possibilities for each first ball color, which equals 22.

There are 22 outcomes in the sample space.

To find the outcomes in the sample space, we can list all the possibilities using the letters Y, W, and G to represent the yellow, white, and green balls, respectively.

The first ball can be any color, so we have three possibilities: Y, W, or G. If the first ball is yellow, the process stops because the desired outcome has been reached.

If the first ball is white, the second ball must also be white to reach the desired outcome. So, the possibilities are as follows: WWYY, WWYW, WWYG, WWGY, WGYY, WGYW, WGYG, WYYG, WYGY, WYYW, WGWY, WGWW, GWYY, GWYW, GWYG, GWWY, GWWY, GYYW, GYYG, GYGY, GYWW, GYWY

There are 22 possibilities in the sample space, which can be verified by calculating the sum of the possibilities for each first ball color: 8 + 6 + 8 = 22.

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The formula that can be used to calculate the average rate of population change between 2010 and 2019 is:

(153,159 - 91,351) / (2019 - 2010)

The expression that can be used to determine the average rate of change in population from 2010 to 2019 is:

(153,159 - 91,351) / (2019 - 2010)

This expression represents the change in population divided by the change in years, giving us the average rate of change in population per year.

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Therefore, when the length is 6 feet, the perimeter of the rectangle is 1242 feet.

To find the perimeter of the rectangle, we need to add up the lengths of all four sides.

The length of the rectangle is given as x, and the width is given as [tex]3x^3 + 3 - x^2.[/tex]

When the length is 6 feet, we can substitute x = 6 into the expressions:

Length = x = 6

Width = [tex]3(6^3) + 3 - 6^2[/tex]

Simplifying the width:

Width = 3(216) + 3 - 36

= 648 + 3 - 36

= 615

Now, we can calculate the perimeter by adding up all four sides:

Perimeter = 2(Length + Width)

= 2(6 + 615)

= 2(621)

= 1242

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(a) 2n is even, and the sum of two even integers is even, we can conclude that m+p is even. Therefore, the statement is true.

To prove that if m+n and n+p are odd integers, then m+p is even, we can use a direct proof.

Assume that m+n and n+p are odd integers. By definition, this means that there exist integers r and s such that:

m+n = 2r+1

n+p = 2s+1

Adding these two equations, we get:

(m+n) + (n+p) = 2r+1 + 2s+1

m+p + 2n = 2(r+s) + 2

Since 2n is even, and the sum of two even integers is even, we can conclude that m+p is even. Therefore, the statement is true.

(b) To prove that for all integers a, b, c, if a divides b and b divides c, then a divides c, we can use a direct proof as well.

Assume that a, b, and c are integers such that a divides b and b divides c. By definition, this means that there exist integers k and l such that:

b = ak

c = bl

Substituting b = ak into the second equation, we get:

c = bl = akl

Since k and l are integers, their product k*l is also an integer. Therefore, we can express c as a product of a and another integer, which means that a divides c. Therefore, the statement is true.

Note that in both parts (a) and (b), we used a direct proof, which involves assuming the premises and using logical deductions to arrive at the conclusion.

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Answer: A

Step-by-step explanation:

We must calculate the mean and compare each score to find the score closest to the standard of the four scores (79, 84, 81, and 73).

Mean = (79 + 84 + 81 + 73) / 4 = 317 / 4 = 79.25

Now, let's compare each score to the mean:

Distance from the standard for 79: |79 - 79.25| = 0.25

Distance from the standard for 84: |84 - 79.25| = 4.75

Distance from the standard for 81: |81 - 79.25| = 1.75

Distance from the standard for 73: |73 - 79.25| = 6.25

The score with the smallest distance from the average is 79, closest to the standard.

Therefore, the correct answer is:

A) 79

The value of the moment generating function Mx (t) = E [ext] for t = -1 is given by:Mx (-1) = (1/3)M1 (-1) + (2/3)M2 (-1)

Moment generating function of a mixture

The moment generating function (MGF) of a mixture is defined as the linear combination of the MGFs of the mixture components with respect to their probabilities. Thus, if there are n components in a mixture, then the MGF of the mixture is expressed as:

Mx (t) = ∑(i=1 to n)PiMi (t)

where Pi and Mi (t) are the probability and MGF of the i-th component, respectively.

The value of the moment generating function

Mx (t) = E[ext] for t = -1 is given as follows:

Let's assume that the mixture contains two components, one with probability 1/3 and MGF M1 (t), and the other with probability 2/3 and MGF M2 (t).

Then the MGF of the mixture is given by:

Mx (t) = (1/3)M1 (t) + (2/3)M2 (t)

Therefore, to calculate Mx (t) = E [ext] for t = -1, we substitute t = -1 in the MGF expression and obtain:

Mx (-1) = (1/3)M1 (-1) + (2/3)M2 (-1)

Thus, the value of the moment generating function Mx (t) = E [ext] for t = -1 is given by:Mx (-1) = (1/3)M1 (-1) + (2/3)M2 (-1)

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The domain of the function f(x) + g(x) = 3x + 4 is (-∞, +∞), representing all real numbers in interval notation.

To find an expression for the indicated function using the given functions f(x) = 2x + 9 and g(x) = x - 5, we need to specify the operation between the two functions.

a) If the indicated function is the sum of f(x) and g(x), we can write it as:

f(x) + g(x) = (2x + 9) + (x - 5)

Simplifying this expression, we combine like terms:

f(x) + g(x) = 2x + x + 9 - 5

= 3x + 4

Therefore, the expression for the indicated function is 3x + 4.

b) To determine the domain of this function, we consider the values of x for which the expression is defined. Since both f(x) = 2x + 9 and g(x) = x - 5 are defined for all real numbers, their sum, 3x + 4, is also defined for all real numbers.

Thus, the domain of the function f(x) + g(x) = 3x + 4 is (-∞, +∞), representing all real numbers in interval notation.

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The given function is f(x) = /1 + e^x. We are to find the derivative of the function.

Using the quotient rule, we have f'(x) = [(1 + e^x)*e^x - e^x*(e^x)] / (1 e^x)^2

Simplifying, we get f'(x) = e^x / (1 + e^x)^2

We used the quotient rule of differentiation which states that if y = u/v,

where u and v are differentiable functions of x, then the derivative of y with respect to x is given byy'

= [v*du/dx - u*dv/dx]/v²

We can see that the given function can be written in the form y = u/v,

where u = e^x and

v = 1 + e^x.

On differentiating u and v with respect to x, we get du/dx = e^x and

dv/dx = e^x.

We then substitute these values in the quotient rule to get the derivative f'(x)

= e^x / (1 + e^x)^2.

Hence, the derivative of the given function is f'(x) = e^x / (1 + e^x)^2.

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The total time you spend working in a day is 4 hours.

If you work from 8 AM to 12 PM and have a one-hour lunch break from 12 PM to 1 PM, the total time you spend at work is 4 hours. However, considering that you have a one-hour lunch break, your actual working hours would be 3 hours.

From 8 AM to 12 PM, you work for 4 hours.

From 12 PM to 1 PM, you have a lunch break and don't work.

Therefore, the total time you spend working in a day is 4 hours.

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The probability of choosing 2 red and one marbles of each other color is 0.268 .

Given,

Green marbles: 4

Red marbles: 4

White marbles: 3

Purple marbles : 2

Now,

Total marbles to be taken = 5.

Out of 5, 2 will be red and 1 each of three different colors.

Total number of marbles = 13

Choosing 5 marbles from 13,

[tex]13C_5\\13!/5!(13 - 5)!\\[/tex]

=  6435.

So,

2 will be red and 1 each of three different colors.

[tex]4C_2 * 4C_1* 3C_1*2C_1[/tex]

= 6*24*6*2

= 1728

So,

Probability = 1728/6435

= 0.268

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The probability that the smartphone will not work properly is 0.041 or 4.1%.

To find the probability that a smartphone will not work properly, we need to consider the probability that at least one of the 22 distinct parts is defective. Since each part is made with an average quality control where only 1 out of 500 is defective, the probability of a part being defective is 0.002.

To find the probability that none of the parts are defective, we subtract the probability that at least one part is defective from 1.

The probability that at least one part is defective can be found using the complement rule, which states that the probability of an event not occurring is 1 minus the probability of the event occurring.

In this case, the probability that at least one part is defective is 1 minus the probability that all parts are not defective.

Since there are 22 parts, the probability that all parts are not defective is (1 - 0.002)^22.

Therefore, the probability that at least one part is defective is 1 - (1 - 0.002)^22.

To calculate this probability, we can use a calculator or spreadsheet.

The rounded probability that at least one part is defective, and thus the smartphone will not work properly, is 0.041 or 4.1%.

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An equation with the steepest graph has the largest absolute value of slope.

The equation with the steepest graph is the equation with the largest absolute value of slope.

A slope is a measure of how steep a line is.

If a line has a positive slope, it is rising to the right.

If a line has a negative slope, it is falling to the right.

If the slope of a line is zero, the line is horizontal.

To multiply the square root of 2 + i and its conjugate, you can use the complex multiplication formula.

(a + bi)(a - bi) = [tex]a^2 - abi + abi - b^2i^2[/tex]

where the number is √2 + i. Let's do a multiplication with this:

(√2 + i)(√2 - i)

Using the above formula we get:

[tex](\sqrt{2})^2 - (\sqrt{2})(i ) + (\sqrt{2} )(i) - (i)^2[/tex]

Further simplification:

2 - (√2)(i) + (√2)(i) - (- 1)

Combining similar terms:

2 + 1

results in 3. So (√2 + i)(√2 - i) is 3.

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The larger rocket will receive 36 kiloliters of fuel, and the smaller rocket will receive 24 kiloliters of fuel.

The Johnsons purchased 2 adult tickets and 5 student tickets.

Let's assume the amount of fuel allocated for the larger rocket is x kiloliters. Since the smaller rocket receives 12 kiloliters less than the larger rocket, the amount of fuel allocated for the smaller rocket is x - 12 kiloliters.

The total amount of fuel is given as 60 kiloliters:

x + (x - 12) = 60

2x - 12 = 60

2x = 72

x = 36

So, the larger rocket will receive 36 kiloliters of fuel, and the smaller rocket will receive 36 - 12 = 24 kiloliters of fuel.

Now, let's calculate the number of adult and student tickets purchased by the Johnsons. They purchased a total of 7 tickets for $22.

Let's assume the number of adult tickets purchased is a, and the number of student tickets purchased is s. The cost of each adult ticket is $4, and the cost of each student ticket is $2.50.

The total number of tickets purchased is given as 7:

a + s = 7

The total cost of the tickets is given as $22:

4a + 2.50s = 22

Solving these two equations simultaneously will give us the values of a and s.

By solving the equations, we find a = 2 and s = 5.

Therefore, the Johnsons purchased 2 adult tickets and 5 student tickets.

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The evidence does not strongly support the claim that women with above-average looks earn significantly more than women with average looks.

To understand the findings, we need to discuss a few key concepts. First, let's clarify the null hypothesis (H0) and the alternative hypothesis (H1). In this case, the null hypothesis states that there is no relationship between physical appearance and income (β2 = 0), while the alternative hypothesis suggests that there is a relationship (β2 ≠ 0).

In this scenario, the one-sided p-value of 0.272 means that there is a 27.2% chance of observing a relationship between physical appearance and income as strong or stronger than what was found in the study, purely by chance, if there is actually no relationship (β2 = 0). Since this p-value is relatively high (greater than the commonly used threshold of 0.05), it implies weak evidence against the null hypothesis.

Therefore, based on the given information, the evidence does not provide sufficient statistical support to reject the null hypothesis that there is no relationship between physical appearance and income (H0: β2 = 0).

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The company purchased 42 pieces of hardware, including inkjet printers, LCD monitors, and memory chips.

The company has decided to upgrade its office technology and has purchased a total of 42 pieces of hardware, which includes inkjet printers, LCD monitors, and additional memory chips. The cost of each individual hardware item is not provided in the given information.

To determine the cost per item, we need additional details about the total cost of the hardware purchase. Without that information, we cannot calculate the cost per item accurately.

However, once we have the total cost of the hardware purchase, we can divide it by the total number of pieces (42) to find the cost per item. This will give us the average cost for each inkjet printer, LCD monitor, and memory chip that was purchased.

It's important to note that without the specific costs for each hardware item or the total cost of the purchase, we cannot provide an exact calculation for the cost per item.

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Answer:

B- 3 1/3 cups

Step-by-step explanation:

4 servings = 2/3

20 = 4×5

20 servings = 2/3 × 5

=3.33

= 3 1/3

(I'm not 100% sure, this is just what I got! Hope it helps :) )

For mileages greater than 30 miles, company A charges less than company B.

Jessica wants to rent a truck for one day.

There are two companies that she can select from Company A charges $102 and allows unlimited mileage. On the other hand, company B has an initial fee of $75 and charges an additional $0.90 for every mile driven.

We need to find out the mileages for which company A charges less than company B.

In Company A, the cost is $102 for unlimited mileage.

In Company B, the cost is $75 plus $0.9 for every mile.

The cost can be represented by the function f(m) = 0.9m + 75, (where m represents the mileage).

Let us find out the mileages for which company A charges less than company B. Cost of company A is less than company B.

102 < 0.9m + 75 (Substituting the value of Company A and Company B)0.9m > 27 (Solving for m) m > 30

So, for mileages greater than 30 miles, company A charges less than company B.

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The probability of getting a sample average of 93 or more customers, if the manager had not offered the discount, is zero.

To calculate the probability of obtaining a sample average of 93 or more customers if the manager had not offered the discount, we need to use the concept of sampling distributions and the Central Limit Theorem.

Given that the underlying distribution of the number of customers is normal, with an average of 22 and a standard deviation of 6, we can calculate the standard deviation of the sample mean (also known as the standard error) using the formula:

Standard Error (SE) = Standard Deviation / √(Sample Size)

In this case, the sample size is 5 (for the 5-weekday period), so the standard error is:

SE = 6 / √5 ≈ 2.683

Next, we can calculate the z-score for a sample average of 93 using the formula:

z = (Sample Average - Population Mean) / Standard Error

z = (93 - 22) / 2.683 ≈ 26.359

Finally, we can use the standard normal distribution table or a calculator to find the probability associated with this z-score:

P(Sample Average ≥ 93) = P(z ≥ 26.359)

Since the z-score is extremely large, the probability associated with it is essentially zero.

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The exact cost of producing the 31st coffee maker is $2962.80. By substituting x = 32, we find the cost at that specific quantity. This represents the approximate total cost incurred from producing and selling 32 coffee makers.

(A) To find the exact cost of producing the 31st coffee maker, we can substitute x = 31 into the cost function C(x) = 1760 - 0.2x^2 + 45x:

C(31) = 1760 - 0.2(31)^2 + 45(31)

      = 1760 - 0.2(961) + 1395

      = 1760 - 192.2 + 1395

      = 2962.80

Therefore, the exact cost of producing the 31st coffee maker is $2962.80.

(B) To approximate the cost of producing the 31st coffee maker, we can use the marginal cost function C'(x) = -0.4x + 45.

The marginal cost represents the rate at which the cost changes with respect to the quantity produced. Since C'(30) = 33, we can use this information to estimate the change in cost from producing 30 to 31 coffee makers:

C'(30) ≈ (C(31) - C(30))/(31 - 30)

33 ≈ (C(31) - 2930)/(31 - 30)

Now, solving for C(31):

33 ≈ (C(31) - 2930)/1

33 ≈ C(31) - 2930

C(31) ≈ 33 + 2930

C(31) ≈ 2963

Therefore, the approximate cost of producing the 31st coffee maker is $2963.

(C) To approximate the total cost from selling 32 coffee makers, we can again use the cost function C(x) = 1760 - 0.2x^2 + 45x. Substituting x = 32:

C(32) = 1760 - 0.2(32)^2 + 45(32)

      = 1760 - 0.2(1024) + 1440

      = 1760 - 204.8 + 1440

      = 2995.20

Therefore, the approximate total cost from selling 32 coffee makers is $2995.20.

(A) To find the exact cost of producing the 31st coffee maker, we substitute x = 31 into the cost function C(x). This gives us the precise value of the cost at that particular quantity.

(B) In this case, we approximate the cost of producing the 31st coffee maker using the marginal cost function C'(x). Since C'(30) is given,

we can estimate the change in cost from producing 30 to 31 coffee makers. By applying the definition of the derivative, we approximate the cost at x = 31 by rearranging the equation and solving for C(31).

(C) To approximate the total cost from selling 32 coffee makers, we once again use the cost function C(x). By substituting x = 32, we find the cost at that specific quantity. This represents the approximate total cost incurred from producing and selling 32 coffee makers.

It's important to note that these calculations involve approximations based on the given information and the assumptions made. For more accurate results, additional data points or a more precise model may be necessary.

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if three diagonals are drawn inside a hexagram, each passing through the center point of the hexagram, a total of 18 triangles are formed.

If three diagonals are drawn inside a hexagram, each passing through the center point of the hexagram, we can determine the number of triangles formed.

Let's break it down step by step:

1. Start with the hexagram, which has six points connected by six lines.
2. Each of the six lines represents a side of a triangle.
3. The diagonals that pass through the center point of the hexagram split each side in half, creating two smaller triangles.
4. Since there are six lines in total, and each line is split into two smaller triangles, we have a total of 6 x 2 = 12 smaller triangles.
5. Additionally, the six lines themselves can also be considered as triangles, as they have three sides.
6. So, we have 12 smaller triangles formed by the diagonals and 6 larger triangles formed by the lines.
7. The total number of triangles is 12 + 6 = 18.

In conclusion, if three diagonals are drawn inside a hexagram, each passing through the center point of the hexagram, a total of 18 triangles are formed.

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Answer:

Step-by-step explanation:

2 is a prime number and is even, therefore F is false

Prime numbers are numbers that only have two factors (1 and itself). Therefore G is false

9 is a composite number having 1, 3, 9 as its factors. 9 is not divisible by 2 and hence H is false.

Prime numbers are numbers with only two factors. Composite numbers are numbers that are not prime. Therefore, all composite numbers have more than two factors. Therefore, J is true

Answer: J

If the area of a trapezoid is 49 square meters, one base is 5 meters long and the other is 2 meters long, then the height of the trapezoid is 14 meters.

To find the height of the trapezoid, follow these steps:

Therefore, the height of the trapezoid is 14 meters.

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Computing Variance & Standard Deviation using the Computational formula. Steps for Computing Variance:

Mean of the Data Set

Mean = (4+3+6+4+8)/5 = 5

For each data point, subtract the mean and square the result. Data Point

Deviation from Mean

Deviation from Mean Squared

4-5=-1

(-1)²=1  

3-5=-2

(-2)²=4  

6-5=1

(1)²=1  

4-5=-1

(-1)²=1  

8-5=3

(3)²=9  

Sum the squares from step 2.

1+4+1+1+9=16

Divide the sum from step 3 by the sample size minus one.16/4=4

Variance = 4

Take the square root of the result from step 4.

Standard Deviation = √4 = 2

In order to find the spread of the given data set, it is essential to calculate the variance and standard deviation of the data set. Variance and Standard Deviation measures the variability of a data set. Variance and standard deviation provide important measures of variability and dispersion in statistical analysis.The formula for variance is given as: variance = ∑(X - μ)²/N-1

And the formula for standard deviation is given as:

standard deviation = √∑(X - μ)²/N-1

Here, we have a sample of data Scores: 4, 3, 6, 4, 8.

Now, we will calculate the variance and standard deviation of this sample by using the computational formula. The mean of this sample is (4+3+6+4+8)/5 = 5.

Using the formula for variance, we get:(4 - 5)² + (3 - 5)² + (6 - 5)² + (4 - 5)² + (8 - 5)² / (5 - 1) = 16/4 = 4

Therefore, the variance of this sample is 4.

Using the formula for standard deviation, we get:√[(4 - 5)² + (3 - 5)² + (6 - 5)² + (4 - 5)² + (8 - 5)²] / (5 - 1) = √16/4 = 2

Therefore, the standard deviation of this sample is 2.

Hence, we can conclude that the variance and standard deviation for the given sample data Scores: 4, 3, 6, 4, 8 are 4 and 2 respectively.

Thus, we can infer that variance and standard deviation are essential measures of variability and dispersion in statistical analysis, which helps in measuring the spread of a data set.

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The solution to the given differential equation is x + y = 2arctan(e^(x+C)) - π.

To solve the differential equation dy/dx = sin(x+y), we can make the substitution A = 1, B = 1, and C = 0.

This substitution allows us to rewrite the equation as dy/dx = f(x+y). Let u = x + y, then differentiate both sides with respect to x using the chain rule: du/dx = 1 + dy/dx.

Rearranging the equation, we have dy/dx = du/dx - 1. Substituting this into the original equation, we get du/dx - 1 = sin(u).

Rearranging, we have du/dx = 1 + sin(u). This is a separable differential equation.

Separating variables and integrating, we have du/(1 + sin(u)) = dx. Integrating both sides, we obtain ln|tan(u/2 + π/4)| = x + C, where C is the constant of integration.

Finally, solving for u, we have u = 2arctan(e^(x+C)) - π. Substituting back u = x + y, we get x + y = 2arctan(e^(x+C)) - π, which is the general solution to the given differential equation.

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To convert the system into an augmented matrix, we can represent the given equations as follows:

1   -5   4   |  22

2   -12  4   |  8

To reduce the system to echelon form, we'll perform row operations to eliminate the coefficients below the main diagonal:

R2 = R2 - 2R1

1   -5   4   |  22

0   -2   -4  |  -36

Next, we'll divide R2 by -2 to obtain a leading coefficient of 1:

R2 = R2 / -2

1   -5   4   |  22

0   1    2   |  18

Now, we'll eliminate the coefficient below the leading coefficient in R1:

R1 = R1 + 5R2

1   0    14  |  112

0   1    2   |  18

The system is now in echelon form. To determine if it is consistent, we look for any rows of the form [0 0 ... 0 | b] where b is nonzero. In this case, all coefficients in the last row are nonzero. Therefore, the system is consistent.

To find the solution, we can express x1 and x2 in terms of the free variable s1:

x1 = 112 - 14s1

x2 = 18 - 2s1

x3 is independent of the free variable and remains unchanged.

Therefore, the solution is (x1, x2, x3) = (112 - 14s1, 18 - 2s1, s1), where s1 is any real number.

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Answer:

Given n = 122 and p = 0.73, we can find the mean, variance, and standard deviation of the binomial distribution using the following formulas:

mean = np

variance = np(1-p)

standard deviation = sqrt(np(1-p))

Substituting the given values into these formulas, we get:

mean = np = 122 x 0.73 = 89.06

variance = np(1-p) = 122 x 0.73 x (1-0.73) = 24.13

standard deviation = sqrt(np(1-p)) = sqrt(122 x 0.73 x (1-0.73)) = 4.91

Therefore, the mean of the binomial distribution is 89.06, the variance is 24.13, and the standard deviation is 4.91.