(a) What is the probability that the digit 1 doesn’t appear among n digits where each digit is one of (0-9) and all sequences are equally likely?
(b) Suppose you randomly permute the numbers 1, 2, . . . , n, that is, you select a permutation uniformly at random. What is the probability that the number k ends up in the i-th position in the resulting permutation?
(c) A fair coin is flipped n times (each outcome in {H, T} n is equally likely). What is the probability that all heads occur at the end of the sequence? (The case that there are no heads is a special case of having all heads at the end of the sequence, i.e. 0 heads.)

The number of ways that the college team can end the season with five wins in ten football games can be calculated using the combination formula.

The formula for a combination is as follows: `C(n,r) = n! / r!(n-r)!` where n is the total number of games played in a season, and r is the number of games won by the team.

The symbol ! is a factorial notation, meaning that the product of all the positive integers up to that number is multiplied. In other words, 5! is equal to 5 x 4 x 3 x 2 x 1, which equals 120.

In this way, the combination formula can be calculated as: C(10,5) = 10! / 5!(10-5)! = (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ((5 x 4 x 3 x 2 x 1) x (5 x 4 x 3 x 2 x 1))= (10 x 9 x 8 x 7 x 6) / (5 x 4 x 3 x 2 x 1) = 252. Therefore, there are 252 ways in which the college team can end the season with five wins in ten football games.

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The degree of a polynomial is the highest exponent of its variable. When multiplying monomials and binomials, we add the exponents of the variables to get the degree of the resulting polynomial.

However, the degree of the product can be different from the degree of the factors because of the distributive property of multiplication.

When we multiply a monomial by a binomial, we need to distribute the monomial to each term of the binomial. This results in new terms that may have different degrees than the original terms. For example, when multiplying the monomial x^2 by the binomial (2x + 3), we get:

x²(2x + 3) = 2x³ + 3x²

The degree of the monomial x² is 2, and the degree of the binomial (2x + 3) is 1. However, the degree of the resulting polynomial 2x³ + 3x² is 3 because the term 2x³ has a higher degree than any of the original terms.

Similarly, when we multiply two binomials, we need to distribute each term of one binomial to each term of the other binomial. This can also result in new terms with different degrees than the original terms.

Therefore, the degree of the product can be different from the degree of the factors when multiplying monomials and binomials because of the distributive property of multiplication.

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Tree M most closely reflects how the three species of  the spiny back are related to one another.

The branches of the tree, which are marked with the species names, demonstrate that the three species are all closely related, but still have distinct differences between them. For example, Species 1 and 2 share a relatively recent common ancestor, while Species 3 has a much more distant one. The species are represented as separate branches of the tree to show the different pathways of evolution that each species has taken.

In conclusion, Tree M most accurately reflects the evolutionary history of the three species of the spiny back. It shows that while they are all related, they still have distinct differences between them.

The complete question is:

A single species of fish, the three-species  spiny back, once inhabited a large lake. At some point in the lake's history, lava flowed from a nearby volcano into the lake cutting it into three completely isolated mini-lakes. Despite the heat from the lava, a few individuals from the original population of spiny back survive in each mini-lake. Three million years later, a researcher finds that each mini-lake is inhabited by its own species of spiny back. Which of the following figures MOST closely reflects how the three species of spiny back are related to one another?

A.) tree H B.) tree L C.) tree M D.) tree K.

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To find equivalent fractions and ratios, one must understand the concept of proportionality and be able to apply it to solve equations, identify common factors, and simplify fractions. Additionally, one should be able to multiply and divide fractions and ratios.

a. To find equivalent fractions without context, you need to use a form of division. Start by dividing the numerator (top number) of the fraction by the denominator (bottom number). The result of this division gives you the number of times the fraction is equivalent to itself. Multiply both the numerator and the denominator of the fraction by this number to get the equivalent fraction. For example, to find the equivalent of ½, divide 1 by 2, which gives you 0.5. Multiply 0.5 by both 1 and 2 to get the equivalent fraction 3/6.

b. To set up equivalent ratios, you need to first identify the ratio given (e.g. a per b). Divide the first number (a) by the second number (b) to get the ratio in fraction form. For example, if the ratio given is 3 per 4, divide 3 by 4 to get the equivalent ratio 3/4.

c. To find equivalent fractions and ratios, the method used depends on the situation. If the fraction or ratio is given in fraction form, use division to find the equivalent fraction or ratio. If the fraction or ratio is given in ratio form, use division to convert it to fraction form and then use division to find the equivalent fraction or ratio.

d. To confidently set up and find equivalent fractions and ratios, it is important to practice and understand the fundamentals of fractions and ratios. It is also important to understand the different forms in which fractions and ratios are presented and how to convert between them. Understanding how to use division to find equivalent fractions and ratios is also essential.

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Answer:

Step-by-step explanation:

-6+21

The required probability of given mean and standard deviation for chloride concentration equals 109,  is less than 109 and is at most 109 is

given by 0, approximately 0.6915 and 0.6915 respectively.

Blood chloride concentration has a normal distribution,

Mean μ = 108

Standard deviation σ = 2.

Probability that chloride concentration equals 109 is,

Since the distribution is continuous, the probability that chloride concentration equals 109 is zero.

As the probability of obtaining any single value in a continuous distribution is zero.

Probability that chloride concentration is less than 109 is,

Area under the normal distribution curve to the left of x = 109.

Use the standard normal distribution and the z-score formula ,

Convert this to a standard normal variable Z,

Z = (X - μ) / σ

  = (109 - 108) / 2

  = 0.5

Using a standard normal table ,

Probability of a standard normal variable being less than 0.5 is approximately 0.6915.

Probability that chloride concentration is at most 109,

Area under the normal distribution curve to the left of x = 109, including the probability of x = 109.

Probability of obtaining any single value in a continuous distribution is zero,

Use the same approach,

Probability that chloride concentration is less than 109.

Add the probability that chloride concentration equals 109 to get the probability that chloride concentration is at most 109,

P(X ≤ 109)

= P(X < 109) + P(X = 109)

≈ 0.6915 + 0 = 0.6915

Therefore, the probability that chloride concentration is equals 109 , less than 109 and at most 109 is zero, approximately 0.6915 and 0.6915 respectively.

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The above question is incomplete, the complete question is:

Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 108 and standard deviation 2. what is the probability that chloride concentration equals 109? is less than 109? is at most 109? (round your answers to four decimal places.)

Answer:

$1,605.6

Step-by-step explanation:

Multiply-

14x1.55=$21.7

7x1.05=$7.35

4x1.10=$4.4

Add 21.7+7.35+4.4=33.45

Now multiply 33.45 by 1200 since there are 1200 students

33.45x1200=1,605.6

$1605.6 should the school expect to spend on all the activities

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

The school surveys a random sample of 25 students

The cost of movie is 1.5 and the number of students is

So total cost of movie is 14x1.55=$21.7

The cost of Planetarium is 1.05 and the number of students is 7

7x1.05=$7.35

4x1.10=$4.4

Add 21.7+7.35+4.4=33.45

The total cost is 33.45

As there are 1200 students

Now multiply 33.45 by 1200

33.45x1200=1,605.6

Hence, $1605.6 should the school expect to spend on all the activities

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Answer:

Just like the other said

Step-by-step explanation:

Add 7 to both like he said. I see you erased some work. what you did was actually correct

The given geometric series is: 6 + 5 + 25 + 125 + 36. First, we need to determine the common ratio of the series.

We can do this by dividing any term by the term before it.

5/6 = 0.8333

25/5 = 5

125/25 = 5

36/125 = 0.288

The common ratio is not consistent throughout the series, so it is not a geometric series.

Therefore, we cannot determine if it is convergent or divergent, and we cannot find its sum.

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The cost of the chocolate for 110 bon-bons  as per given radius , density and cost is equal to $8.80 to the nearest cent.

Volume of a sphere with radius r is ,

V = ( 4/3 )πr³

Radius 'r' of each chocolate bon-bon is 0.8 cm,

Volume of each bon-bon is equals to,

V = ( 4/3 )π (0.8)³

   = 2.144 cm³

Density 'ρ' of the chocolate is 1.3 g/cm^3,

Mass 'm'  of each bon-bon is equals to,

m = ρV

   = ( 1.3 ) ( 2.144 )

   = 2.78grams

The cost of the chocolate used in each bon-bon is $0.03 per gram,

This implies,

Cost of the chocolate used in one bon-bon is ,

= ( 0.03 ) × 2.78

= 0.0834

Rounding to the nearest cent, the cost of the chocolate for one bon-bon is $0.08.

Cost of the chocolate for 110 bon-bons,

= 110 × 0.08

= 8.8

Therefore, the chocolate for 110 bon-bons would cost $8.80 to the nearest cent.

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Answer:

Step-by-step explanation:

To solve the system of equations by combining the equations, we can use the method of elimination:

x + 2y = -14 (equation 1)

-4x - y = 28 (equation 2)

We can eliminate y by multiplying equation 1 by 4 and adding it to equation 2:

4x + 8y = -56 (4 * equation 1)

-4x - y = 28 (equation 2)

7y = -28

Solving for y, we get:

y = -4

Substituting this value into equation 1, we get:

x + 2(-4) = -14

Solving for x, we get:

x - 8 = -14

x = -6

Therefore, the solution to the system of equations is x = -6 and y = -4.

(a) To prove by contrapositive that if x+y is rational, then either x or y (or both) is rational. (b) To prove by contradiction that if x*y > 100 for positive real numbers x and y, then either x > 10 or y > 10.

(a) To prove by contrapositive, we assume that x+y is rational and x is irrational. Then, we can write y = (x+y) - x. Since x+y and x are both rational, their difference y must also be rational, which contradicts the assumption that y is irrational. Therefore, if x is irrational and y is irrational, then x+y must be irrational.

(b) To prove by contradiction, we assume that xy > 100 and both x and y are less than or equal to 10. Then, we can write x = 10-a and y = 10-b, where a and b are positive numbers. Substituting these values in the inequality, we get (10-a)(10-b) > 100, which simplifies to ab > 1.

This contradicts the assumption that both a and b are positive numbers. Therefore, if x*y > 100, then either x > 10 or y > 10.


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 The equation for the plane that is perpendicular to the line l(t) = (8, 0, 4)t + (5, −1, 1) and passes through (6, −1, 0) is 0x + 8y - 12z + 8 = 0

The equation for the plane that is perpendicular to the line l(t) = (−3, −6, 9)t + (0, 7, 1) and passes through (2, 2, −1) is 24x + 18y - 54z - 138 = 0

1. The equation for the plane that is perpendicular to the line l(t) = (8, 0, 4)t + (5, −1, 1) and passes through (6, −1, 0) is given by: ax + by + cz + d = 0 where a, b, c, and d are constants, and the normal vector to the plane is (a,b,c).

We can determine a, b, and c by taking the cross product of the line's vector direction with the vector joining the point on the line to the point on the plane. So,

Vector direction of line: (8, 0, 4)Vector joining point on line to point on plane: (-1, -2, -4)

We take the cross product of these two vectors,

Cross product = (b, c, a) = (0, 8, -12)

Substituting these values into the equation, we get

ax + by + cz + d = 0

0x + 8y - 12z + d = 0

Now, we have to find the value of d. We know the coordinates of the point on the plane, which is (6, -1, 0). Substituting these coordinates into the equation above gives us:

0x + 8(-1) - 12(0) + d = 0-8 + d = 0d = 8

So, the equation of the plane is: 0x + 8y - 12z + 8 = 0

2. The equation for the plane that is perpendicular to the line l(t) = (−3, −6, 9)t + (0, 7, 1)and passes through (2, 2, −1) is given by: ax + by + cz + d = 0 where a, b, c, and d are constants, and the normal vector to the plane is (a,b,c).

We can determine a, b, and c by taking the cross product of the line's vector direction with the vector joining the point on the line to the point on the plane. So,

Vector direction of line: (-3, -6, 9)Vector joining point on line to point on plane: (-3, 4, -2)

We take the cross product of these two vectors,

Cross product = (b, c, a) = (-54, 18, 24)

Substituting these values into the equation, we get

ax + by + cz + d = 0

24x + 18y - 54z + d = 0

Now, we have to find the value of d. We know the coordinates of the point on the plane, which is (2, 2, -1). Substituting these coordinates into the equation above gives us:

24(2) + 18(2) - 54(-1) + d = 048 + 36 + 54 + d = 0138 + d = 0d = -138

So, the equation of the plane is: 24x + 18y - 54z - 138 = 0

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After 116 years, οnly abοut 9.375 g οf the οriginal 150 g sample οf strοntium-90 will remain.

A linear equatiοn is an equatiοn that describes a straight line in a twο-dimensiοnal plane οr a plane in higher dimensiοns. In its simplest fοrm, a linear equatiοn in twο variables (x and y) can be written as:

y = mx + b

The half-life οf strοntium-90 is 29 years, which means that after 29 years, half οf the initial amοunt οf strοntium-90 will decay. After anοther 29 years (i.e., a tοtal οf 58 years), half οf the remaining amοunt will decay again, and sο οn. Therefοre, we can use the fοllοwing fοrmula tο calculate the remaining amοunt οf strοntium-90 after 116 years:

Remaining Amοunt =[tex]$Initial Amount \times(1/2)^{(number of half- lives)[/tex]

Tο find the number οf half-lives that have elapsed in 116 years, we can divide 116 by 29:

Number οf half-lives = 116/29 = 4

Sο, after 116 years, there have been fοur half-lives οf strοntium-90 decay. Plugging this intο the fοrmula abοve, we get:

Remaining Amοunt[tex]= 150 x (1/2)^4[/tex]

= 150 x 0.0625

= 9.375 g

Therefοre, after 116 years, οnly abοut 9.375 g οf the οriginal 150 g sample οf strοntium-90 will remain.

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$495. Late, but hopefully you got it right.

Javier would have to pay $240 for the ambulance ride. Javier's insurance company pays for 70% of the cost of the ambulance ride, which means that he is responsible for the remaining 30%.

First, we need to calculate the amount that the insurance company will pay:

70% of $1300 = 0.7 * $1300 = $910

Next, we subtract the insurance payment from the total cost of the ambulance ride to find out how much Javier is responsible for:

$1300 - $910 = $390

However, we also need to subtract the $150 deductible that Javier is responsible for paying:

$390 - $150 = $240

Therefore, Javier would have to pay $240 for the ambulance ride.

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Proved, A' is the inverse of A, AA' = A'A = I3.

Let A and B be two matrices of size 3. To find AB and BA, we need to multiply the matrices A and B in the given order.

AB = A*B = [a11*b11 + a12*b21 + a13*b31, a11*b12 + a12*b22 + a13*b32, a11*b13 + a12*b23 + a13*b33]

BA = B*A = [b11*a11 + b12*a21 + b13*a31, b11*a12 + b12*a22 + b13*a32, b11*a13 + b12*a23 + b13*a33]

To show that AB = BA if and only if B is a scalar multiple of I3, we need to prove that AB - BA = 0 if and only if B = k*I3 for some scalar k.

AB - BA = [a11*b11 + a12*b21 + a13*b31 - b11*a11 - b12*a21 - b13*a31, a11*b12 + a12*b22 + a13*b32 - b11*a12 - b12*a22 - b13*a32, a11*b13 + a12*b23 + a13*b33 - b11*a13 - b12*a23 - b13*a33]

If B = k*I3, then AB - BA = [a11*k - k*a11, a12*k - k*a12, a13*k - k*a13] = [0, 0, 0]

Therefore, AB = BA if and only if B is a scalar multiple of I3.

(c) Let D be a square matrix of size 3 and D2 = 0. To show that (I3 + D)(Is - D) = Is, we need to multiply the matrices (I3 + D) and (Is - D) and simplify the result.

(I3 + D)(Is - D) = I3*Is - I3*D + D*Is - D*D = Is - D + D - D2 = Is - D + D - 0 = Is

Therefore, (I3 + D)(Is - D) = Is.

Let A' be the inverse of A. To show that AA' = A'A = I3, we need to multiply the matrices A and A' in the given order and simplify the result.

AA' = A*A' = [a11*a'11 + a12*a'21 + a13*a'31, a11*a'12 + a12*a'22 + a13*a'32, a11*a'13 + a12*a'23 + a13*a'33]

A'A = A'*A = [a'11*a11 + a'12*a21 + a'13*a31, a'11*a12 + a'12*a22 + a'13*a32, a'11*a13 + a'12*a23 + a'13*a33]

Since A' is the inverse of A, AA' = A'A = I3.

Therefore, AA' = A'A = I3.

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To convert between millimeters and kiloliters, you would need to know the dimensions of the object in millimeters and then calculate its volume in liters or cubic meters, which can then be converted to kiloliters.

Conversion rate is a way to express the relationship between two units of measurement. However, millimeters and kiloliters are units of length and volume, respectively, and cannot be directly converted into one another.

For example, if you have a rectangular tank with dimensions of 1 [tex]000mm x 500mm x 2000mm,[/tex] you can calculate its volume in liters by multiplying the three dimensions together and then dividing by  [tex]1000[/tex] to convert to liters:

[tex]1000mm x 500mm x 2000mm = 1000000000[/tex] cubic millimeters

[tex]1000000000 / 1000 = 1000000[/tex] liters

[tex]1000000 / 1000 = 1000[/tex] To convert liters to kiloliters, divide by [tex]1000[/tex] :

kiloliters

Therefore, the conversion rate from millimeters to kiloliters for this tank is [tex]1000[/tex] kiloliters per [tex]1000000000[/tex] cubic millimeters, or 1 kiloliter per  [tex]1,000,000[/tex] cubic millimeters.

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The true statements about the proportional relation are A and C.

Here we can see the graph of a proportional relation (because it passes through the origin), so we can write it as:

y = k*x

Where k is a constant and represents the cost of each gallon.

We can see the point (7, 21), replacing these values we will get:

21 = k*7

21/7 = k

3 = k

Then the proportional relation is:

y = 3*x

Then the cost is $3, the first statement is true.

And the statement C is also true, (0, 0) means no gas was purchased.

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The function that satisfies the given initial value problem (IVP) is y = f(x) = 5e 1 3 x + 2xe 1 3 x + 1 6 x2. This can be obtained by integrating the given equation using integration by parts.
Integrating the given equation y0 = (2 + 5x)e 1 3 x, we have y = ∫(2 + 5x)e 1 3 xdx.

Using integration by parts,
Let u = 2 + 5x, v' = e 1 3 x
Then u' = 5, and v = - 1 3 e 1 3 x
So, y = ∫(2 + 5x)e 1 3 xdx = uv - ∫uv' dx
Substituting the values of u and v, we get y = (2 + 5x) (- 1 3 e 1 3 x) - ∫(5) (- 1 3 e 1 3 x) dx

Integrating the right-hand side, we get y = (2 + 5x) (- 1 3 e 1 3 x) + 1 6 xe 1 3 x + c
Given that y(0) = 5, when x = 0, we have 5 = (2 + 5x) (- 1 3 e 1 3 x) + 1 6 xe 1 3 x + c
Equating to c, we get c = 5 - (2 - 1 3) = 5 + 1 6

Therefore, the function that satisfies the given initial value problem is
y = f(x) = 5e 1 3 x + 2xe 1 3 x + 1 6 x2 + 5 + 1 6
= 5e 1 3 x + 2xe 1 3 x + 1 6 x2

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The required dimensions length and width of the rectangle with the given condition is equal to 6 units and 3 units respectively.

Let us consider the length of the rectangle be x in any unit.

Then the width of the rectangle is x - 3 in the same unit.

The perimeter of a rectangle = 2(length + width)

According to the problem statement, the rectangle is equal to its perimeter we have,

x(x-3) = 2(x + x-3)

Simplifying the above equation, we get,

⇒ x^2 - 3x = 4x - 6

⇒ x^2 - 7x + 6 = 0

⇒ (x-6)(x-1) = 0

So, x = 6 or x = 1.

However, the length of the rectangle cannot be 1

This implies width would be -2 not possible.

The length of the rectangle is 6 inches

And the width is 6-3=3 inches.

Therefore, the dimensions of the rectangle are length 6 inches  and width 3 inches .

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The canonical algebraic expression in sum of products form is F(A,B,C) = AB + AB' + ABC

The canonical algebraic expression in sum of products form is a representation of a Boolean function in which each term is the product of a set of literals, and the entire expression is the sum of these products.

For example, let's consider the Boolean function F(A,B,C) = (A+B)C. We can represent this function in sum of products form by first distributing the C term to each of the terms in the parenthesis:

F(A,B,C) = AC + BC

Now, each term is the product of a set of literals (A and C in the first term, B and C in the second term), and the entire expression is the sum of these products. This is the canonical algebraic expression in sum of products form.

Another example is the Boolean function F(A,B,C) = A + BC. We can represent this function in sum of products form by first multiplying each term by 1 in the form of (A+A') or (B+B'):

F(A,B,C) = A(B+B') + BC(A+A')

Then, we can distribute the terms:

F(A,B,C) = AB + AB' + ABC + ABC'

Finally, we can simplify the expression by combining like terms:

F(A,B,C) = AB + AB' + ABC . This is the canonical algebraic expression in sum of products form.

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The chances that two or fewer of ten tests from this sample have positive results is 50.1%.

To calculate this, we can use the binomial probability formula:

P(x) = n! / (x!(n-x)!) * px * q(n-x)

Where n is the total number of trials (10 in this case),

x is the number of successes desired (2 in this case),

p is the probability of success (50% in this case) and q is the probability of failure (50% in this case).

Plugging the values into the formula, we get:

P(2) = 10! / (2!(10-2)!) * 0.52 * 0.5(10-2)

This simplifies to P(2) = 0.501

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The company needs to sell 1.5 units of merchandise to make the maximum profit that implies 300.

What is profit ?

Profit is the financial gain that a business or an individual derives from a transaction or business activity, after deducting all costs and expenses. It is the difference between the amount earned and the amount spent to produce, sell, or provide goods or services. Profit is an important measure of business success and is used to evaluate the performance of a company, as well as to attract investors and secure financing.


According to the question:
To find the number of units of merchandise the company needs to sell to make the maximum profit, we need to locate the vertex of the parabola. Since the parabola opens downward, the vertex represents the maximum point on the graph.

From the graph, we can see that the vertex is located at (1.5, 7.5). This means that the company will make the maximum profit by selling 1.5 units of merchandise.

Therefore, the company needs to sell 1.5 units of merchandise to make the maximum profit.

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The relationship between the median and the mean scores is the mean is greater than the median. (option c).

Mathematically speaking, the relationship between the median and the mean can vary depending on the distribution of the data. In a symmetrical distribution, where the data is evenly distributed around the center point, the mean and median will be equal.

Conversely, if the data is skewed to the left, meaning there are a few low values that pull the average down, the mean will be less than the median.

In summary, the relationship between the median and the mean depends on the distribution of the data. If the data is symmetrical, the median and mean will be equal. If the data is skewed, the mean will be pulled towards the direction of the skewness, while the median represents the midpoint of the data regardless of the skewness.

Hence the option (c) is correct.

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Answer:

1/5

Step-by-step explanation:

40 / 8 = 5 therefore, it equals 1/5, because 1/5 of the classes are year 7.

Answer:

1/5

Step-by-step explanation:

We have 40 classes total. 8 of the 40 classes are Year 7 classes.

So we can say that [tex]\frac{8}{40}[/tex] classes are Year 7 classes.

The problem statement asks for the answer in simplest form, so we reduce our fraction by a factor of 8.

[tex]\frac{8}{40} = \frac{1}{5}[/tex]

So 1/5 (one fifth) of the classes are the secondary school are Year 7 classes.

The expression to calculate the total gallons of water used in a 5-minute shower is:

total gallons = 7.5 gallons/minute × 5 minutes

Multiplying the rate of water usage (7.5 gallons/minute) by the duration of the shower (5 minutes) gives:

total gallons = 37.5 gallons

Therefore, a 5-minute shower would take about 37.5 gallons of water.

The given information is an angle relationship between angles 6 and 7, which states that the sum of angles 6 and 7 is 180 degrees. This implies that angles 6 and 7 are supplementary angles.

If two lines are intersected by a transversal and a pair of corresponding angles are congruent, then the lines are parallel. However, the given information does not include any information about the relationship between the lines that angles 6 and 7 lie on. Therefore, we cannot determine whether any lines are parallel based on this information alone.

In summary, the given information does not allow us to determine whether any lines are parallel, and there is no theorem that justifies such a conclusion.

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Figure of the question is given in the attachment.

Using the theorem of angle at a point, opposite angles and angle on a straight line, the value of x and y are 7 and 5 respectively.

In geometry, an angle at a point refers to the measure of the angle formed by two rays that originate from the same endpoint, which is called the vertex of the angle. The endpoint of the rays can be any point on a plane, and the angle formed by the rays is often represented using the symbol ∠.

An angle at a point is typically measured in degrees or radians, with 360 degrees or 2π radians representing a full circle. The measure of an angle at a point can range from 0 degrees (or 0 radians) to 180 degrees (or π radians), with a measure of 90 degrees (or π/2 radians) representing a right angle.

7y - 10 = 25

Reason: opposite angles are equal

solve for y

7y = 25 + 10

7y = 35

y = 35/7

y = 5

a + 25 = 180

reason: angles on a straight line is equal to 180

a = 180 - 25

a = 155°

15x + 10y = 155

reason: opposite angles are equal

but y = 5

15x + 10(5) = 155

15x + 50 = 155

15x = 155 - 50

15x = 105

x = 105/15

x = 7

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Solve the following exercises by finding the variable or the angle that indicates.

Someone help me answer it please! ..

The required sum of the positive integers x and y with the given condition of product of x and y as  square and cube is equal to 85.

Two positive integers are x and y.

Product of 360 and x is a square.

Product of 360 and y is a cube.

Prime factors of 360 equals to ,

360 = 2³ × 3² × 5

To make a product of 360 and x as square ,

Multiply by 2 and 5 and 3 is already a perfect square.

x = 2 × 5

 = 10

360 × 10 = 3600

Now , to make product of 360 and y as cube.

Multiply by 3  and 5² and 2 is already a perfect cube.

y = 3 × 5²

   = 75

360 × 75 = 27000

Sum of the values of x and y is equal to,

x + y

= 10 + 75

= 85

Therefore, the sum of the positive integers x and y are 85.

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The magnitude and angle of the complex number e^j2x/e^-jy are sqrt(e^(j4x) + e^(-j2y)) and 2.3562, respectively.

To determine the magnitude and angle of the given complex numbers, we need to use the following formulas:

Magnitude: |z| = sqrt(x^2 + y^2)

Angle: theta = tan^-1(y/x)

(a) 2e^-5jx

In this case, x = 2 and y = -5j

Magnitude: |z| = sqrt(2^2 + (-5j)^2) = sqrt(4 + 25j^2) = sqrt(29)

Angle: theta = tan^-1((-5j)/2) = -1.1903

Therefore, the magnitude and angle of the complex number 2e^-5jx are sqrt(29) and -1.1903, respectively.

(b) e^x + 1

In this case, x = e^x and y = 1

Magnitude: |z| = sqrt((e^x)^2 + 1^2) = sqrt(e^(2x) + 1)

Angle: theta = tan^-1(1/e^x) = 0.7854

Therefore, the magnitude and angle of the complex number e^x + 1 are sqrt(e^(2x) + 1) and 0.7854, respectively.

(c) e^j2x/e^-jy

In this case, x = e^j2x and y = e^-jy

Magnitude: |z| = sqrt((e^j2x)^2 + (e^-jy)^2) = sqrt(e^(j4x) + e^(-j2y))

Angle: theta = tan^-1((e^-jy)/(e^j2x)) = 2.3562

Therefore, the magnitude and angle of the complex number e^j2x/e^-jy are sqrt(e^(j4x) + e^(-j2y)) and 2.3562, respectively.

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