find the exact value of all six trigonometric functions of an acute angle, in standard position if the radius is 6 and the x coordinate is 5

Answer: x = 41

Step-by-step explanation:

When you add supplementary angles together, you should have a total of 180 degrees. You can set up and equation like this...

(2x - 25) + 3x = 180

Combine like terms.   5x - 25 = 180

Add 25 to both sides to help isolate x     5x = 205

Divide by 5 to both sides to get x by itself.   x = 41

The size of the augmented matrix for this system will be 2x3. Option B is the correct answer.

Augmented matrix is a mathematical tool developed in matrix theory to solve linear equations that combine two matrices or vectors to make a new matrix type.

For solving linear equations, the coefficients of the linear equations are written in a matrix form with a column beside it containing its constant values, which makes it an augmented matrix.

one example of an augmented matrix is,

x+y=1

x-y=2

can be represented as

= [tex]\left[\begin{array}{ccc}42&61&|393\\59&78&|529\\\end{array}\right][/tex]

In the given question the linear equations contain two variables, hence the number of rows in the augmented matrix will be 2.

Since it is an augmented matrix it will contain 3 columns with 2 columns of the coefficient matrix and 1 column of the constant matrix.

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The total length of the tiles that Chris needs to cover the floor is 63/8 inches.

Multiplication is a method of finding the product of two or more numbers

We need to find the total length of the tiles, we need to multiply the number of tiles by the length of each tile.

The number of tiles Chris needs is 9.

The length of each tile is 7/8 of an inch.

To find the total length of the tiles, we can multiply the number of tiles by the length of each tile as follows:

Total length of tiles = Number of tiles x Length of each tile

Total length of tiles = 9 x (7/8) inches

Total length of tiles = 63/8 inches

Therefore, the total length of the tiles that Chris needs to cover the floor is 63/8 inches.

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Complete question is given below.

Chris needs 9 tiles to cover length of floor, each tile is 7/8 of an inch what is the total length of tiles?

The probability of exactly 90 defective washers is approximately 0.021, and the probability of at least 2 defective items is approximately 0.985.

A shipment of 1500 washers contains 400 defective and 1100 non-defective washers. The probability of exactly 90 defective washers being chosen and the probability of at least 2 defective items being found can be calculated using the hypergeometric distribution formula.

A. The probability of exactly 90 defective washers being chosen can be calculated using the hypergeometric distribution. The probability mass function of the hypergeometric distribution is given by:

P(X = 90) = (400 choose 90) * (1100 choose 110) / (1500 choose 200)

Substituting the given values, we get:

P(X = 90) ≈ 0.021

B. The probability of at least 2 defective items can be calculated as 1 minus the probability of 0 or 1 defective item. The probability of 0 defective items is given by:

P(X = 0) = (1100 choose 200) / (1500 choose 200)

The probability of 1 defective item is given by:

P(X = 1) = (400 choose 1) * (1100 choose 199) / (1500 choose 200)

Substituting these values, we get:

P(X >= 2) = 1 - P(X = 0) - P(X = 1) ≈ 0.985.

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The complete question is:

A shipment of 1500 washers contains 400 defective and 1100 non-defective washers. Two hundred washers are chosen at random (without replacement) and classified as defective or non-defective_ a) What is the probability that exactly 90 defective washers are found? (Do NOT compute out:) b) What is the probability that at least 2 defective items are found?

The distance between point D and E is 17.89 units

Distance between two points is the length of the line segment that connects the two points in a plane. The formula to find the distance between the two points is usually given by d=√((x2 – x1)² + (y2 – y1)²). This formula is used to find the distance between any two points on a coordinate plane or x-y plane.

d=√((x2 – x1)² + (y2 – y1)²).

d = √ (11-3)² + 13-(-3)²

d = √ 8²+ 16²

d = √ 64 + 256

d = √320

d = 17.89 units

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The Cartesian Equation of the given parametric equation is written below

: x^2 + y^2=4

To eliminate the parameter t and find the Cartesian equation, we can use the trigonometric identity: cos^2(t) + sin^2(t) = 1. We can rearrange the given equations to get:

x(t) = 2cos(t)

y(t) = 2sin(t)

Dividing both sides of each equation by 2, we get:

cos(t) = x/2

sin(t) = y/2

Squaring both equations and adding them, we get:

cos^2(t) + sin^2(t) = (x/2)^2 + (y/2)^2

Using the identity, we can simplify the right-hand side to get:

1 = (x/2)^2 + (y/2)^2

Multiplying both sides by 4, we get the Cartesian equation:

x^2 + y^2 = 4

This is the equation of a circle with radius 2 centered at the origin. The domain of this equation is all real numbers, and the range is from -2 to 2 (inclusive) in both the x and y directions, represented as [-2,2].

In summary, the Cartesian equation of the given parametric equations is x^2 + y^2 = 4, with domain of all real numbers and range of [-2,2].

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A system may or may not be:

a.) Memoryless: Yes, Time Invariant: Yes, Linear: Yes, Casual: No, Stable: Yes

b.) Memoryless: No, Time Invariant: No, Linear: No, Casual: No, Stable: Yes

c.) Memoryless: No, Time Invariant: No, Linear: No, Casual: Yes, Stable: Yes

d.) Memoryless: Yes, Time Invariant: No, Linear: No, Casual: No, Stable: Yes

e.) Memoryless: Yes, Time Invariant: No, Linear: No, Casual: No, Stable: Yes

f.) Memoryless: No, Time Invariant: Yes, Linear: Yes, Casual: No, Stable: Yes

g.) Memoryless: Yes, Time Invariant: Yes, Linear: No, Casual: Yes, Stable: Yes

A memoryless, time invariant, linear, casual, and stable system is an idealized system in which the output is directly proportional to the input and the output at any given time is independent of the outputs at any other time.

The system is stable in the sense that the output does not grow indefinitely as the input increases, and it is casual in the sense that the output at any given time is only affected by the inputs at previous times.

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The lines L1 and L2 are not parallel and intersect at a single point. We have also found the point of intersection.

If the direction vectors are parallel, then the lines are parallel. If the direction vectors are not parallel but do not intersect, then the lines are skew. If the direction vectors are not parallel and do intersect, then the lines intersect at a single point.

To determine whether the given lines are parallel, skew, or intersecting, we need to compare their direction vectors. The direction vectors of the lines are the coefficients of the parameters t and s in the respective equations. Thus, the direction vector of L1 is <2,-1,3> and the direction vector of L2 is <4,-2,5>.

To check whether the direction vectors are parallel, we can compute their cross product. If the cross product is the zero vector, then the direction vectors are parallel.

<2,-1,3> × <4,-2,5> = (7,2,-10)

Since the cross product is not zero, the direction vectors are not parallel. Thus, the lines are either skew or intersecting.

To determine whether the lines are intersecting, we can set the parametric equations of the lines equal to each other and solve for t and s. This will give us the point of intersection.

3 + 2t = 1 + 4s

4 - t = 3 - 2s

1 + 3t = 4 + 5s

Rearranging the equations, we get:

2t - 4s = -2

t + 2s = 1

3t - 5s = 3

Using Gaussian elimination or other methods, we can solve for t and s:

t = 11/13

s = 2/13

Substituting these values back into either equation gives us the point of intersection:

x = 3 + 2(11/13) = 37/13

y = 4 - (11/13) = 41/13

z = 1 + 3(11/13) = 40/13

Thus, the lines intersect at the point (37/13, 41/13, 40/13).

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Complete Question:

Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.

L1: x = 3 + 2t, y = 4 – t, z = 1 + 3t

L2: x = 1 + 4s, y = 3 – 2s, z = 4 + 5s

The similar doll's quilt has a width of (1/4) yards.

Two or more than two shapes are said to be similar if they have equal corresponding measures of angles and the ratio of corresponding sides should be the same.

Given, The daughter's quilt has a length of 2 yards and a width of 1 yard

and the other similar quilt ahs a length of (1/2) yards.

Now, (1/2)/(2) = 1/4.

Therefore, The width of a similar quilt is (1/4)×1 = 1/4 yards.

Q. A woman sews similar quilts for her daughter's doll. If the daughter's quilt has a length of 2 yards and a width of 1 yard, and the doll's quilt has a length of 1/2 yard, what is the width of the doll's quilt?

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The required minimal sample size would be 26. This is because in order to estimate the mean score with a margin of error of 10 and a confidence level of 90%, the sample size must be large enough for the Central Limit Theorem to apply.  Therefore, in this case, the sample size would be (1.96*100)/10 = 19.6, which is approximately 26.

1. Calculate the sample size, n: n = (1.96*σ)/m

2. Plug in the values: n = (1.96*100)/10

3. Calculate: n = 19.6

4. Round up: n = 26

The required minimal sample size of 26 is necessary in order to estimate the mean SAT score for a population of students with a 90% confidence interval and a margin of error of 10. This is because, in order for the Central Limit Theorem to apply, the sample size must be large enough. The formula for the sample size is (1.96*σ)/m, where σ is the population standard deviation and m is the margin of error. Plugging in the values, we get (1.96*100)/10, which is approximately 19.6. Therefore, the required minimal sample size is 26. This sample size is necessary in order to make sure that the estimates have a high degree of accuracy and reliability.

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Answer:

the logistic function that models the population of this species as a function of time is P(t) = 7000 / (1 + 0.05 * e^(-0.1682 * t)).

Step-by-step explanation:

We can use the logistic growth equation to model the population of this species as a function of time:

P(t) = K / (1 + A * e^(-r * t))

Where:

P(t) is the population at time t

K is the carrying capacity of the environment

A is the initial population as a proportion of the carrying capacity (A = P(0)/K)

r is the growth rate of the population

We are given that the initial population A is 350/7000 = 0.05 (since the carrying capacity is 7000). We are also given that after 5 years the population has grown to 900. So we can use this information to find the growth rate r:

P(5) = 900 = K / (1 + A * e^(-r * 5))

We also know that the carrying capacity K is 7000. Substituting these values, we get:

900 = 7000 / (1 + 0.05 * e^(-r * 5))

Multiplying both sides by the denominator and simplifying, we get:

1 + 0.05 * e^(-r * 5) = 7.777...

Subtracting 1 from both sides, we get:

0.05 * e^(-r * 5) = 6.777...

Dividing both sides by 0.05, we get:

e^(-r * 5) = 135.555...

Taking the natural logarithm of both sides, we get:

-ln(135.555...) = -r * 5

Solving for r, we get:

r = 0.1682...

Now that we have the growth rate r, we can use the logistic growth equation to find the function that models the population as a function of time:

P(t) = 7000 / (1 + 0.05 * e^(-0.1682 * t))

Therefore, the logistic function that models the population of this species as a function of time is P(t) = 7000 / (1 + 0.05 * e^(-0.1682 * t)).

The given relationship is a function, as of the given condition it consists of 3 sub-functions,

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

here,

Let's assume that, 3-time intervals

1 . 0 to 5

2. 5 to 7  

3. 7 to 10

So for intervals, we have 3 different equation of functions which represents the distance traveled by Camilla in time t.

Thus,  the given relationship is a function, as of the given condition it consists of 3 sub-functions.

When she increased her pace, she principally made up for lost time standing at the position. assuming that she made it in the same quantum of time if she did not stop, you can assume that she didn't waste presently time because she picked up the pace.

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The expected value of cream-filled donuts the store City Donuts is selling should be 2.

We are aware of In the realm of probability theory, the expected value, commonly referred to as the expected average, is a generalization of the weighted average.

Given, City Donuts recently sold 14 donuts, of which 4 were cream-filled donuts.

So, The probability of cream-filled donuts is = 4/14 = 2/7.

Therefore, The expected number of cream-filled donuts is,

= (2/7)×7.

= 2.

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Answer: 97.5% ( look at the image for the solution )

Step-by-step explanation:

x=-1/2 and  x=-3 are the zeros of the function y =2x²+7x+3

A quadratic equation is a second-order polynomial equation in a single variable x , ax²+bx+c=0. with a ≠ 0 .

The given function is y= 2x²+7x+3

Factor out the equation

2x²+6x+x+3

2x(x+3)+(x+3)

(2x+1)(x+3)=0

So the roots we get

2x+1=0

x=-1/2 is one  zero and x=-3 is another zero.

Hence, x=-1/2 and  x=-3 are the zeros of the function y =2x²+7x+3

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The solution to the proportion  4/(x+1) = 3/(x+2) is x = -5 and

The solution to the proportion  x/(x+3) = (x-2)/x is x = 2.

A quadratic polynomial is a polynomial of degree 2, which means that the highest power of the variable in the polynomial is 2. A quadratic polynomial can be expressed in the general form:

a[tex]x^2[/tex] + bx + c

where a, b, and c are constants, and x is the variable.

In this form, coefficient a is the leading coefficient, and it determines the concavity of the quadratic function. If a is positive, the function is concave up, meaning that it opens upward and has a minimum value. If a is negative, the function is concave down, meaning that it opens downward and has a maximum value.

The quadratic polynomial is named as such because it is a polynomial of degree 2, and the term "quad" means "square." The quadratic function is commonly seen in geometry, algebra, and physics, and it has many applications in real-life problems, such as finding the maximum or minimum values of a function, determining the trajectory of a projectile, and modeling certain natural phenomena.

Quadratic polynomials can be solved by various methods, including factoring, completing the square, and using the quadratic formula.

To solve the proportion 4/(x+1) = 3/(x+2), we can cross-multiply to get:

4(x+2) = 3(x+1)

Expanding the brackets gives:

4x + 8 = 3x + 3

Subtracting 3x from both sides gives:

x + 8 = 3

Subtracting 8 from both sides gives:

x = -5

To solve the proportion x/(x+3) = (x-2)/x, we can cross-multiply to get:

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

Expanding the brackets gives:

[tex]x^2[/tex]- 2x = [tex]x^2[/tex] + x - 6

Subtracting[tex]x^2[/tex] from both sides gives:

-2x = x - 6

Subtracting x from both sides gives:

-3x = -6

Dividing both sides by -3 gives:

x = 2

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Round each number to the nearest whole

Compare the estimated sum to the actual sum

Look at the tenths place to round to the nearest whole number

The estimated answer is close to the actual answer

Two or more expressions with an Equal sign is called as Equation.

The key phrases used are:

Round each number to the nearest whole

Compare the estimated sum to the actual sum

Look at the tenths place to round to the nearest whole number

The estimated answer is close to the actual answer

These phrases are all related to the process of approximating a sum by rounding the addends to the nearest whole number and comparing the resulting estimate to the exact sum.

Hence, Round each number to the nearest whole

Compare the estimated sum to the actual sum

Look at the tenths place to round to the nearest whole number

The estimated answer is close to the actual answer

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If a certain affects virus 0.2% of the population:

a. The probability that a person has the virus given that they have tested positive is  1.7%.

b. The probability that a person does not have the virus given that they test negative is 99.8%.

a) Using Bayes' theorem, we can find the probability that a person has the virus given that they have tested positive:

P(Infected | Positive Test) = P(Positive Test | Infected) * P(Infected) / P(Positive Test)

where P(Positive Test) is the probability of a positive test result, regardless of whether the person is infected or not. This can be calculated as:

P(Positive Test) = P(Positive Test | Infected) * P(Infected) + P(Positive Test | Not Infected) * P(Not Infected)

= 0.88 * 0.002 + 0.1 * 0.998

= 0.10064

Now we can plug in the given values:

P(Infected | Positive Test) = 0.88 * 0.002 / 0.10064 ≈ 0.0174

So the probability that a person has the virus given that they have tested positive is approximately 1.7%.

b) Similarly, we can use Bayes' theorem to find the probability that a person does not have the virus given that they test negative:

P(Not Infected | Negative Test) = P(Negative Test | Not Infected) * P(Not Infected) / P(Negative Test)

where P(Negative Test) is the probability of a negative test result, regardless of whether the person is infected or not. This can be calculated as:

P(Negative Test) = P(Negative Test | Infected) * P(Infected) + P(Negative Test | Not Infected) * P(Not Infected)

= 0.12 * 0.002 + 0.998 * 0.998

= 0.99796

Now we can plug in the given values:

P(Not Infected | Negative Test) = 0.998 * 0.99796 / 0.998 = 0.99796

So the probability that a person does not have the virus given that they test negative is approximately 99.8%.

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Answer:

the number is 14.

Step-by-step explanation:

Let's start by using algebra to solve the problem.

We can translate the given information into an equation:

5x + 2 = 72

where x is the unknown number.

To solve for x, we need to isolate it on one side of the equation. We can do this by first subtracting 2 from both sides:

5x = 70

Then, we can divide both sides by 5:

x = 14

Therefore, the number is 14.

The probability of being dealt a flush is 0.001980,  probability of being dealt a one pair is 0.422569,  probability of being dealt two pairs is 0.04753.

a) First select a suit: 4 choices

then, we can have to select 5 cards out of 13:  C(13, 5)

required probability: 4*C(13, 5)/C(52, 5)

= 0.001980

b)- We have to select two cards to make a pair: C(4, 2) choices (we have to select 2 from 4 because a suit can't have 2 cards    of  same number, so we are selecting 2 suits from 4),

then select a number for the pair: 13 choices,

now for remaining three cards: each card can be of same suit but number must be different, i.e. 4³*C(12, 3)

required probability: 13*C(4, 2) * C(12, 3) *4³ / C(52, 5)

= 0.422569

c)- First, we have to select two numbers for two pairs: C(13, 2) ,

Select suits for both pairs: C(4, 2) * C(4, 2) ,

For the remaining one card: 11 choices for the number and 4 choices for the suit,

required probability: C(13, 2)*C(4, 2)*C(4, 2)*11*4/C(52, 5)

= 0.04753

Probability means possibility. It's a branch of mathematics that deals with the circumstance of a arbitrary event. The value is expressed from zero to one. Probability has been introduced in Maths to prognosticate how likely events are to be. The meaning of probability is principally the extent to which commodity is likely to be.

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Complete question:

If it is assumed that all (52 5) poker hands are equally likely, what is the probability of being dealt (a) a flush? (A hand is said to be a flush if all 5 cards are of the same suit.)

(b) one pair? (This occurs when the cards have denominations a, a, b, c, d, wherea, b, c, and d are all distinct.)

(c) two pairs? (This occurs when the cards have denominations a, a, b, b, c, wherea, b, andc are all distinct.)

Answer:

Step-by-step explanation:

The sampling distribution of the mean will be approximately normal, regardless of the underlying population distribution.

Population sampling is the process of selecting a representative sample from a larger population for the purpose of making statistical inferences about the population.

a. The sampling distribution for samples of 25 customer receipts for a supermarket for a year would likely be approximately normally distributed. This assumes that the receipts are randomly selected from the entire population of receipts and that the population distribution of receipts is roughly normal.

The central limit theorem suggests that for a sufficiently large sample size (usually at least 30), the sampling distribution of the mean will be approximately normal, regardless of the underlying population distribution.

b. The sampling distribution for samples of 25 insurance payouts in a particular geographical area in a year would likely be positively skewed, assuming that most payouts are relatively small and a few are very large. The central limit theorem may still apply, but the distribution of the sample means would not be exactly normal.

c. The sampling distribution for samples of 25 call centre logs of inbound calls tracking handling time for a credit card company during the year would likely be approximately normally distributed. This assumes that the call centre logs are randomly selected from the entire population of logs and that the population distribution of handling times is roughly normal.

The central limit theorem suggests that for a sufficiently large sample size (usually at least 30), the sampling distribution of the mean will be approximately normal, regardless of the underlying population distribution.

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The transformed function is g ( x ) = ( -1/2 ) | x - 1 | + 2

The transformation of a function may involve any change.

Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs), etc.

If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:

Horizontal shift (also called phase shift):

Left shift by c units: y=f(x+c) (same output, but c units earlier)

Right shift by c units: y=f(x-c)(same output, but c units late)

Vertical shift:

Up by d units: y = f(x) + d

Down by d units: y = f(x) - d

Stretching:

Vertical stretch by a factor k: y = k × f(x)

Horizontal stretch by a factor k: y = f(x/k)

Given data ,

Let the parent function be represented a f ( x )

Now , the value of f ( x ) = | x |

Let the function be plotted on the graph and

when the function is transformed by a reflection on y axis , we get

h ( x ) = - | x |

Let the function be shifted left by 1 unit , we get

h' ( x ) = - | x - 1 |

Now , when the function is transformed by a vertical stretch of ( 1/2 ) units , we get

j ( x ) = - ( 1/2 ) | x - 1 |

Now , when the function is shifted vertically up by 2 units , we get

g ( x ) = ( -1/2 ) | x - 1 | + 2

Hence , the transformed function is g ( x ) = ( -1/2 ) | x - 1 | + 2 and it is plotted

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The statement (i) and (iv) is true.

An equation containing variables with its highest power as one is called linear equation.

The cost of 2 game apps is $3.50; the cost of 5 game apps is $8.75.

The graph below represents the linear relationship.

By the graph, the equation of the graph is y=1.75x

And the point (0,0) lies on the graph.

The points (7.3,6) and (31.75,51) is not on the line.

Hence, the statement (i) and (iv) is true.

Question:

The cost of 2 game apps is $3.50; the cost of 5 game apps is $8.75.The graph below represents this linear relationship. Select all true statements about the graph.

(i) The equation of the graph is y=1.75x.

(ii) The point (7.3,6) lies on the line.

(iii) The point (31.75,51) lies on the line.

(iv) The point (0,0) lies on the line.

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Buying 36 bottles for $4 is cheaper than buying 3 bottles for 24.

A unitary method is a mathematical way of obtaining the value of a single unit and then deriving any no. of given units by multiplying it with the single unit.

Given, Target sells 24 bottles for 3$,

Therefore, The unit price of a bottle is $(3/24) = $0.125.

Target also sells 36 bottles for $4,

Therefore, The unit price of a bottle is,

= $(4/36) = $0.11.

So, Buying 36 bottles for $4 is cheaper.

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A variable can be both Nominal and ordinal as shown in the explanation below.

Nominal data is defined as data that's used for naming or labeling variables, without any quantitative value. It's occasionally called “ named ” data – a meaning chased from the word nominal.

Ordinal data is a type of categorical data with an order. The variables in ordinal data are listed in an ordered manner. The ordinal variables are generally numbered, so as to indicate the order of the list.

Age can be both nominal and ordinal data depending on the question types. I.e “ How old are you ” is used to collect nominal data while “ Are you the firstborn or What position are you in your family ” is used to collect ordinal data.

An interval variable is analogous to an ordinal variable, except that the intervals between the values of the numerical variable are inversely spaced.

A categorical variable( occasionally called a nominal variable) is one that has two or further orders, but there's no natural ordering to the orders.

separate variables are innumerable in a finite quantum of time. For illustration, you can count the change in your fund. You can count the plutocrat in your bank account. You could also count the quantum of plutocrat in everyone’s bank accounts. It might take you a long time to count that last item, but the point is it’s still innumerable.

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Answer:

there are 458640 different ways to award the six medals.

Step-by-step explanation:

There are two distinct groups in the competition, boys and girls. The order in which the medals are awarded to the groups does not matter, so we can count the number of ways to award medals to the boys and the girls separately, and then multiply the results.

For the boys, there are 7 possible candidates for the gold medal. Once the gold medalist is chosen, there are 6 remaining candidates for the silver medal, and 5 candidates for the bronze medal. So, there are 7 x 6 x 5 = 210 ways to award medals to the boys.

Similarly, for the girls, there are 14 possible candidates for the gold medal. Once the gold medalist is chosen, there are 13 remaining candidates for the silver medal, and 12 candidates for the bronze medal. So, there are 14 x 13 x 12 = 2184 ways to award medals to the girls.

To get the total number of ways to award the six medals, we multiply the number of ways to award medals to the boys by the number of ways to award medals to the girls:

Total number of ways = 210 x 2184 = 458640

Therefore, there are 458640 different ways to award the six medals.

The circumstances in which the median is the preferred measure of central tendency is given as follows:

There are three measures of central tendency of a data-set, which are defined as follows:

The median represents the middle value of the distribution, hence it should be used in these following cases:

They should be used in this case as the median is less affected by outliers compared to the mean.

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In the diagram, the area of the square c is 2009

The area of the square is solved using the formula

length x length

The lenght is caluclalted using Pythagoras theorem given by the  the formula

hypotenuse² = opposite² + adjacent²

plugging the values as in the problem

let x be the required dimensions

x² = 35² + 28²

x² = 1225 + 784

x² =  2009

x = √2009

x = 44.82

Area of the square is x² =  2009

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