I really need help and there also is a part c and d

a) An automorphism of a group G is a bijective map φ:G→G that preserves the group structure. That is, φ(ab) = φ(a)φ(b) and φ(a⁻¹) = φ(a)⁻¹ for all a, b ∈ G.

b) The set-theoretic composition φψ of any two automorphisms φ, ψ is an automorphism, as it preserves the group structure and is bijective.

c) The set Aut(G) of all automorphisms of G, with the operation of composition of maps, is a group. This is because it satisfies the four group axioms: closure, associativity, identity, and inverses. Therefore, Aut(G) is a group under composition of maps.

An automorphism of a group G is a bijective map φ:G→G that preserves the group structure, meaning that for any elements a,b∈G, we have φ(ab) = φ(a)φ(b) and φ(a⁻¹) = φ(a)⁻¹. In other words, an automorphism is an isomorphism from G to itself.

To show that the set-theoretic composition φψ is an automorphism, we need to show that it satisfies the two conditions for being an automorphism. First, we have

(φψ)(ab) = φ(ψ(ab)) = φ(ψ(a)ψ(b)) = φ(ψ(a))φ(ψ(b)) = (φψ)(a)(φψ)(b)

using the fact that ψ and φ are automorphisms. Similarly,

(φψ)(a⁻¹) = φ(ψ(a⁻¹)) = φ(ψ(a))⁻¹ = (φψ)(a)⁻¹

using the fact that ψ and φ are automorphisms. Therefore, φψ is an automorphism.

To show that Aut(G) is a group, we need to show that it satisfies the four group axioms

Closure: If φ,ψ∈Aut(G), then φψ is also in Aut(G), as shown above.

Associativity: Composition of maps is associative, so (φψ)χ = φ(ψχ) for any automorphisms φ,ψ,χ of G.

Identity: The identity map id:G→G is an automorphism, since it clearly preserves the group structure and is bijective. It serves as the identity element in Aut(G), since φid = idφ = φ for any φ∈Aut(G).

Inverses: For any automorphism φ∈Aut(G), its inverse φ⁻¹ is also an automorphism, since it is bijective and preserves the group structure. Therefore, Aut(G) is closed under inverses.

Since Aut(G) satisfies all four group axioms, it is a group under composition of maps.

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Answer:

30, rational

Step-by-step explanation:

[tex]3\sqrt{20}\cdot\sqrt{5}=3\sqrt{4}\sqrt{5}\cdot\sqrt{5}=(3\cdot2)\cdot5=6\cdot5=30[/tex]

The result is rational because it can be written as a fraction of integers.

Answer:

area of semi circle =pi r^2/2

3.14*6*6/2=56.2

area of rectangle=lb

=20*12=240

240+56.2=296.2

rounding it it will become 300 ft sqr

perimeter of rectangle without including 4th side=20+12+20=52

perimeter of semicircle=pi r+d (d is not needed here)

3.14*6=18.84

so total perimeter=52+18.84=70.84ft

Step-by-step explanation:

Sure, let's solve this step-by-step:

First, we need to solve for x in the equation x + 1/2 = 5.

We can do this by subtracting 1/2 from both sides, giving us x = 4 1/2.

Now, we can substitute x = 4 1/2 into the equation 2*x^2 - 3x + 6 - 3/x +2/x^2.

We can simplify the equation by multiplying both sides by x^2, giving us:

2*x^2 - 3x + 6 - 3/x +2 = 10*x^2 - 3x + 6.

Now, we can combine all of the terms with x:

10*x^2 - 6x + 6 = 0.

Finally, we can solve the equation using the quadratic formula:

x = 3/5 or x = 2.

Therefore, the answer to the equation is 10*(3/5)^2 - 6(3/5) + 6 = 4.8, or 10*2^2 - 6(2) + 6 = 16.

Answer:

Step-by-step explanation:

The given sequence is geometric.

To find the common ratio (r) of the sequence, we need to divide any term by its preceding term. Let's divide the second term (8) by the first term (32):

r = 8/32 = 1/4

Now, we can use the formula for a geometric sequence to find any term:

an = a1 * r^(n-1)

where:

an = nth term of the sequence

a1 = first term of the sequence

r = common ratio

n = position of the term we want to find

Let's use this formula to find the third term:

a3 = 32 * (1/4)^(3-1) = 2

So, the common ratio of the sequence is 1/4, and each term is obtained by multiplying the preceding term by 1/4. The sequence is decreasing rapidly because the ratio is less than 1.

The constraint w1 w2 w3 < 1, where w1, w2, and w3 are 0-1 integer variables, is often called a packing constraint. The packing constraint w1 w2 w3 < 1 limits the number of variables that can be set to 1 and is used to control the number of items that can be selected in integer programming problems.

A packing constraint is a type of constraint used in integer programming that limits the number of items that can be selected from a set of items. In this case, the constraint limits the number of variables that can take the value 1 to be less than 1.

The term "packing" comes from the idea of packing items into a container. In the context of integer programming, packing constraints are used to limit the number of items that can be "packed" into a container (i.e., set to 1), while ensuring that the total value of the packed items meets certain requirements or constraints.

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Answer:

Fai's wallet contained $30.

Step-by-step explanation:

$9 = 30%

x = 100%

If you cross multiply:

x * 30% = $9 * 100%

0.3x = $9

x = $9/0.3

x = $30

The asymptotes of the function f(x) = (2x² - 5x + 3)/(x - 2) are given as follows:

The function for this problem is defined as follows:

f(x) = (2x² - 5x + 3)/(x - 2)

The vertical asymptote is the value of x for which the function is not defined, hence it is at the zero of the denominator, and thus it is given as follows:

x - 2 = 0

x = 2.

The oblique asymptote is at the quotient of the two functions, hence:

(mx + b)(x - 2) = 2x² - 5x + 3

mx² + (b - 2m) - 2b = 2x² - 5x + 3.

Hence the values of m and b are given as follows:

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The math method that returns the nearest whole number that is greater than or equal to its argument is the "ceiling" function, denoted by ⌈x⌉ in mathematics.

The Ceiling function is denoted by ⌈x⌉, is a mathematical function that takes a real number x as an input and returns the smallest integer that is greater than or equal to x.

The ceiling function takes a real number x as an argument and returns the smallest integer that is greater than or equal to x.

For example, if x = 3.7, then ⌈x⌉ = 4, since 4 is the smallest integer that is greater than or equal to 3.7.

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Answer:

1

Step-by-step explanation:

slope = rise/run = 1/1 = 1

The following are the values for the angles and side of the right triangle using the trigonometric ratio: P = 28.6137, Q = 61.3863, and r = 14.0112

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.

tan P= 6.71/12.3 {opposite/adjacent}

P = tan⁻¹(6.71/12.3)

P = 28.6137

Q = 180 - (90 + 28.6137) {sum of interior angles of a triangle}

Q = 61.3863

sin P = 6.71/r {opposite/hypotenuse}

sin 28.6137 = 6.71/r

r = 6.71/sin 28.6137 {cross multiplication}

r = 14.0112

Therefore, the values for the angles and side of the right triangle using the trigonometric ratio are: P = 28.6137, Q = 61.3863, and r = 14.0112

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Answer:

So the lengths of the legs are approximately 8.6 cm and 15.6 cm.

Step-by-step explanation:

Let's call the smaller leg "x" and the larger leg "x + 7". According to the Pythagorean theorem, we know that:

x^2 + (x + 7)^2 = 17^2

Expanding the square on the left side and simplifying, we get:

2x^2 + 14x - 210 = 0

Dividing both sides by 2, we get:

x^2 + 7x - 105 = 0

Now we can solve for x using the quadratic formula:

x = (-7 ± sqrt(7^2 - 4(1)(-105))) / 2(1)

x = (-7 ± sqrt(649)) / 2

x ≈ -15.6 or x ≈ 8.6

Since we're dealing with lengths of sides in a triangle, we can't have a negative value for x. So we discard the negative solution and conclude that the smaller leg is approximately 8.6 cm.

To find the larger leg, we add 7 to x:

x + 7 ≈ 15.6 cm

Answer:

  2.06

Step-by-step explanation:

You want the value of the numerical expression √36÷5×12÷∛343.

This is a straightforward calculator problem. Your pocket calculator, or any of numerous calculator apps, online calculators, or spreadsheets can evaluate this expression for you.

The attachment shows the result is 2.06.

__

Additional comment

As expressed in this problem statement, the expression is ...

  [tex]\dfrac{\sqrt{36}\times12}{5\times\sqrt[3]{343}}=\dfrac{6\cdot12}{5\cdot7}=\dfrac{72}{35}[/tex]

If you mean something else, you need to identify the quantities that need to be considered as a unit.

Answer:

3

Step-by-step explanation:

you would need to conduct a survey on random people to know what the would like

Answer:

12x-4y=12

-4y= -12x+12
___________ [divide everything by -4]

-4

Y=3x+3

Step-by-step explanation:

on the y axis is 3 and the slope is 3

By the values of w and l we can plot the graph as shown in figure.

A graph in x-y axis plot is a visual representation of mathematical functions or data points on a Cartesian coordinate system. The x-axis represents the horizontal or independent variable, while the y-axis represents the vertical or dependent variable. A line or curve is drawn connecting the plotted points to show the relationship between the two variables.

Our equation is;

[tex]l = w +\frac{1}{2}[/tex]

Table between w and l can be draw as;

w            l

1         [tex]1+\frac{1}{2}[/tex] = 1.5

2        [tex]2+\frac{1}{2}[/tex] = 2.5

3        [tex]3+\frac{1}{2}[/tex] = 3.5

4        [tex]4+\frac{1}{2}[/tex] = 4.5

By these values we can plot the graph as shown in figure.

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Answer:

84 [cm³]

Step-by-step explanation:

if to imagine the given figure as parallelepiped, then the required volume can be calculated as V[1]-V[2], where V[2] is the additional part.

finally, V=10*2*7-7*2*4=84.

All the details are in the attachment.

In light of the solution, Vince is correct in believing that he should only save up to $103.55.

1. a: to formally decide (something), particularly because of facts or evidence: to establish (something) precisely or with authority. Title of the land has now been determined by the town. The land's legal owner has been established by the town. A particular committee will choose the new policy.

Vince is accurate because he perceived the solution correctly.

The inequality 225.90 ≤ 122.35 + a show that the cost of a phone ($225.90) is lower than or equal to the sum of Vince's saved money ($122.35) plus the remaining amount he needs to save (a).

The inequity can be made simpler by taking 122.35 off of both sides to get 103.55 ≤ a. Hence, Vince now needs to put aside at least $103.55 in order to buy the cheapest phone he loves.

In light of the solution, Vince is correct in believing that he should only save up to $103.55.

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Answer: 84

Step-by-step explanation:

Area of whole rectangle = lb = 11x9 = 99

Area of Inner rectangle = lb = 5x3 = 15

Area of Shaded region = Area of whole rectangle - Area of Inner rectangle
                                      = 99 - 15

                                      = 84

Answer:

Area of shaded region = 84 units²

Step-by-step explanation:

Area = Area of bigger - Area of smaller

[tex] { \tt{area = (11 \times 9) - (5 \times 3)}} \\ \\ { \tt{area = 99 - 15}} \\ \\ { \tt{area = 84 \: units {}^{2} }}[/tex]

Answer:j+309=313

Step-by-step explanation:

313-309=4

J=4

Answer:

For a client who is new to strength training and learning new movement patterns, it is ideal to start with bodyweight exercises and light resistance training. This will help them focus on proper form and technique without risking injury. Additionally, exercises that engage multiple muscle groups and involve functional movements, such as squats, lunges, and push-ups, are recommended as they provide a good foundation for overall strength and fitness. It is important to progress slowly and gradually increase resistance over time as the client becomes more comfortable with the movements and their strength improves. A certified personal trainer or strength coach can provide guidance and create a tailored program to meet the individual needs and goals of the client.

Answer:

Yes

Step-by-step explanation:

Yes because 1+14=15 hours and that is more than two

Circle A externally tangent to circle B, a common internal tangent is line r. so, the correct answer is A).

In a configuration where Circle A is externally tangent to Circle B, a common internal tangent is a line that is tangent to both circles and lies between them. This line is commonly referred to as the "direct common tangent" or "line of centers" and is denoted by letter r.

Line r is the direct common tangent because it passes through the centers of both circles and is perpendicular to the line segment that connects their centers. It also separates the two circles into distinct regions. Therefore, line r is the correct answer.

Line s is a common external tangent because it is tangent to both circles but lies outside of them, while segment AB is not a tangent but rather a chord of Circle A. So, the correct option is A).

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___The given question is incomplete, the complete question is given below:

Given: Circle A externally tangent to Circle B.

A common internal tangent is

A. line r

B. Line s

C. Segment AB

Answer: 56

Step-by-step explanation:

One possible method to estimate the population of snapping turtles in the area is by using the mark and recapture method, also known as the Lincoln-Petersen index.

According to this method, the population size can be estimated by dividing the number of marked individuals in the second sample by the proportion of marked individuals in the combined sample. In other words:

Estimated population size = (Number of individuals in sample 1 × Number of individuals in sample 2) / Number of marked individuals in sample 2

Using the information provided in the problem, we can fill in the formula as follows:

Estimated population size = (15 × 15) / 4

Estimated population size = 56.25

Rounding to the nearest whole number, we get an estimated population size of 56 snapping turtles in the area.

Answer:

∠1 = 50

∠2 = 50

∠3 = 80

∠4 = 130

∠5 = 130

Step-by-step explanation:

∠1 = 180 - 130 = 50

∠2 = ∠1 = 50

∠3 = 180 - ∠1 - ∠2 = 180 - 50 - 50 = 80

∠4 = 180 - ∠2 = 180 - 50 = 130

∠5 = ∠4 = 130

The height of the pole is approximately 17.75 meters.

The main trigonometric functions are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively. They are used to relate the angles of a right triangle to the lengths of its sides. The sine function gives the ratio of the length of the side opposite an angle to the length of the hypotenuse of the triangle. The cosine function gives the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent function gives the ratio of the length of the opposite side to the length of the adjacent side.

Let's denote the height of the pole as h, and let's denote the distance between the pole and the student's original position (due west of the pole) as x.

From the student's original position, we have a right triangle with the pole being the hypotenuse. The angle opposite to the height of the pole is 40°. So, we have:

tan(40°) = h/x

From the student's new position (10 m due south of the original position), we have another right triangle with the pole being the hypotenuse. The angle opposite to the height of the pole is 35°. The distance between the pole and the student's new position is (x+10) meters (the student moved 10 m south). So, we have:

tan(35°) = h/(x+10)

Now we have two equations with two unknowns (h and x). We can solve for x in terms of h from the first equation:

x = h/tan(40°)

Substitute this expression for x into the second equation:

tan(35°) = h/((h/tan(40°))+10)

Simplify and solve for h:

h = (10 tan(35°) tan(40°)) / (tan(40°) - tan(35°)) ≈ 17.75 m

Therefore, the height of the pole is approximately 17.75 meters.

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Therefore, the probability of getting exactly 3 successes in 10 independent trials of a binomial probability experiment with probability of success 0.35 is approximately 0.213.

Probability is a branch of mathematics that deals with the measurement and analysis of random events. It is a way of quantifying the likelihood of an event occurring. Probability can be expressed as a number between 0 and 1, with 0 indicating that an event is impossible and 1 indicating that an event is certain to occur.

The concept of probability is used in a wide range of fields, including statistics, physics, engineering, economics, and finance. It helps us make predictions about the likelihood of future events and make informed decisions based on those predictions.

by the question.

To compute the probability of X successes in N independent trials of a binomial experiment, we use the following formula:

[tex]P(X = x) = (N choose x) * p^x * (1-p)^(N-x)[/tex]

where "N choose x" is the binomial coefficient, given by:

[tex](N choose x) = N! / (x!(N-x)!)[/tex]

In this case, we are given that:

N = 10 (number of independent trials)

p = 0.35 (probability of success in each trial)

X = 3 (number of successes)

Therefore, we can compute the probability of X successes as:

[tex]P(X = 3) = (10 choose 3) * 0.35^3 * (1-0.35)^(10-3)[/tex]

Using a calculator, we can calculate:

[tex](10 choose 3) = 120 / (3! * 7!) = 120 / (6 * 5040) = 0.1666666670.35^3 = 0.042875(1-0.35)^(10-3) = 0.338915[/tex]

Putting it all together:

[tex]P(X = 3) = 0.166666667 * 0.042875 * 0.338915 = 0.00240157[/tex]

Substituting these values into the formula, we get:

[tex]P(X = 3) = (10 choose 3) * 0.35^3 * (1-0.35)^(10-3)[/tex]

[tex]= (10! / (3! * 7!)) * 0.35^3 *0.65^7[/tex]

≈ 0.213

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