At a soda fountain 22 quarts of ice cream were used to make 105 milk shakes. How many quarts are needed to make 922 milk shakes?

The only option that represents an independent event is: Option C: "Spinning a Spinner with eight evenly spaced sections, then spinning it again.".

Independent events are defined as those events whose occurrence is not dependent on any other event. For example, if we flip a coin in the air and get the outcome as Head, then again if we flip the coin but this time we get the outcome as Tail. In both cases, the occurrence of both events is independent of each other.

Looking at the given options, the only one that represents an independent event is "Spinning a Spinner with eight evenly spaced sections, then spinning it again.".

This is because each event does not depend on another one of the events being described.

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Answer:

1/4 > 3/16

Step-by-step explanation:

1/4 = 4/16

4/16 > 3/16

1/4 > 3/16

hope this helps:)

~~~Wdfads~~~

The value of the missing side length is 2√6. Option C

We have that the six trigonometric identities are;

From the information given, we have that;

Using the tangent identity,

tan 60 = 6√6/y

find the value

y = 6√6 /√3

Using the sine identity

sin 45 =  6√6 /√3/x

sin45 =6√18/3 × 1/x

√2/2x = 6√18/3

x = 2√18/√2

x = 2√36

x = 2√6

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The tent occupies 640 cubic feet of space.

Volume is a measurement of the amount of three-dimensional space occupied by an object or substance. It is the measure of the amount of space that an object or substance occupies in three dimensions, typically expressed in cubic units such as cubic meters, cubic centimeters, or cubic feet.

For example, the volume of a cube can be calculated by multiplying its length by its width by its height (or side cubed):

V = l x w x h = [tex]s^3[/tex],

where V is the volume, l is the length, w is the width, h is the height, and s is the length of a side.

Volume is an important measurement in fields such as physics, engineering, architecture, and chemistry, where it is used to calculate quantities such as the amount of fluid that can be held by a container, the amount of material needed to construct a building, or the amount of a substance that can react with another substance in a chemical reaction.

The volume of the tent can be found by calculating the volume of the square pyramid and then adding the volume of the rectangular box formed by the tent's base.

The base of the tent is an 8-ft by 8-ft square, so its area is 8 x 8 = 64 square feet. The height of the pyramid is 3 ft, so its volume is:

V(pyramid) = (1/3) x base area x height

V(pyramid) = (1/3) x 64 x 3

V(pyramid) = 64 cubic feet

The rectangular box formed by the tent's base has dimensions of 8 ft by 8 ft by 9 ft, so its volume is:

V(box) = length x width x height

V(box) = 8 x 8 x 9

V(box) = 576 cubic feet

Therefore, the total volume of space the tent occupies is:

V(total) = V(pyramid) + V(box)

V(total) = 64 + 576

V(total) = 640 cubic feet

So the tent occupies 640 cubic feet of space.

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A piecewise function is a function that is defined by different equations or formulas on different intervals or pieces of its domain. In other words, a piecewise function is a function that consists of multiple functions "stitched" together.

When looking at the line graph, it is clear that there is a error because the scales don't begin at 0.

An example of a chart that shows data trends over time or other continuous periods is a line graph. The x-axis represents time or another continuous variable, while the y-axis represents the value of the data. It consists of a sequence of data points connected by a line.

The visualisation of trends and patterns in data, such as variations in temperature, stock prices, or sales numbers over time, is frequently done using line graphs. Due to the ability to draw many lines in a variety of colors and patterns, they are particularly helpful for comparing different data series on the same graph.

When looking at the line graph, it is clear that there is a flaw because the scales don't begin at 0.

It presents information that is incongruous, as if it were saying that the highest hour on Friday is 7, but that it is precisely 1 to 7, making it equivalent to 6.

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Answer: 1 is .4

2. is .6

3 is .5

4 is .375

5 is .18

6 is .71

7 is .16

8 is .66 repeating

9 is .91

and 10 is .25

Step-by-step explanation:

to get all these and fractions to decimals in the future just divide the numerator by the denominator in other words the top number by the bottom number

1): 0.4

2): 0.67 or 0.7

3): 0.5

4): 0.37 or 0.4

5): 0.18

6): 0.42 or 0.5

7): 0.17

8): 0.7

9): 0.97 or 1

10): 0.25

Answer:

Step-by-step explanation:

the correct form of g multiplied by 67 is 67g

The type of variable for,

a. The number of counties in California: Quantitative discrete.

b. The stages of childhood: Qualitative ordinal.

c. In the scenario above, 23 minutes is a statistic, because 28 students is a sample.

d. Between 6 minutes and 8 minutes: Approximately 68% of the athlete's mile times are expected to be between 6 and 8 minutes, according to the empirical rule.

e. Between 4 minutes and 7 minutes: Approximately 68% of the athlete's mile times are expected to be between 4 and 10 minutes, according to the empirical rule.

f. Less than 9 minutes: Approximately 84% of the athlete's mile times are expected to be less than 9 minutes, according to the empirical rule.

In statistics, variables can be categorized into two types: qualitative and quantitative.

Qualitative variables describe characteristics or qualities that cannot be measured numerically, such as gender or hair color.

Quantitative variables, on the other hand, represent numerical values that can be measured or counted.

There are two types of quantitative variables: continuous and discrete. Continuous variables can take any numerical value within a range, such as age or weight.

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Step-by-step explanation:

Total people surveyed =    246

  total female =  122    

      divorced = 15 + 13   BUT the 13 was already counted in 'female'

  soooo  

        female OR divorced =  122 + 15 out of 246 = 137/246 = .5569

A statistician would reply that in order to determine if the generic drug is less effective than the name-brand drug, a hypothesis test needs to be conducted.

The null hypothesis (H0) would be that there is no difference in the average blood pressure reduction between the two drugs, while the alternative hypothesis (H1) would be that the name-brand drug has a higher average reduction in blood pressure than the generic drug.

To test these hypotheses, a t-test would be appropriate since we have two independent samples (control and treatment groups) with known means, standard deviations, and sample sizes. The t-test will provide a p-value, which can be compared to a chosen significance level (e.g., α = 0.05).

If the p-value is less than the significance level, we reject the null hypothesis and conclude that there is a significant difference in the average blood pressure reduction between the two drugs. If the p-value is greater than the significance level, we fail to reject the null hypothesis, meaning we do not have enough evidence to claim that the name-brand drug is more effective than the generic drug.

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the length of the ladder is approximately 54.01 feet.We can use trigonometric ratios to solve this problem

what is trigonometric ratios ?

Trigonometric ratios are mathematical functions used to relate the angles and sides of a right triangle. In a right triangle, there are three sides: the hypotenuse (the longest side, opposite the right angle), the opposite side (opposite to the angle we are interested in), and the adjacent side

In the given question,

We can use trigonometric ratios to solve this problem. Let's label the ladder as the hypotenuse of a right triangle, the height of the electric box as the opposite side, and the distance from the base of the ladder to the house as the adjacent side. Then we have:

sin(62°) = opposite/hypotenuse

cos(62°) = adjacent/hypotenuse

We know the height of the electric box is 27 feet, and we want to find the length of the ladder (the hypotenuse). Let's use the cosine ratio to solve for the hypotenuse:

cos(62°) = adjacent/hypotenuse

hypotenuse = adjacent/cos(62°)

Now we need to find the adjacent side. We know the height of the electric box is the opposite side, and we can use the tangent ratio to find the adjacent side:

tan(62°) = opposite/adjacent

adjacent = opposite/tan(62°)

opposite = 27 feet

Substituting in the values, we get:

adjacent = 27/tan(62°) ≈ 24.39 feet

Then we can find the hypotenuse:

hypotenuse = adjacent/cos(62°) ≈ 54.01 feet

Therefore, the length of the ladder is approximately 54.01 feet.

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The survey question that would produce a qualitative response is one that asks about qualities or attributes.

A qualitative response is a non-numeric response that describes the characteristics or qualities of something.

In a survey, a qualitative question typically asks about opinions.

Attitudes, behaviors, or other non-measurable factors.

Out of the following survey questions, the one that would produce a qualitative response is:

"What is your favorite food in the cafeteria?"

This question is asking for a subjective opinion, which is a qualitative response. The response could vary from student to student and cannot be quantified numerically.

A qualitative response is a response that is based on qualities or attributes, rather than numerical or measurable data.

Out of the listed questions, the survey question that would produce a qualitative response is:

What is your favorite color?

This question asks about a personal preference, which cannot be measured numerically. The response would be a word or phrase indicating the color, which is a qualitative attribute.

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The dimension for the largest area is 181 sq ft.

The rectangle has width. [tex]$w$[/tex] and height [tex]$h$[/tex],and let the radius of the semicircle be [tex]$r$[/tex]. We want to maximize the area [tex]$A$[/tex] of the window, which is given by:

[tex]A= 21 \pi r 2 +wh[/tex]

We also have the constraint that the total length of framing material is 12 feet:

[tex]2r+2w+h=12[/tex]

Solving this equation for [tex]$h$[/tex] we get:

[tex]h=12-2r-2w[/tex]

Substituting this expression for [tex]$h$[/tex] into the equation for [tex]$A$[/tex], we get:

[tex]A= 21 \pi r ^2+w(12-2r-2w)[/tex]

Expanding this expression, we get:

[tex]A= 21\pi r^2 +12w-2rw-2w^2[/tex]

To find the values of [tex]$r$[/tex], [tex]$w$[/tex], and [tex]$h$[/tex] that maximize[tex]$A$[/tex], we take the partial derivatives of [tex]$A$[/tex] with respect to each variable, set them equal to zero, and solve for the variables.

We get:

[tex]\partial r/\partial A=\pi r-2w=0 \Rightarrow r= 2w/\pi[/tex]

[tex]\partial w/\partial A=12-2r-4w=0 \Rightarroww= (6-r)/2= (6-2w/\pi)/2=3- w/\pi[/tex]

Substituting the expression for [tex]$w$[/tex] into the equation for [tex]$r$[/tex], we get:

[tex]r= 2/\pi (3-w/\pi )= 6/\pi - 2/\pi^2w[/tex]

Substituting these expressions for [tex]$r$[/tex] and [tex]$w$[/tex] into the equation for [tex]$h$[/tex], we get:

[tex]h=12-2r-2w=12- (4/\pi)w[/tex]

So the dimensions that maximize the area are:

[tex]w= \pi/2,r= 3\pi /4 ,h= (48-8\pi) / \pi[/tex]

The area of the window is:

[tex]A= (1/2)\pir^2 +wh= (1/2)\pi ( 3\pi/4)^2 + (\pi/2)\cdot (48-8\pi)/\pi = 9\pi/8+24-4\pi\approx 18.1 sq ft[/tex]

​

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Therefore, the dimensions of the largest area rectangular window with a semicircle top that can be made with 12' of framing material are approximately 1.55' * 3.38'. The maximum area is: Area = 4.82 square feet.

To find the dimensions for the largest area of a rectangular window with a semicircle top using 12' of framing material, we need to use optimization techniques.

Let's denote the width of the rectangular window as "w" and the height of the rectangular portion as "h". We also know that the semicircle top will have a radius equal to the width of the rectangular portion, so its diameter will be 2w.

The perimeter of the window is given by:

Perimeter = 2w + h +[tex]\pi w[/tex]

Since we have 12' of framing material available, we can write:

2w + h + [tex]\pi w[/tex] = 12

We can rearrange this equation to solve for h:

h = 12 - 2w - [tex]\pi w[/tex]

The area of the window is given by:

Area = w * h + 1/2 * [tex]\pi w^2[/tex]

Substituting the expression for h from the perimeter equation, we get:

Area = w(12 - 2w - [tex]\pi w[/tex]) + 1/2 * [tex]\pi w^2[/tex]

Expanding and simplifying, we get:

Area = 12w - [tex]2\pi w^2[/tex] - [tex]w^2[/tex]

To find the dimensions that maximize the area, we need to take the derivative of the area equation with respect to w and set it equal to zero:

d(Area)/dw = 12 - [tex]4\pi w[/tex] - 2w = 0

Solving for w, we get:

w = 12/(4\pi+2) ≈ 1.55'

Substituting this value back into the perimeter equation, we can find the height:

h = 12 - 2w - \piw ≈ 3.38'

Therefore, the dimensions of the largest area rectangular window with a semicircle top that can be made with 12' of framing material are approximately 1.55' * 3.38'. The maximum area is:

Area ≈ 4.82 square feet.

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Answer:

The nth term rule for an arithmetic sequence with a first term of -10 and a common difference of 0.5 is given by:

an = a₁ + (n-1)d

where a₁ = -10 and d = 0.5.

Substituting these values into the formula gives:

an = -10 + (n-1)0.5

Simplifying this expression gives:

an = -10 + 0.5n - 0.5

an = -10 + 0.5n - 0.5

an = -10 + 0.5n - 0.5

an = -10 + 0.5n - 0.5

an = -10 + 0.5n - 0.5

an = -10 + 0.5n - 0.5

an = -0.5n - 9.5

I hope that helps! Let me know if I'm wrong!

Step-by-step explanation:

Answer:

Table D represents the linear function

y = x + 1.

the probability that the emergency center will receive five calls on a given day is approximately 0.156, or 15.6%.

To find the probability of receiving five calls on a given day at the emergency center, we need to use the Poisson distribution formula:

[tex]P(x = 5) = (e^{-4})*\frac{(4^5)}{(5!)}[/tex]
Where:
- x = number of phone inquiries
- e = Euler's number (approximately 2.71828)

- ! = factorial (i.e. 5! = 5*4*3*2*1)

Given that the average number of phone inquiries per day is four, we can use that as our lambda (λ) value in the Poisson distribution formula, since lambda represents the mean number of events in a specific time interval:

λ = 4

Now we can substitute these values into the formula and solve:

P(x = 5) = (e^-4)*(4^5)/(5!) = (2.71828^-4)*(1024)/(120) ≈ 0.156

Therefore, the probability that the emergency center will receive five calls on a given day is approximately 0.156, or 15.6%.

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The probability that the emergency center will receive five calls on a given day is approximately 0.1755 or 17.55%.

To find the probability that the emergency center will receive five calls on a given day, given that the average number of phone inquiries per day is four, we can use the Poisson distribution formula.

Identify the average rate (λ): In this case, λ = 4 calls per day.

Identify the desired number of events (k): In this case, k = 5 calls.

Use the Poisson distribution formula: [tex]P(X = k) = (e^(-λ) \times (λ^k)) / k![/tex]

e is the base of the natural logarithm (approximately 2.71828)

λ is the average rate

k is the desired number of events

k! is the factorial of k

Plug in the values and calculate the probability:

[tex]P(X = 5) = (e^{(-4)} \TIMES (4^5)) / 5![/tex]

[tex]P(X = 5) = (0.0183 \times 1024) / 120[/tex]

P(X = 5) ≈ 0.1755

So, the probability that the emergency center will receive five calls on a given day is approximately 0.1755 or 17.55%.

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Answer:

565.486677646 inches squared

Step-by-step explanation:

Let's recall the formula for the surface area of a cylinder:

[tex]A=2\pi rh+2\pi r^2[/tex]

Where r is the radius and h is the height.

We are given that the radius is 6 inches and the height is 9 inches.

Substitute the values and solve the equation, like so:

[tex]A=2\pi (6)(9)+2\pi (6)^2=\\A=2\pi (54) +2\pi(36)=\\A=108\pi +72\pi =\\A=180\pi[/tex]

Thus, in terms of pi, the surface area is equal to [tex]180\pi[/tex].

180 times pi is equal to approximately 565.486677646 inches squared.

Note that the function that is graphed below is y = x + 7.

The equation y = x + 7 represents a linear function, where x is the independent variable and y is the dependent variable.

It has a slope of 1, which means that for every unit increase in x, y increases by 1. The y-intercept is 7, which is the value of y when x is zero. This equation can be graphed as a straight line on a coordinate plane, with a positive slope and y-intercept of (0, 7).

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The best answer is (c) understanding others' emotions.

Women tend to have a more empathetic communication style compared to men, and they tend to prioritize relationships and emotions in their interactions. This communication style involves actively listening to others, being supportive, and expressing empathy towards their feelings. Women often seek to connect with others on an emotional level and understand their perspective, rather than simply giving orders or being self-centered.

Research has shown that women's communication style can be more effective in building and maintaining relationships, as it fosters trust, mutual respect, and understanding. This style also creates a supportive environment where individuals can express their emotions freely and receive validation for their feelings.

In conclusion, women's communication style involves understanding others' emotions, listening actively, and showing empathy towards their feelings.

This approach promotes positive relationships and creates an environment of mutual respect and trust.

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The residual for the given student is approximately 2.77.

This indicates that the student performed better on the English test than what would have been expected based on their IQ score according to the line of best fit.

What is Linear regression?

Linear regression is a statistical technique used to model the relationship between two variables by fitting a linear equation to observed data points.

To find the residual for the given student, we can substitute the student's IQ score (s = 96) into the equation for the best-fit line, y = -26.7 + 0.9346s.

Plugging in s = 96 into the equation, we get:

y = -26.7 + 0.9346(96)

y = -26.7 + 89.6304

y ≈ 62.93 (rounded to two decimal places)

So, the predicted English test score for a student with an IQ score of 96 is approximately 62.93.

To calculate the residual, we subtract the actual English test score (65.7) from the predicted English test score (62.93):

Residual = Actual English test score - Predicted English test score

Residual = 65.7 - 62.93

Residual ≈ 2.77 (rounded to two decimal places)

The residual for the given student is approximately 2.77.

Interpretation:

The positive residual of 2.77 suggests that the actual English test score for the student (65.7) is higher than the predicted English test score based on their IQ score (62.93).

Hence, This indicates that the student performed better on the English test than what would have been expected based on their IQ score according to the line of best fit.

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The area of the intersection of the sphere and the plane in sq cm is 286.46 square centimeters.

The sphere is divided into two equal halves by the plane. The closest point on the plane is 3 cm away from the sphere's center. As a result, a right triangle is formed, with one leg equal to the sphere's radius and the other to the sphere's radius from the plane.

We may calculate the radius of the sphere using the Pythagorean theorem:

r^2 = (20/2)^2 - 3^2

r = sqrt(91) 9.54 cm and r2 = 91

A circle with a radius of 9.54 cm forms the point where the sphere and the plane cross. Its area is therefore: A = r2 A = (9.54)2 A 286.46 sq cm.

The sphere and plane's intersection therefore has a surface area of about 286.46 square centimeters.

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The average home size in the United States in 2004 was 7,000 square feet.

To solve this problem, we can use the concept of ratios. Let x be the average home size in 2004.

The concept of ratios involves comparing two or more quantities by expressing them as a fraction or a division of one quantity by another. It is used to determine how much larger or smaller one quantity is than another, and is often used in mathematics, science, finance, and other fields

We know that the average home size in 1970 was 1,400 square feet, and that the average size in 2004 was about 5 times greater. This can be expressed as:

x = 5(1,400)

Simplifying the right side of the equation, we get

x = 7,000 square feet

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Answer:

Let x be the measure of the angle we are trying to find, in degrees.

The complementary angle to x is 90° - x, since the sum of complementary angles is 90 degrees.

The problem tells us that x is 3.4 degrees less than its complementary angle, so we can set up the following equation:

x = (90 - x) - 3.4

Simplifying and solving for x, we get:

x = 86.6/2

x = 43.3

Therefore, the angle we are trying to find has a measure of 43.3 degrees, and its complementary angle has a measure of 90 - 43.3 = 46.7 degrees.

The drive-thru with typically more wait time is Burger Quick, because it has a larger median. The Option A.

The median is a measure of central tendency that represents the middle value of a set of data. In this case, the median wait time at Burger Quick is 15.5 minutes, while the median wait time at Super Fast Food is 12 minutes.

This indicates that, on average, customers at Burger Quick experience a longer wait time compared to customers at Super Fast Food. The larger median at Burger Quick suggests that there may be some longer wait times skewing the data towards the higher end which could be due to various factors such as slower service, or other operational issues at Burger Quick resulting in a longer wait time for customers at their drive-thru.

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If observer decides to count traffic in 30-second time intervals, then the probability of observer counting exactly 1 vehicles in an interval is 0.344.

The Vehicle headway follows an exponential distribution and the vehicle arrival follows a poison distribution.

We know that that, probability of getting headway greater than or  equal to 9 seconds is 65%,

So, P(h≥9) = 0.65         ...equation(1)

We know that, P(h≥t) = [tex]e^{-\lambda \times t}[/tex],

where, h is headway , t is time and λ is vehicle "arrival-rate",

So, P(h≥9) = [tex]e^{-\lambda \times 9}[/tex],      ....equation(2)

Equating both equation(1) and equation(2),

We get,

⇒ [tex]e^{-\lambda \times 9}[/tex] = 0.65,

⇒ -9λ = ln(0.65),

⇒ λ = 0.047 vehicles per second.

Since, the vehicle arrival follows poison distribution,

The probability of having "n" vehicles arrive in "t" time is P(n) ,

⇒ P(n) = {(λt)ⁿ[tex]e^{-\lambda \times t}[/tex]}/n!,

where, λ = average vehicle arrival rate = 0.047,

"t" = duration of time interval is = 30,

To find the probability of getting exactly 1 vehicle, we put n = 1,

So,

⇒ P(1) = {(0.047×30)¹ [tex]e^{-0.047 \times 30}[/tex]}/1!

⇒ P(1) = 0.344.

Therefore, the required probability is 0.344.

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The given question is incomplete, the complete question is

An observer has determined that the time headways between successive vehicles on a section of highway are exponentially distributed and that 65% of the headways between vehicles are 9 seconds or greater. If the observer decides to count traffic in 30-second time intervals,

Estimate the probability of the observer counting exactly 1 vehicles in an interval.

The plant is not in the sprinkler's range since the distance between the sprinkler and the plant is greater than the radius of the sprinkler's range.

The distance between the sprinkler and the plant can be calculated using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the distance between the sprinkler and the plant is the hypotenuse of a right triangle with legs of length 4 feet and 6 feet:

distance = sqrt(4^2 + 6^2)

distance = sqrt(16 + 36)

distance = sqrt(52)

distance ≈ 7.21 feet

Since the distance between the sprinkler and the plant is greater than the radius of the sprinkler's range (8 feet), the plant is not in the sprinkler's range. Therefore, the plant will not receive water from the sprinkler.

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The perimeter of the rectangular mural after increasing the dimensions by Hailey is equal to 7826 centimeters.

Measures of rectangular mural are,

length = 890 centimeters

Width = 2891 centimeters

New mural is 66 centimeters longer.

The new dimensions of the  rectangular mural will be,

New  length = 890 cm + 66 cm

                     = 956 cm

New width = 2891 cm + 66 cm

                  = 2957 cm

The perimeter of the new mural can be calculated by adding up the lengths of all four sides,

Perimeter = 2(length + width)

⇒Perimeter = 2(956 cm + 2957 cm)

⇒Perimeter = 2(3913 cm)

⇒ Perimeter = 7826 cm

Therefore, the perimeter of Hailey's new mural will be 7826 centimeters.

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The above question is incomplete , the complete question is:

A rectangular mural measures 890 centimeters by 2891centimeters. Hailey creates a new mural that is 66 centimeters longer. What's is the perimeter of Hailey new mural?

The height of the parabolic balloon arch for the prom is 12.25 feet.

Using these assumptions, we can find the equation of the parabola that the arch follows as x = a(y-k)² + h, where (h,k) is the vertex and a is a constant that determines the shape of the parabola. We can find the value of a by using one of the points that the arch passes through, say (3,0):

3 = a(0-k)² + h h = 3 - a(k²)

Similarly, using the other point that the arch passes through, say (7,0):

7 = a(0-k)² + h h = 7 - a(k²)

Equating the expressions for h, we get:

3 - a(k²) = 7 - a(k²) a = -1/4

Substituting this value of a into one of the equations for h, say h = 7 - a(k²), we get:

h = 7 + 1/4(k²)

So the vertex of the parabola is at (h,k) = (7,0), and the equation of the parabola is x = -1/4(y² - 28y + 49).

To find the height of the arch, we need to find the y-coordinate of the vertex, which is k = 0. So the height of the arch is given by the distance between the vertex and the lowest point of the arch, which is the x-intercept of the parabola. To find the x-intercept, we set y = 0 in the equation of the parabola:

x = -1/4(0² - 28(0) + 49)

x = -1/4(49) = -12.25

However, since we are dealing with a physical object, the height cannot be negative. Therefore, we take the absolute value of the x-intercept, which gives us:

| -12.25 | = 12.25 feet

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The equation for the sinusoidal function that represents Jane's motion would be d(t) = 25 x sin(π/4 x (t + 4)) - 5

Let d(t) be the distance Jane is from the bank (in feet) at time t (in seconds). Since Jane's motion is sinusoidal, we can express it as:

d(t) = A x sin(B x (t - C)) + D

We need to find the values of A, B, C, and D that satisfy these conditions.

Since the amplitude is half the peak-to-peak amplitude, we have:

A = 50 / 2 = 25

The vertical shift (D) is the average of the maximum and minimum distances:

D = (20 + (-30)) / 2 = -10 / 2 = -5

Since the sine function is -1 at 3π/2 (270 degrees), we have:

π/4 x (2 - C) = 3π/2

2 - C = 6

C = -4

So, the equation of the sinusoidal function representing Jane's motion is:

d(t) = 25 x sin(π/4 x (t + 4)) - 5

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The question is:

Find the sinusoidal function that represents Jane's motion.

The two options that represent circles with diameter 12 units and center on the y-axis are:

x² + (y - 6)² = 36

x² + (y + 6)² = 36

The diameter of a circle is a straight line segment that passes through the center of the circle and connects two points on its circumference. It is twice the length of the circle's radius.

The equations that represent circles that have a diameter of 12 units and a center that lies on the y-axis are:

x² + (y - 6)² = 36

x² + (y + 6)² = 36

Explanation:

For a circle with diameter 12 units, the radius is half of the diameter, which is 6 units.

Since the center of the circle lies on the y-axis, the x-coordinate of the center is 0.

The general equation for a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².

Using the given information, we substitute h = 0, k = ±6, and r = 6 to get the two equations:

(x - 0)² + (y - 6)² = 6² => x² + (y - 6)² = 36

(x - 0)² + (y + 6)² = 6² => x² + (y + 6)² = 36

Therefore, the two options that represent circles with diameter 12 units and center on the y-axis are:

x² + (y - 6)² = 36

x² + (y + 6)² = 36

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