can i have ten dollahairs

Answer:

G = 109

Step-by-step explanation:

The exterior angle is equal to the sum of the opposite interior angles.

12x-10 = 73+ 7x-3

Combine like terms.

12x-10 = 70+ 7x

Subtract 7x from each side.

12x-10-7x= 70+7x-7x

5x-10 = 70

Add 10 to each side.

5x-10+10 = 70+10

5x= 80

Divide each side by 5.

5x/5 = 80/5

x = 16

Now we can find angle G.

7x-3

7*16-3

112-3

109

G = 109

The numbers can be placed in the number line which is mentioned in the attached image:

Real numbers are represented by a straight line known as a number line. The distance between the dots on the line, which correlates to the magnitude of the numbers, is how the numbers are typically represented. Positive or negative numerals can be used, and they are normally marked at regular intervals.

Given numbers,

A. 3/2

B. 9/5

C. 14/10

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a. The population density of Monaco is 19,307 people per square kilometer.

b. The population density of Vatican City is 37 people per square kilometer.

c. The Vatican City has the highest population density

What is population density?

Population density is measured by dividing the total area of a region in question by the total number of people that live there.

a. To find the population density of Monaco, we need to divide the population by the area, and then convert acres to square kilometers (since Vatican City's area is given in square kilometers).

499 acres x 0.00404686 square kilometers/acre = 2.02 square kilometers

The population density of Monaco is therefore:

39,000 people / 2.02 square kilometers = 19,307 people per square kilometer.

Rounded to the nearest whole number, the population density of Monaco is 19,307 people per square kilometer.

b. To find the population density of Vatican City, we first need to convert its area to acres:

0.44 square kilometers x 247.105 acres/square kilometer = 108.7 acres

The population density of Vatican City is then:

1,000 people / 108.7 acres = 9.2 people per acre

To compare with Monaco's population density, we need to convert this to people per square kilometer:

9.2 people per acre x 0.00404686 square kilometers/acre = 37.3 people per square kilometer

Rounded to the nearest whole number, the population density of Vatican City is 37 people per square kilometer.

c. Therefore, Vatican City has the highest population density, with a density of 37 people per square kilometer compared to Monaco's density of 19,307 people per square kilometer.

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the correct option is A. 333 cm (tiny 3).

Radius is a term used in geometry to refer to the distance from the center of a circle, sphere, or other curved shape to its edge or surface. It is a key measurement used to calculate the area, circumference, volume, and surface area of these shapes. The radius is often represented by the symbol "r" and can be calculated using various formulas depending on the shape in question. In general, the radius is half the diameter of a circle or sphere, which is the distance across the shape through its center.

Given by the question.

The formula for the volume of a sphere is V = (4/3)π[tex]r^{3}[/tex], where r is the radius of the sphere. Since the diameter of the sphere is given, we can find the radius by dividing it by 2:

radius = diameter/2 = 8.6 cm/2 = 4.3 cm

Now we can plug this value into the volume. and solve for V:

V = (4/3)π[tex](4.3cm)^{3}[/tex]≈ 333.06 [tex]cm^{3}[/tex]

To the nearest whole, the volume is approximately 333. [tex]cm^{3}[/tex].

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The given sentence (exist t elementof R) (t middot x = 20) is an open sentence because it contains a variable x that is not specified. A statement, on the other hand, is a sentence that can be classified as either true or false, without any variables or unknowns.

If 5 is substituted for x, the resulting sentence is a statement: (exist t elementof R) (t middot 5 = 20). The statement is false because there is no real number t that can be multiplied by 5 to get 20.

If pi is substituted for x, the resulting sentence is a statement: (exist t elementof R) (t middot pi = 20). The statement is false because there is no real number t that can be multiplied by pi to get 20.

If 0 is substituted for x, the resulting sentence is a statement: (exist t elementof R) (t middot 0 = 20). The statement is false because any number multiplied by 0 is always 0, not 20.

The truth set of the open sentence (exist t elementof R) (t middot x = 20) is the set of all real numbers t such that their product with x is equal to 20. In other words, the truth set is {t elementof R | t middot x = 20}.

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Answer:

[tex]x=8.3[/tex]

The first option listed

Step-by-step explanation:

We can use the cosine function to evaluate [tex]x[/tex].

The definition of the cosine function is

[tex]\cos \theta=\frac{A}{H}[/tex]

Note

[tex]\theta[/tex] is the angle

[tex]A[/tex] is the side adjacent to the angle

[tex]H[/tex] is the hypotenuse

In this example we are given the hypotenuse and the angle.

Knowing these 2 values we can evaluate the adjacent side ([tex]x[/tex]).

Lets solve for [tex]A[/tex].

[tex]\cos \theta=\frac{A}{H}[/tex]

Multiplying both sides by [tex]H[/tex] lets us isolate [tex]A[/tex] ([tex]x[/tex]).

[tex]A=H*\cos \theta[/tex]

Numerical Evaluation

We are given

[tex]\theta = 41\textdegree\\H=11[/tex]

Inserting those values into our equation for [tex]A[/tex] ([tex]x[/tex]) yields

[tex]A=11*\cos 41[/tex]

[tex]A=8.30180534[/tex]

Rounding to the nearest tenth gives us

[tex]A=8.3[/tex]

[tex]x=8.3[/tex]

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The equation that Helena thought of will be written as 8x² - 41x - 15 = 0.

In mathematics, expression is defined as the relationship of numbers, variables, and functions using mathematical signs such as addition, subtraction, multiplication, and division.

Since the two solutions to the equation are x=-3/8 and x=5, the factors of the equation must be:

[tex](x + \dfrac{3}{8})(x - 5) = 0[/tex]

Expanding the equation we get:

[tex]x^2 - \dfrac{41}{8}x - \dfrac{15}8 = 0[/tex]

Multiplying the equation by 8 to get rid of the fractions, we get:

8x² - 41x - 15 = 0

Comparing this to the standard form of ax + bx + c = 0, we can see that a = 8, b = -41, and c = -15.

Therefore, the equation that Helena thought of is 8x² - 41x - 15 = 0.

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Answer:

To prove that △PRT ∼ △QRS, we need to show that the corresponding angles of the two triangles are equal and that the corresponding sides are proportional.

We can use the given information that Q lies on line PR and S lies on line RT to show that the triangles are similar.

The condition that proves △PRT ∼ △QRS is:

Angle-angle-angle (AAA) similarity condition: If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles must also be congruent, and the triangles are similar.

In this case, we can see that:

∠PRT and ∠QRS are corresponding angles (both are opposite to segment RS)

∠PTR and ∠QSR are corresponding angles (both are opposite to segment QS)

Therefore, we have two pairs of corresponding congruent angles, and by the AAA similarity condition, we can conclude that △PRT ∼ △QRS.

Step-by-step explanation:

The equation of the function of the given graph is y = (x - 5)² + 3 shown.

The parabola equation into the vertex form:

(y-k) = a(x-h)²

Where (h,k) is the x and y-coordinates of the vertex.

According to the given graph, we have data as follows:

Points on the x and y-axis = (8, 12).

Vertex (h, k)= (5, 3).

Substitute the values of h = 5 and k = 3 in the above equation

y - 3 = a(x - 5)²

y = a(x - 5)² + 3

If the parabola contains the point (0, 0)

Substitute the point (8, 12) in the above equation

12 = a((8-5)² + 3

12 = a(3)² + 3

12 - 3 = 9a

9a = 9

a = 1

4a = -3

a = -3/4

So, the equation becomes y = (x - 5)² + 3

Hence, the required equation is y = (x - 5)² + 3 which represents the given parabola.

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According to ASA regulation, AC should be equal to XZ for ABC ≅ XYZ.

Two triangles are said to be congruent if their sides are equal in length, the angles are of equal measure, and they can be superimposed on each other.

As per the given data:

∠A = ∠X

∠C = ∠Z

To show that ΔABC ≅ ΔXYZ by ASA:

∠A = ∠X [Given]

AC = XZ [Required condition for ΔABC ≅ ΔXYZ by ASA]

∠C = ∠Z [Given]

∴ For ΔABC ≅ ΔXYZ by ASA rule AC = XZ.

AC should be equal to XZ for ΔABC ≅ ΔXYZ by ASA rule.

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Based on the stress-strain curves shown, the most likely statement is that Material A is likely a ceramic material because it shows brittle behavior.

Brittle behavior is characterized by low ductility and little plastic deformation before failure, which is evident in the stress-strain curve for Material A. The curve for Material B shows some plastic deformation before fracture, which indicates a higher level of ductility than Material A. The stress-strain curve for Material C shows a high level of ductility, with a long plastic deformation region before fracture. This behavior is typical of polymeric materials, so Material C is most likely a polymeric material.

It is not possible to determine the specific type of material based solely on stress-strain curves, as many materials can exhibit similar behaviors. Additionally, it is not accurate to say that Material B or Material C are likely ceramic materials based on their stress-strain curves, as both exhibit significant plastic deformation before failure, which is not typical of brittle ceramics.

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The explicit rule is aₙ = 7+5(n-1), where n is the position of the term.

The complete table is given below.

An explicit formula is a formula that allows us to find the nth term of an arithmetic sequence. Based on this formula, the sum of the arithmetic series can be derived.

The first term a₁=7 and the fourth term a₄=22.

The arithmetic explicit formula for the nth term is aₙ=a₁+(n-1)d, where d is a common difference.

So, find the value of d using a₁=7 and a₄=22 in the formula.

22=7+(4-1)d

15=3d

d=5

Therefore, the explicit formula becomes aₙ = 7+5(n-1).

Substitute n=2 and n=3 into the obtained formula.

a₂=7+5(2-1)

=7+5

=12

a₃=7+5(3-1)

=7+10

=17

Therefore, the required answers are a₂=12 and a₃=17.

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Answer:Yes, they are all correct

Step-by-step explanation:

From the attached image, we see that the sign is showing;

Parking lot: ¾ miles

Summit: 1½ miles

Summit distance can also be expressed as an improper fraction = 3/2 miles

Now, since they started hiking from the park to the summit, it means that they had moved ¾ mile from the parking lot and had 3/2 miles left to get to the summit.

Thus, total distance from parking lot to summit = ¾ + 3/2 = 9/4 miles

Now, let's analyze each of their statements;

Mai said they are a third of their way there.

They had covered 3/4 mile and the total distance is 9/4 miles.

Thus, fraction of total distance covered is;

9/4 ÷ 3/4 = 1/3.

So, Mai is correct

Clare said they have to go twice as far as they had already gone.

They had covered 3/4 miles.

Therefore, twice this = 2 × 3/4 = 6/4 = 3/2 which is same as the sign distance left to the summit.

Thus, Clare is correct

Tyler said that the total hike is three times as long as what we have already gone.

They had gone 3/4 miles

3 times this = 3 × 3/4 = 9/4 miles.

This tallies with the total distance calculated earlier.

Thus, they are all correct

The probability that the arrow will land on a section labeled S or T is 7/8.

Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is the measure of the likelihood of an event occurring divided by the number of possible outcomes. Probability is used to determine the chances of a particular outcome occurring and can range from 0 to 1.

Therefore, it is likely that the arrow will land on one of these sections around 56 to 57 times out of 75 spins. It is impossible to predict exactly how many times the arrow will land on S or T, as this is a probability-based outcome. Generally speaking, with such a large number of spins, the result should be close to the probability of 7/8.

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0.306 is  the z-score. Round to two decimal places. Include any leading zeros.

What does "z-score" mean?

The Z-score provides information on how far a given value deviates from the standard deviation. The amount of standard deviations a given data point is above or below the mean is represented by the Z-score, also known as the standard score.

                           Essentially, standard deviation is a measure of the degree of variability within a given data collection.

Let X the random variable that represent the rating score of a population, and for this case we know the distribution for X is given by

     X  `N(667,65 )

Where μ = 667 and σ = 65

We are interested on this probability

P(X > 700 )

And the best way to solve this problem is using the normal standard distribution and the z score given by

    Z= X - μ/σ

If we apply this formula to our probability we got this

  P(X > 700 )

 = P(X - μ/σ > 700 -μ/σ)

 = P(Z > 700 - 667/65 )

= P(Z > 0.508 )

And we can find this probability using the complement rule and excel or a calculator and we got

  P( z > 0.508 ) = 1 - P(Z - 0.508 ) = 1 - 0.694  = 0.306

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The correct answer is Option 3 (0,2). The medians of a triangle intersect at the midpoint of the opposite side of the triangle.

Triangle is a three-sided geometric shape with three angles and three vertices. Triangles can be classified into different types based on the lengths of their sides and the angles between them. The three most commonly referred to types include the equilateral, isosceles and scalene triangle. An equilateral triangle has three sides of equal length, while an isosceles triangle has two sides of equal length. A scalene triangle has no equal sides or angles. All three types of triangles have interior angles that add up to 180 degrees and all three sides must be connected. All triangles are two-dimensional shapes, meaning they have no thickness or depth.

In triangle ABC, the opposite side is the line segment BC, which has endpoints at (-3,2) and (1,2). The midpoint of this line segment is (0,2).

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The given two triangles are not similar. The correct answer would be an option (A).

Similar Triangles are defined as two triangles with the same shape, equal pair of corresponding angles, and the same ratio of the corresponding sides.

As per the figure, two sides of the triangles are given.

∠HMG = ∠JMK (vertically opposite angles)

HM/MK = 8/12 = 2/3

GM/MJ = 12/16 = 3/4

Since the two sides of one triangle are not proportional to the corresponding sides in the other, and the angle in the midpoint is equal, the above triangles are not similar.

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Leasing and buying are two popular options for acquiring a car or other assets. Both options have their own advantages and disadvantages, and the decision between leasing and buying will depend on individual needs and preferences. Below are some pros and cons of leasing and buying.

Initial and Monthly Payments: When it comes to initial and monthly payments, leasing typically requires lower upfront costs and lower monthly payments compared to buying. This is because leasing only covers the cost of the car's depreciation during the lease term, while buying involves financing the full cost of the vehicle.

Total Cost: When considering the total cost of leasing and buying, it's important to look beyond just the initial and monthly payments. While leasing may have lower monthly payments, the total cost of the lease over the long term may be higher than buying. This is because at the end of the lease term, the lessee has no equity in the car and must either lease a new car or purchase a new car.

Ownership and Customization: One of the main advantages of buying is that the car is owned outright, giving the owner the freedom to customize the vehicle as desired. With leasing, the lessee is limited to the terms of the lease agreement and cannot make any major modifications to the car.

Flexibility: Leasing provides greater flexibility in terms of upgrading to a newer car at the end of the lease term, while buying requires selling or trading in the old car to acquire a new one.

In summary, leasing and buying both have their own advantages and disadvantages, and the decision between the two will depend on individual needs and preferences.

Leasing may be a good option for those who prioritize lower monthly payments and want the flexibility to upgrade to a newer car every few years, while buying may be a good option for those who prioritize ownership, customization, and long-term cost savings.

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Answer: Your welcome!

Step-by-step explanation:

The new balance after the first minimum payment is made is $328.5. This is calculated by taking the balance of $350 and subtracting 7% of the balance, which is $24.50.

The minimum payment that is due the next month is $23.04. This is calculated by taking 7% of the new balance of $328.5.

The measure of side ML of the triangle MLK is ML = 6 units

If two triangles' corresponding angles are congruent and their corresponding sides are proportional, they are said to be similar triangles. In other words, similar triangles have the same shape but may or may not be the same size. The triangles are congruent if their corresponding sides are also of identical length.

Corresponding sides of similar triangles are in the same ratio. The ratio of area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides

Given data ,

Let the first triangle be represented as ΔMLK

Let the second triangle be represented as ΔHJI

The measure of side HJ = 20 units

The measure of side HI = 36 units

The measure of side JI = 24 units

And ,

The measure of side LK = 5 units

The measure of side MK = 9 units

The measure of side ML = A units

Now , the triangles are similar

So , the corresponding sides of similar triangles are in the same ratio

The measure of side MK / measure of side HI = The measure of side ML / measure of side JI

Substituting the values in the equation , we get

A / 24 = 9 / 36

Multiply by 24 on both sides of the equation , we get

A = ( 24 x 9 ) / 36

A = 24 / 4

A = 6 units

Hence , the measure of side ML of triangle is 6 units

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Answer:

1) y² - 5y = 750

2) 750 -y(y -5) = 0

3) (y + 25)(y -30) = 0

Step-by-step explanation:

Area of rectangle = 750 ft²

                 length = y ft

                width = (y - 5) ft

Area of rectangle = 750

       length * width = 750

           y (y -5) = 750

          y*y - 5*y = 750

          y² - 5y = 750

                    0 = 750 - y(y -5)

          750 - y(y-5) = 0

        y² - 5y - 750 = 0

        Sum = -5

      Product = -750

      Factors = -30 , 25   {-30 + 25 = -5  &  (-30)*25 = -750}

y² - 30y + 25y - 750 = 0      {Rewrite the middle term using the factors}

y(y - 30) +25(y - 30) = 0

      (y - 30)(y + 25) = 0        

The given passage is telling the life story of Emest Cline, therefore, this is a Biography.

A biography is simply the story of a real person's life. It could be about a person who is still alive, someone who lived centuries ago, someone who is globally famous, an unsung hero forgotten by history, or even a unique group of people.

The given passage is about the famous American Screenwriter and author, Emest Cline,

The passage is telling his works, his career achievements, his first work, his latest work, the most successful work.

We know that, a biography is usually written history of a person's life.

The value of the passage is the same as a biography.

Therefore, the given passage is telling the life story of Emest Cline, therefore, this is a Biography.

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a) The number of ways to deal the cards without getting one of the designated cards are equals to the 2250829575120.

b) The number of ways to deal each player 7 cards, regardless of whether the designated cards come out are equals to the 21945588357420.

c) The probability of a successful hand that will go all the way till everyone gets 7 cards is equals to the 0.1025.

Five friends group are playing poker one night. They have a standard 52-card deck. So, here total possible outcomes

= 52

Now, the designated cards are 6 of Spades, Jack of Diamonds. So,

a) Number of cards are in the deck that are not one of the designated cards = 52 - 2 = 50

Number of cards that need to be dealt in order for each player to have 7 cards

= 5× 7 = 35

Thus total possible number of ways

= ⁵⁰C₃₅ = 2250829575120, which are ways to deal the cards without getting one of the designated cards.

b) Number of cards are in the deck = 52

Number of cards that need to be dealt in order for each player to have 7 cards

= 5× 7 = 35

Thus total possible number of ways

= ⁵²C₃₅ = 21945588357420

Which are ways to deal each player 7 cards, regardless of whether the designated cards come out.

c) The probability of a successful hand that will go all the way till everyone gets 7 cards is = Number of ways to deal the cards without getting one of the designated cards/Total number of ays to deal the cards

= 2250829575120/21945588357420

= 0.10256410256

Hence, required probability is 0.10256.

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Complete question:

A group of 5 friends are playing poker one night, and one of the friends decides to try out a new game. They are using a standard 52-card deck. The dealer is going to deal the cards face up. There will be a round of betting after everyone gets one card. Another round of betting after each player gets a second card, etc. Once a total of 7 cards have been dealt to each player, the player with the best hand will win. However, if any player is dealt one of the designated cards, the dealer collects all cards, shuffles, and starts over. The designated cards are: 6 of Spades, Jack of Diamonds. The players wish to determine the likelihood of actually getting to play a hand without mucking the cards and starting over.

a) In how many ways can you deal the cards WITHOUT getting one of the designated cards? (Hint: Consider how may cards are in the deck that are NOT one of the designated cards and consider how many cards need to be dealt in order for each player to have 7 cards.)

b) In how many ways can you deal each player 7 cards, regardless of whether the designated cards come out?

c) What is the probability of a successful hand that will go all the way till everyone gets 7 cards? (Round your answer to 4 decimal places.) Recall, while using your calculator, that E10 means to move the decimal place 10 places to the right.

The surface area of the pyramid is 119 inches².

The area of a three dimensional object on it's outer surface is called the surface area of the object.

A square pyramid has a square base and 4 triangular faces.

Surface area of the pyramid is the base area plus 4 times the area of each triangular face.

Base is a square.

Base area = a², where a is the side length of the square.

Base edge of square = 7 inches

Base area = 7² = 49 inches²

Slant height of triangle = 5 inches

Area of a triangular face = [tex]\frac{1}{2}[/tex] × base × height

                                         = [tex]\frac{1}{2}[/tex] × 7 × 5

                                         = 17.5 inches²

Area of 4 triangular faces = 4 × 17.5 inches²

                                           = 70 inches²

Surface area of the pyramid = 49 inches² + 70 inches² = 119 inches²

Hence the total surface area is 119 inches².

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Your question is incomplete. The complete question is probably the one given below.

Find the surface area of the pyramid. The side lengths of the base are equal. Square pyramid with base edge of 7 inches, and a slant height measuring 5 inches, on a triangular face.

The probability that the random variable X will be less than or equal to x. A uniform distribution's CDF is a linear function that increases from 0 to 1 across the range [a, b].

The uniform probability density function is given as follows:

If a = x = b = 0, then fx (x) = 1/(b-a), otherwise

where a and b are the uniform distribution's lower and upper bounds, respectively.

This means that the probability density function fx(x) is constant and equal to 1/(b-a) for any value of x between a and b, indicating that the probability of x taking a value in that interval is proportional to the width of the interval.

The probability density function fx (x) is zero outside of this range, It implies that the probability of x taking a value outside of [a, b] is zero. This is because the uniform distribution is only defined within the range [a, b].

It's the cumulative distribution function (CDF) used to express the probability density function for a uniform distribution:

Fx (x) = (x-a)/(b-a)

where a = x = b = 0, x a = 1, and x > b.

It shows the probability that the random variable X will be less than or equal to x. A uniform distribution's CDF is a linear function that increases from 0 to 1 across the range [a, b].

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The solution is, 2240 for each product the firm should produce in order to maximize profit.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

here, we have,

Explanation:

a.

Decision variables:

Let

X1 = no of units of product X1

X2 = no of units of product X2

X3 = no of units of product X3

Objective function is to maximize profits

Max Z = 20X1 + 6X2 + 8X3

Constraints:

8X1 + 2X2 + 3X3 <= 800

4X1 + 3X2 <= 480

2X1 + X3 <= 320

X1, X2, X3>=0

b.

please see attachment for the excel solutions.

c.

X1 = 0

X2 = 160

X3 = 160

Z = 2240

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The number of muffins in each bag is 6

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

5 boxes of muffins with m number in each box equals 30​

Using the above as a guide, we have the following equation

5m = 30

Divide both sides by 5

m = 6

Hence, there are 6 muffins in each bag

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Complete question

5 boxes of muffins with m number in each box equals 30​

What is the number of muffin in each box

The six trigonometric functions of theta are: sine (sinθ) = -7/3, cosine (cosθ) = -3/3, tangent (tanθ) = -7/3, cotangent (cotθ) = -3/7, secant (secθ) = -3/7, and cosecant (cscθ) = -7/3.

sinθ = -7/3

cosθ = -3/3

tanθ = -7/3

cotθ = -3/7

secθ = -3/7

cscθ = -7/3

The point (3,-7) is located on the terminal side of an angle θ in standard position, thus allowing us to calculate the six trigonometric functions of theta. The six trigonometric functions of θ are sine (sinθ), cosine (cosθ), tangent (tanθ), cotangent (cotθ), secant (secθ), and cosecant (cscθ). To calculate each of these functions, we must first calculate the ratio of the vertical and horizontal components of the point. In this case, the vertical component is -7 and the horizontal component is 3. Therefore, the ratio of these two components is -7/3. This ratio is equal to the value of sinθ, tanθ, and cscθ. To calculate the cosθ, cotθ, and secθ, we must take the reciprocal of the ratio of the vertical and horizontal components. In this case, the reciprocal is -3/7. Therefore, the exact values of the six trigonometric functions of theta are sinθ = -7/3, cosθ = -3/3, tanθ = -7/3, cotθ = -3/7, secθ = -3/7, and cscθ = -7/3.

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(a) If A, B and C are independent, the Pr (A|B Intersection C) = Pr (A) - The given statement is False  

(b) The events S, phi, A are independent (S is the certain event, phi is the impossible event and A is an arbitrary event here) - True  

(c)  If the events A and B are mutually exclusive and Pr (A), Pr (B) are both positive, then they cannot be independent - True  

(d) Pr (A_1 Intersection A_2 Intersection Intersection A_n) = Pr (A_n) Pr [tex](A_n-1 | A_n) Pr (A_1|A_n[/tex] Intersection Intersection A_2)- True  

(e) sigma^n_k=1 (n k)p^k (1 - p)^n-k = 1 for 0 less than or equal to k less than or equal to n, n greater than or equal to 1 - True  

(f) Let X be a continuous random variable with pdf fx (x). Then Pr (X = x_0) = fx (x_0) - False  

(g) Let X be a continuous random variable with pdf fx (x). Then (for a less than or equal to b) Pr (a less than or equal to X less than or equal to b) = integral [tex]^b_a f_X (x) dx[/tex] - True    

(h) Let X be a discrete random variable with pmf px (x_i). Then Pr (X = x_0) = px (x_0). - True    

(i) Let X be a discrete random variable with pmf p_X (x_i). Then (for a less than or equal to b) Pr (a less than or equal to X less than or equal to b) = sigma_x_i: a less than or equal to x_i less than or equal to b p_X (x_i) - True  

(j) Let X be a random variable with cdf F_X (x). Then (for a less than or equal to b) Pr (a less than or equal to X less than or equal to b) = F_X (b) -F_X (a) - True

(a) False

Counter example:

Let A, B, and C be events such that

P(A) = 1, P(B) = 0.5, P(C) = 0.5, and P(A | B ∩ C) = 0

Then, P(A) ≠ P(A | B ∩ C)

So A, B, and C are not independent.

(b) True

The definition of independence of events states that if A and B are independent, then P(A | B) = P(A).

Since S is the certain event and phi is the impossible event, it follows that P(S) = 1 and P(phi) = 0, and for any arbitrary event A, P(A | S) = P(A) and P(A | phi) = 0.

Therefore, S, phi, and A are independent.

(c) True

The definition of mutually exclusive events states that if A and B are mutually exclusive, then P(A ∩ B) = 0.

If P(A) and P(B) are both positive, then A and B cannot be independent, because if A and B are independent, then P(A ∩ B) = P(A) P(B) > 0, which contradicts the fact that A and B are mutually exclusive.

(d) True

This is the definition of the multiplication rule for independent events.

If A1, A2, ..., An are independent events, then P(A1 ∩ A2 ∩ ... ∩ An) = P(A1) P(A2 | A1) P(A3 | A1 ∩ A2) ... P(An | A1 ∩ A2 ∩ ... ∩ An-1) = P(An) P(An-1 | An) ... P(A1 | A2 ∩ ... ∩ An)

(e) True

This is the binomial theorem, which states that the sum of the probabilities of all possible outcomes of a binomial experiment is equal to 1.

(f) False

Counterexample:

Let X be a continuous random variable with pdf fX(x) = 0 for all x except x = 0

where fX(0) = 1.

Then,

P(X = 0) = 0

but fX(0) ≠ 0.

So, it is not true that P(X = x0) = fX(x0) for a continuous random variable.

(g) True

This is the definition of the cumulative distribution function (CDF) for a continuous random variable.

(h) True

This is the definition of a probability mass function (PMF) for a discrete random variable.

(i) True

This is the definition of the cumulative distribution function (CDF) for a discrete random variable.

(j) True

This is the definition of the cumulative distribution function (CDF) for a random variable.

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For y(t)=(4+t)⁽⁻¹²⁾ to be a solution to the initial value problem y!+αyⁿ=0, y(0)=y0, α must be 12y0² and n must be (ln(y0²) - ln(4)) / (12 ln(2)) + 1/2, provided that y0 ≠ 0 and y0 ≠ 2.

To determine the values of y0, α, and n for which y(t) = (4+t)⁽⁻¹²⁾ is a solution to the initial value problem
(4) y! + αyⁿ = 0, y(0) = y0,
we first need to find the value of y! and y'(t).
Using the chain rule, we can write
y'(t) = -12(4+t)⁽⁻¹³⁾.
To compute y!, we can use the formula
y! = dy/dt * dt/dy,
where dt/dy is the inverse of dy/dt. In this case, dt/dy = 1/y', so we have
y! = y'(t) / dt/dy = -12(4+t)⁽⁻¹³⁾ * (dt/dt)⁽⁻¹⁾ = -12(4+t)⁽⁻¹³⁾ * y(t)².
Substituting y(t) = (4+t)⁽⁻¹²⁾ into the differential equation (4), we get
y! + αyⁿ = -12(4+t)⁽⁻¹³⁾ * (4+t)⁽⁻²⁴⁾ + α(4+t)⁽⁻¹²ⁿ⁾ = (-12 + α(4+t)⁽⁻¹²ⁿ⁺¹³⁾)(4+t)⁽⁻²⁴⁾.
For y(t) to be a solution to the differential equation (4), we need y! + αyⁿ to be identically zero, so we must have
-12 + α(4+t)⁽⁻¹²ⁿ⁺¹³⁾ = 0.
Setting t = 0 and using the initial condition y(0) = y0, we have
y!(0) + αyⁿ(0) = -12y0² + α = 0,
which implies α = 12y0². Substituting this into the previous equation, we get
-12 + 12y0²(4⁽⁻¹²ⁿ⁺¹³⁾) = 0,
or equivalently,
-1 + y0²(4⁽⁻¹²ⁿ⁺¹³⁾) = 0.
Solving for n, we get
n = (ln(y0²) - ln(4)) / (12 ln(2)) + 1/2,
which is valid for y0 ≠ 0 and y0 ≠ 2. For α, we have α = 12y0², and for any n that satisfies the above equation, y(t) = (4+t)⁽⁻¹²⁾ is a solution to the initial value problem (4) with y(0) = y0.

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Complete question:

For what values of the constants y0, α and integer n is the function y(t)=(4+t)⁻¹² a solution of the initial value problem below?
(4) y!+αyⁿ = 0, y(0)=y0.