Let f be a permutation on the set {1,2,3,4,5,6,7,8,9}, defined as follows f =1 2 3 4 5 6 7 8 9 4 1 3 6 2 9 7 5 8
(a) Write f as a product of transpositions (not necessarily disjoint), separated by commas (e.g. (1,2), (2,3), ... ). f = (b) Write f-l as a product of transpositions in the same way. f-1 = Assume multiplication of permutations f,g obeys the rule (fg)(x) = f(g(x)so (1,3)(1, 2) = (1,2,3) not (1,3,2).

Answers

Answer 1

P(Billy and not Bob) = (3 choose 1) * (18 choose 5) / (19 choose 5)

= 54/323

(a) We can write f as a product of transpositions as follows:

f = (1,4,6,9,8,5,2)(3,6,9)(2,1)(7,9,5,8)

Note that this is just one possible way of writing f as a product of transpositions, as there can be multiple valid decompositions.

(b) To find f-1, we need to reverse the order of the elements in each transposition and then reverse the order of the transpositions themselves:

f-1 = (2,1)(5,8,9,7)(1,2)(9,6,3)(2,5,8,9,6,4,1)

Again, note that there can be multiple valid ways of writing f-1 as a product of transpositions.

(c) To find the probability that either Bob or Billy is chosen among the 5 students, we can use the principle of inclusion-exclusion. The probability of Billy being chosen is 1/4, and the probability of Bob being chosen is also 1/4. However, if we simply add these probabilities together, we will be double-counting the case where both Billy and Bob are chosen. The probability of both Billy and Bob being chosen is (2/19) * (1/18) = 1/171, since there are 2 ways to choose both of them out of 19 remaining students, and then 1 way to choose the remaining 3 students out of the remaining 18. So the probability that either Billy or Bob is chosen is:

P(Billy or Bob) = P(Billy) + P(Bob) - P(Billy and Bob)

= 1/4 + 1/4 - 1/171

= 85/342

(d) To find the probability that Bob is not chosen and Billy is chosen, we can use the fact that there are (18 choose 5) ways to choose 5 students out of the remaining 18 after Bob has been excluded, and (3 choose 1) ways to choose the remaining student from the 3 that are not Billy. So the probability is:

P(Billy and not Bob) = (3 choose 1) * (18 choose 5) / (19 choose 5)

= 54/323

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Related Questions

Find the radius of convergence, R, of the series. [infinity] (7x − 4)nn7nn = 1R =Find the interval, I, of convergence of the series. (Enter your answer using interval notation. )I =

Answers

The radius of convergence R is 1/7 and the interval of convergence I is: I = (-1/7, 5/7)

To find the radius of convergence R, we can apply the ratio test:

[tex]lim_n→∞ |(7x-4)(n+1)/7(n+1)| = lim_n→∞ |7x-4|/7 = |7x-4|[/tex]

The series converges when the limit is less than 1, so we have: |7x - 4| < 1

Solving for x,

we get: -1/7 < x < 5/7

This means that the series converges for all values of x within the interval (-1/7, 5/7) and diverges for values of x outside that interval. The interval is open on the left endpoint and closed on the right endpoint because the limit at x=-1/7 and x=5/7 needs to be tested separately.

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Help me what’s the answer?I need the answer asap

Answers

Answer:

Step-by-step explanation:

The question and answer are in the picture

Answers

Answer:

15.8

Step-by-step explanation:

mean is the average in math so 9 + 14 + 11 + 31 + 14 =79 then you have to count how many numbers there is and minus it from the total which there is 5 numbers so 79 divided by 5 = 15.8

imagine that you are at an eighteenth century coffee shop, engaged in a lively conversation with your friend pierre. pierre wants to know the probability that the sun will rise tomorrow. what is the most reasonable response to this question? group of answer choices 1/2 1/365 1 it depends on the probability model used

Answers

Pierre's question is a common philosophical and scientific question about the nature of prediction and probability.

In the 18th century, there was not a comprehensive understanding of the scientific laws that govern the natural world as we have today. Therefore, the most reasonable response to Pierre's question would be that it depends on the probability model used. The probability of the sun rising tomorrow would be based on various factors such as astronomical observations, scientific knowledge of celestial mechanics, and weather patterns.

While we cannot predict the future with absolute certainty, we can use available data and knowledge to make informed predictions about the likelihood of the sun rising tomorrow.

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Consider the family of functions f(x)=1/x^2-2x k, where k is constant

Answers

The value of k, for k > 0, such that the slope of the line tangent to the graph off at x = 0 is -2.

We are given a family of functions f(x) = x² - 2x + k, where k is a constant. This family of functions includes all the possible quadratic functions of the form x² - 2x + k. To find the value of k, we need to use the given condition that the slope of the tangent line to the graph of the function at x = 0 equals 6.

To find the slope of the tangent line at x = 0, we need to take the derivative of the function f(x) and evaluate it at x = 0. Taking the derivative of f(x), we get:

f'(x) = 2x - 2

Evaluating f'(x) at x = 0, we get:

f'(0) = 2(0) - 2 = -2

This gives us the slope of the tangent line to the graph of the function at x = 0, which is -2.

Therefore, the answer to the problem is that there is -2 of k, for k > 0, such that the slope of the line tangent to the graph of the function at x = 0 equals 6.

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Complete Question;

Consider the family of functions f(x) = where k is a constant. x^2 - 2x +k

Find the value of k, for k > 0, such that the slope of the line tangent to the graph off at x = 0

rewrite 6 2/7 as an improper fraction

Answers

The improper fraction can be written as:

44/7

How to rewrite this as an improper fraction?

An improper fraction is a fraction where the numerator is larger than the denominator.

Here we want to write.

6 + 2/7 as an improper fraction, to do so, we just need to write 6 as a fraction with a denominator of 7 and then add them.

We know that:

6 = 6*(7/7) = 42/7

Then we can write:

6 + 2/7 = 42/7 + 2/7 = 44/7

That is the improper fraction.

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i need help with this slide...

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The calculated slopes of the relations are -5, 9, -1/3 and 3

Finding the slopes of the relations

The table 1

The slope is calculated as

Slope = change in y/x

So, we have

Slope = (13 - 18)/(-2 + 3)

Evaluate

Slope = -5

Ordered pair 2

The slope is calculated as

Slope = change in y/x

So, we have

Slope = (9 + 27)/(1 + 3)

Evaluate

Slope = 9

Graph 3

The slope is calculated as

Slope = change in y/x

So, we have

Slope = (0 - 1)/(3 - 0)

Evaluate

Slope = -1/3

The table 4

The slope is calculated as

Slope = change in y/x

So, we have

Slope = (2 + 1)/(-2 + 3)

Evaluate

Slope = 3

Hence, the values of the slopes are -5, 9, -1/3 and 3

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Find the surface area

Answers

The surface area of the pyramid is 179 sq. m.

What is surface area of a shape?

The surface area of a given shape is the summation of all the area of each figure that forms its sides called surfaces.

The given pyramid has triangular shaped surfaces, so that;

area of a triangle = 1/2 *base*height

To determine the area of one of the surfaces, we have;

area of the triangular surface = 1/2x base x height

base = 8 m, and slant height of the surface = 11.2 m

So that;

the area of one triangular surface = 1/2*8*11.2

                                                    = 44.8 sq. m.

Thus since the pyramid has 4 equal triangular surfaces, then;

the surface area of the pyramid = 4 x 44.8

                                            = 179.2

The surface area of the pyramid is 179 sq. m.

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Let m € Rn and r> 0 be given and define the ball C := {r € Rn: ||x - m|| ≤r}. In this exercise, we want to compute the projection Pc(r) for x ER", i.e., we want solve the optimization
problem
min /Y€Rn. 1/2 ||y–x||² subject to ||y–m||²≤r²
a) Write down the KKT conditions for problem (3).
b) Show that the KKT conditions have a unique solution and calculate the corresponding, KKT pair explicitly.

Answers

The KKT pair is given by:

λ = 0, y = x (when x is inside the ball)

λ = 1/2, y = m + (x – m)/2 (when x is outside the ball)

a) The Lagrangian function for the optimization problem is given by:

L(y, λ) = 1/2 ||y – x||² + λ (r² – ||y – m||²)

where λ is the Lagrange multiplier.

The KKT conditions for the problem are:

Stationarity condition: ∇y L(y, λ) = 0

∇y L(y, λ) = y – x – 2λ (y – m) = 0

Primal feasibility condition: ||y – m||² ≤ r²

Dual feasibility condition: λ ≥ 0

Complementary slackness condition: λ (r² – ||y – m||²) = 0

b) To show that the KKT conditions have a unique solution, we can use the second-order sufficiency conditions. The Hessian matrix of the Lagrangian function is given by:

∇²L(y, λ) = I – 2λ I = (1 – 2λ)I

where I is the identity matrix. Since λ ≥ 0, we have 1 – 2λ ≤ 1, which means that the Hessian matrix is positive definite. Therefore, the KKT conditions have a unique solution.

To calculate the KKT pair, we need to solve the stationarity and primal feasibility conditions. From the stationarity condition, we have:

y – x – 2λ (y – m) = 0

y – 2λy = x – 2λm

y = (I – 2λ)⁻¹(x – 2λm)

Substituting this into the primal feasibility condition, we have:

||(I – 2λ)⁻¹(x – 2λm) – m||² ≤ r²

Expanding this expression, we get:

||x – m||² – 4λ (x – m)ᵀ(I – λ(I – 2λ)⁻¹)(x – m) + 4λ² ||(I – 2λ)⁻¹(m – x)||² ≤ r²

Let A = (I – λ(I – 2λ)⁻¹). Then, the above expression can be written as:

||x – m||² – 4λ (x – m)ᵀA(x – m) + 4λ² ||A(m – x)||² ≤ r²

Since λ ≥ 0, we have A = (I – λ(I – 2λ)⁻¹) ≥ 0, which means that A is positive semidefinite. Therefore, the minimum value of the expression on the left-hand side is achieved when λ = 0 or λ = 1/2.

If λ = 0, then we get:

y = x

If λ = 1/2, then we get:

y = m + (x – m)/2

Therefore, the KKT pair is given by:

λ = 0, y = x (when x is inside the ball)

λ = 1/2, y = m + (x – m)/2 (when x is outside the ball)

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For each of the following ions, indicate the noble gas that has the same lewis structure as the ion.

express your answer as a chemical symbol?

BR-

O2-

Rb+

Ba2+

Answers

if in not mistaken
BR- has the same Lewis structure as Kr.

O2- has the same Lewis structure as Ne.

Rb+ has the same Lewis structure as Kr.

Ba2+ has the same Lewis structure as Xe.

there are 26 members of a basketball team. (3) from the 14 players who will travel, the coach must select her starting line-up. she will select a player for each of the five positions: center, right forward, left forward, right guard, left guard. however, there are only 4 of the 14 players who can play center. otherwise, there are no restrictions. how many ways are there for her to select the starting line-up?

Answers

The number of ways there are for her to select the starting line-up is 68,640 ways.

To determine the number of ways for the coach to select the starting line-up, we need to consider the choices for each position:

1. Center: There are 4 players who can play this position, so there are 4 choices.
2. Right Forward: Since one player has been selected as Center, there are now 13 players remaining. So, there are 13 choices for this position.
3. Left Forward: After selecting the Center and Right Forward, 12 players remain, resulting in 12 choices for this position.
4. Right Guard: With three players already chosen, there are 11 players left to choose from, giving us 11 choices.
5. Left Guard: Finally, after selecting players for the other four positions, 10 players remain, providing 10 choices for this position.

Now, we can calculate the total number of ways to select the starting line-up using the counting principle by multiplying the number of choices for each position:

4 (Center) × 13 (Right Forward) × 12 (Left Forward) × 11 (Right Guard) × 10 (Left Guard) = 68,640 ways

So, there are 68,640 ways for the coach to select the starting line-up.

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(5) Let р and q be two distinct primes. Show that p9-1+qp-1 is congruent to 1 (mod pq).

Answers

By using the Chinese Remainder Theorem separate the statement into two congruences we have x(p-1)(q−1)+1 (mod pq) for all x € Z.

Under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1), the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can uniquely determine the remainder of the division of n by the product of these integers.

It suffices to show pq divides x(x(p-1)(q-1) - 1) for all x e Z. We consider three cases. Consider gcd (x, pq). It has 3 possibilities.

Case 1: If gcd(x, pq) 1. Then applying using Euler's Theorem we have

= x(pq) = 1 (mod pq)

= x(p-1)(−1) = 1 (mod pq)

= x(p-1)(q-1)+1 (mod pq)

and so the result holds if gcd(x, pq) = 1. EX

Case 2: If gcd(x, pq) p. This means x = 0 (mod p). In this case we have

= 0 = x (mod p).

Since gcd(x, pq) = p therefore qx and = 1 (mod q) by Fermat's Little Theorem. This gives us that x(p-1)(q-1)+1 so we have x9-1 x(p−1)(q−1) = 1 (mod q) = x(p-1)(q-1)+1 = x (mod q).

We have shown that x(p-1)(q-1)+1 = x (mod p) and x(p-1)(q-1)+1 = x (mod q). Using the Chinese Remainder Theorem we get x(p-1)(q−1)+1 = x (mod pq).

Case 3: If gcd(x, pq) = q. This case is same as Case 2, with p being replaced by q.

Thus we have extinguished all cases and we have shown x(p-1)(q−1)+1 (mod pq) for all x € Z.

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Complete question:

Let р and q be distinct primes. Show that for all x € Z, we have the congruence x(p-1)(9–1)+1 x (mod pq). (Hint: Use the Chinese Remainder Theorem/Sun Ze's Theorem to separate the statement into two congruences.)

What’s the answer ? I need help pls answer

Answers

The distance between (2 + i) and (4 +3i) would be,

⇒ d = 2√2

We have to given that;

To find distance between (2 + i) and (4 +3i).

Now, We can formulate;

Two points are (2, 1) and (4, 3).

We know that;

The distance between two points (x₁ , y₁) and (x₂, y₂) is,

⇒ d = √ (x₂ - x₁)² + (y₂ - y₁)²

Hence, The distance between (2 + i) and (4 +3i) would be,

⇒ d = √(4 - 2)² + (3 - 1)²

⇒ d = √4 + 4

⇒ d = √8

⇒ d = 2√2

Thus, The distance between (2 + i) and (4 +3i) would be,

⇒ d = 2√2

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show that in a sequence of m integers there exists one or more consecutive terms with a sum divisible by m.

Answers

The sum of the m integers between ai and aj is si - s(i-1) + s(i-1) - s(j-1) = si - sj, which is divisible by m since si, sj have same remainder. So, there exists a consecutive subsequence of the original sequence with a sum divisible by m, namely the integers between ai and aj.

We can prove this using the Pigeonhole Principle.

Consider the sequence of m integers a1, a2, ..., am. Let's compute the prefix sums of this sequence, which we'll denote by s0, s1, s2, ..., sm. That is, we define si = a1 + a2 + ... + ai-1 for i = 1, 2, ..., m, and s0 = 0.

Note that there are m + 1 prefix sums, but only m possible remainders when we divide a sum by m (namely, 0, 1, 2, ..., m-1).

Therefore, by the Pigeonhole Principle, at least two of the prefix sums must have the same remainder when divided by m. Let's say these are si and sj, where i < j.

Then, the sum of the m integers between ai and aj (inclusive) is si - s(i-1) + s(i-1) - s(j-1) = si - sj, which is divisible by m since si and sj have the same remainder when divided by m.

Therefore, there exists a consecutive subsequence of the original sequence with a sum divisible by m, namely the integers between ai and aj.

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If X is an exponential random variable with parameter λ, and c>0, show that cX is exponential with parameter λ/c.CDF Method:Let X be a continuous random variable and let Y=g(X)be a function of that random variable, where g(X) is some function of X. Let fX(x) be the probability density function (PDF) of X and fY(y) be the PDF of Y. Recall that the cumulative distribution function (CDF) of X is defined as the probability that X is less than or equal to some value x, for any real value of x. Mathematically,FX(x)=P(X≤x)Similarly, FY(y)=P(Y≤y).To find the distribution of Y, we can use the CDF method. We start by expressing the CDF of Y (FY(y)) in terms of X. We do this by using the fact that Y=g(X)and then solving the resulting inequality for X. Mathematically,FY(y)=P(Y≤y)=P(g(X)≤y)=⋯=P(X ???⋯)We isolate X in the inequality and we get an inequality which can be changed into CDF terms (the CDF of X).After we find the CDF of Y, we can differentiate it to get the PDF of Y. Recall that for any random variable, the first derivative of its CDF is equal to its PDF. In mathematical terms,fY(y)=ddyFY(y)We do this using the CDF of Y we obtained earlier. After completing this step, you will have the PDF of Y.

Answers

We have shown that cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ and c > 0.

To show that cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ, and c>0, we will use the CDF method:

1. Define the transformation: Let Y = cX be a function of the random variable X, where c > 0.

2. Find the CDF of Y: We want to find P(Y ≤ y), which is equal to P(cX ≤ y) or P(X ≤ y/c).

3. Express CDF of Y in terms of X: Since P(X ≤ y/c) is the CDF of X at y/c, we have FY(y) = FX(y/c).

4. Find the PDF of X: The exponential distribution has the PDF fX(x) = λ * exp(-λx) for x ≥ 0.

5. Differentiate the CDF of Y to find its PDF: To find fY(y), we differentiate FY(y) with respect to y. Using the chain rule, we have:

fY(y) = d(FX(y/c))/dy = fX(y/c) * (1/c)

6. Substitute the PDF of X: Now, we replace fX(y/c) with its exponential form λ * exp(-λ(y/c)):

fY(y) = (λ * exp(-λ(y/c))) * (1/c)

7. Simplify the expression: fY(y) = (λ/c) * exp(-λ(y/c))

This is the PDF of an exponential distribution with parameter λ/c. Therefore, cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ and c > 0.

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Place its midpoint I.
Draw the circle C of diameter [AB].
Draw the perpendicular bisector of the segment [AB]. It intersects circle C at points E and F.
Draw the half-lines [AE) and [BE).
Draw the arc of a circle with center A, radius [AB] and origin B. It intersects the half line [AE) at point H.
Draw the arc of a circle with center B, radius [BA] and origin A. It intersects the half line [BE] at point G.
Draw the quarter circle with center E, radius [EG] and bounded by points G and H.

Answers

Answer:

To complete the construction described:

Place the midpoint I of segment [AB]. Draw the circle C of diameter [AB]. Draw the perpendicular bisector of segment [AB]. Label the point where it intersects circle C as E and F. Draw half-lines [AE) and [BE). Draw an arc with center A and radius [AB] that passes through point B. Label the points where the arc intersects half-line [AE) as H and J. Draw an arc with center B and radius [BA] that passes through point A. Label the points where the arc intersects half-line [BE) as G and K. Draw the quarter circle with center E and radius [EG] that is bounded by points G and H. This completes the construction.

The final figure should consist of circle C, perpendicular bisector EF, half-lines [AE) and [BE), arcs passing through points B and A, and the quarter circle with center E, radius [EG], and bounded by points G and H.

Step-by-step explanation:

A die has six sides, with the numbers 1, 2, 3, 4, 5, and 6.
What is the probability of rolling an integer? (An integer is a whole number.)
Fraction:
Percent:
Likelihood:

Answers

The probability of rolling an integer is

Fraction: 6/6Percent: 100%Likelihood: Likely

What is the probability of rolling an integer?

From the question, we have the following parameters that can be used in our computation:

Sides = 6

Integer sides = 6

using the above as a guide, we have the following:

P(Integer) = Integer sides/Total sides

So, we have

P(Integer) = 6/6

Evaluate

P(Integer) = 1

Hence, the probability is 1

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From the attachment, what is the missing side

Answers

From the attachment, the missing side is B. 21.0.

Trigonometric functions.

Trigonometric functions are basic functions which can be used to determine the missing value of a right angled triangle when given the value of one of its internal angles. Some of these functions are; sine, cosine, tangent etc.

To determine the value of the missing side x, we have to apply the appropriate trigonometric function. Thus we have;

Sin θ = opposite/ hypotenuse

Sin 65 = 19/ x

So that;

x = 19/ 0.9631

  = 20.964

x = 21

From the attachment, the missing side is 21.0. Thus option B.

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the normal force is equal to the perpendicular component of object's weight, which decreases as the angle of inclination increases.
true or false

Answers

The statement "The normal force is equal to the perpendicular component of the object's weight, which decreases as the angle of inclination increases" is true.

As the angle of inclination increases, the object's weight can be divided into two components: one perpendicular to the inclined surface (the normal force) and one parallel to it. As the angle increases, the perpendicular component (normal force) decreases, while the parallel component increases.

So to directly answer your question, the normal force is never equal to the weight of the object on an inclined plane (unless you count the limiting case of level ground). It is equal to the weight of the object times the cosine of the angle the inclined plane makes with the horizontal.

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Two concentric circles form a target. The radii of the two circles measure 8 cm and 4 cm. The inner circle is the bullseye of the target. A point on the target is randomly selected.

What is the probability that the randomly selected point is in the bullseye?

Enter your answer as a simplified fraction

Answers

The probability that the randomly selected points is in the bullseye is 1/4

What is a concentric circles?

Concentric circles are circles with the same or common center.

To calculate the probability that the randomly selected points is in the bullseye, we use the formula below

Formula:

P = r²/R²............................. Equation 1

Where:

r = Radius of the inner cencentric circleR = Radius of the outer circle P = Probability that the selected point is in the bullseye

From the question,

Given:

R = 8 cmr = 4 cm

Substitute these values into equation 1

P = 4²/8²P = 16/64P = 1/4

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The following table shows the number of lemons that grew on Mary’s lemon tree each season last year. Number of lemons. Winter 3. Spring 15 summer 21 fall 13 find the mean number of lemons

Answers

Therefore, the mean number of lemons that grew on Mary's lemon tree is 13.

The sum of the data divided by the total amount of data determines the mean of a set of numbers, also known as the average. Only numerical variables—regardless of whether they are discrete or continuous—can be used to determine the mean. Simply dividing the total number of values in a data collection by the sum of all of the values yields it.

Mean number of lemons that grew on Mary's lemon tree, we need to sum up the number of lemons from all four seasons and then divide by the total number of seasons. So, the sum of the number of lemons from all four seasons is:

3 + 15 + 21 + 13 = 52

Since there are four seasons, the mean number of lemons is:

=52/4

= 13

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PLEASE HELP I HAVe TO SUBMIT THIS NOW!! my current grade in math is a 28 :( and if I do this assignment my grade will go 40% percent up :)

Answers

You need to graph, for the first one start at (0,1) (1 on the line that goes up and down) then go up 1 and to the right one. Shade above the line, but you might want to wait till the next equation is graphed.

For the 2nd equation start at 6, again on the Y up and down line and go down 2 and I believe to the left, 1. Continue these lines till there is no more room left on the graph. Then shade below.

Both of these should have the same shaded area. Check if those points are in the shaded area if their not, the answer is no if they are, the answer is yes.

(Remember on when you have to go down and over because it’s a negative, you do the opposite to go up from the starting point)

Hope this helps :)

The histogram displays the ages of 50 randomly selected users of an online music service. Based on the data, is advertising on the service more likely to reach people who are younger than 30 or people who are 30 and older?

Answers

Answer:

Based on the histogram displaying the ages of 50 randomly selected users of an online music service, it is more likely that advertising on the service will reach people who are younger than 30, as the frequency (or height) of the bars appears to be higher in the younger age group compared to the 30 and older age group.

Step-by-step explanation:

The sample space refers to
a. both any particular experimental outcome and the set of all possible experimental outcomes are correct
b. any particular experimental outcome
c. the set of all possible experimental outcomes
d. the sample size minus one

Answers

The sample space refers to option (c) the set of all possible experimental outcomes. In probability theory and statistics, a sample space represents all possible outcomes of an experiment or a random event.

The correct answer is c. The sample space refers to the set of all possible experimental outcomes. This includes every possible outcome that could occur in an experiment, whether or not it actually occurs. For example, if you flip a coin, the sample space would be {heads, tails}. This encompasses every possible outcome of the experiment. It provides a foundation for calculating probabilities and understanding the range of results that may occur in a given situation. Sample spaces can vary in size and complexity, depending on the nature of the experiment or event being studied. Understanding the sample space is crucial for making accurate predictions and informed decisions based on data.

Option a is also correct to some extent, as any particular experimental outcome can be considered a part of the sample space. However, it is not a complete definition of the sample space as it only focuses on one outcome and not all possible outcomes.

Option b is incorrect, as the sample space is not limited to just one particular experimental outcome. It is the set of all possible outcomes.

Option d is also incorrect as the sample space has nothing to do with the sample size or the number of participants in the experiment. It is solely based on the set of all possible outcomes of the experiment.

In conclusion, the sample space is the set of all possible experimental outcomes, including both successful and unsuccessful outcomes. It is an important concept in probability theory and is used to calculate the probability of specific events occurring in an experiment.

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Use the Intermediate Value Theorem to identify the location of the first positive root in f(x)=x²-3

Answers

The first positive root of the function f(x) = x² - 3 is located between x = 1 and x = 2.

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on values f(a) and f(b) with opposite signs, then there exists at least one root (zero) of the function between a and b.

In this case, we have f(x) = x² - 3. To find the first positive root of the function, we need to look for a positive value of x where f(x) = 0.

We can start by evaluating f(0) and f(2), which are the values of the function at the endpoints of the interval [0, 2]:

f(0) = 0² - 3 = -3

f(2) = 2² - 3 = 1

Since f(0) is negative and f(2) is positive, by the Intermediate Value Theorem, there must be at least one root of the function between x = 0 and x = 2.

To further narrow down the location of the root, we can evaluate f(1), which is the midpoint of the interval [0, 2]:

f(1) = 1² - 3 = -2

Since f(1) is negative, we know that the root is between x = 1 and x = 2.

To summarize, the first positive root of the function f(x) = x² - 3 is located between x = 1 and x = 2.

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How do you solve this problem step by step please hurry I will get anxious if someone don’t answer quickly.

The equation is in the photo I took off of my phone that I do for fun I really love math so this is what I do for fun so please help me solve this problem please and thank you.

Answers

The value of the expression is 5.

We have,

(|-52 + 1| (-1) + 4²) / (-84 ÷ 7 + 5)

Now,

PEMDAS is an acronym used to remember the order of operations in arithmetic and algebraic expressions. It stands for:

Parentheses: Simplify expressions inside parentheses first.

Exponents: Simplify any expressions involving exponents or powers.

Multiplication and Division: Perform multiplication and division in order from left to right.

Addition and Subtraction: Perform addition and subtraction in order from left to right.

Now,

(|-52 + 1| (-1) + 4²) / (-84 ÷ 7 + 5)

We solve | | first and exponents second.

|-52 + 1| = |-51| = 51

4² = 16

And,

-84 ÷ 7 = -84/7 = -12

So,

(|-52 + 1| (-1) + 4²) / (-84 ÷ 7 + 5)

= 51 x -1 + 16 / -12 + 5

= -51 + 16 / -12 + 5

= -35/-7

= 5

Thus,

The value of the expression is 5.

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HEY GUYS NEED SOME HELP!
When would the vertex of an angle have the same coordinates after a rotation?

Answers

The vertex of an angle would have the same coordinates after a rotation if it is rotated at angle of 360 degrees.

What is a rotation?

In Mathematics and Geometry, a rotation refers to a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

Generally speaking, when a point (x, y) is rotated about the center or origin (0, 0) in a counterclockwise (anticlockwise) direction by an angle θ, the coordinates of the new point (x′, y′) formed include the following:

x′ = xcos(θ) − ysin(θ)

y’ = xsin(θ) + ycos(θ).

(x′, y′) → (x, y) ⇒ (360 degrees rotation).

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Una caja de 25 kg se encuentra en reposo sobre un plano inclinado de 30 grados. Si la fuerza de rozamiento es de 50 N, ¿cuál es la magnitud de la fuerza que se debe aplicar paralela al plano para que la caja suba el plano con una aceleración de 2 m/s²?

Answers

The magnitude of the force that must be applied parallel to the plane is 92 N, under the condition that the box moves up the plane with an acceleration of 2 m/ s².

The force needed to move a box up an inclined plane can be evaluated as
Fp = W sin α = m ag sin α
Here
Fp = pulling force (N),
W = weight of the box (N),
α = angle of incline (degrees),
m = mass of the box (kg),
a = acceleration of the box (m/s²), and
g = acceleration due to gravity (9.8 m/s²).

For the given case, we possess a 25 kg box at rest on a 30 degree incline with a friction force of 50 N acting on it.
Now
We have to evaluate the weight of the box using
W = mg
= 25 kg x 9.8 m/s²
= 245 N.

Then, we have to calculate the force required to overcome friction using Ff = μFn where μ is the coefficient of friction and Fn is the normal force acting on the box. Since the box is at rest on an incline, Fn can be calculated as Fn = W cos α = 245 N cos(30°) ≈ 212 N. Therefore, Ff = μFn = 0.2 x 212 N ≈ 42 N.

Now we can evaluate the force applied to move the box up the incline utilizing
Fp = ma + Ff
Here,
a = desired acceleration
Ff = frictional force acting on the box.
Staging the values
Fp = ma + Ff
Fp = (25 kg)(2 m/s²) + 42 N
Fp ≈ 92 N

Hence, a force of approximately 92 N must be applied parallel to the plane so that the box moves up the plane with an acceleration of 2 m/s².
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The complete question is
A 25 kg box is at rest on a 30 degree incline. If the friction force is 50 N, what is the magnitude of the force that must be applied parallel to the plane so that the box moves up the plane with an acceleration of 2 m/ s²?

‼️WILL MARK BRAINLIEST‼️

Answers

The probabilities are given as follows:

a. P(number greater than 10) = 1/6.

b. P(number less than 5) = 1/3.

c. The solid is fair, as each side of the dice has the same probability of coming up.

How to calculate a probability?

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The total number of outcomes for this problem is given as follows:

12.

2 of the numbers are greater than 10, which are 11 and 12, hence the probability is given as follows:

p = 2/12

p = 1/6.

4 of the numbers, which are 1, 2, 3 and 4, are less than 5, hence the probability is given as follows:

p = 4/12

p = 1/3.

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Draw a right triangle with a tangent ratio of 3/2 for one of the acute angles.
Then find the measure of the other acute angle to the nearest tenth of a degree.
cosine

Answers

The measure of the other acute angle to the nearest degree is 34°, since the trigonometric tangent ratio of one acute angle is 3/2.

What is trigonometric ratios?

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

we shall call the acute angles X and Y such that;

tan X = 3/2 {opposite/adjacent}

X = tan⁻¹(3/2) {cross multiplication}

X = 56° approximately to the nearest degree

Y = 180° - (56 + 90)° {sum of interior angles of a triangle}

Y = 180° - 146°

Y = 34°

Therefore, the measure of the other acute angle to the nearest degree is 34°, since the trigonometric tangent ratio of one acute angle is 3/2.

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