A sample of college students participated in a sleep study. The amount of REM sleep was recorded before and after an experimental drug was administered. The two samples of REM sleep time are compared to determine how much of a difference the new drug made.

\(check \: photo \: for \: detailed \: analysis\)

This question is based on the point of intersection.Therefore, the points of intersection of the graphs of the equations at 0 ≤ θ < 2π are:

\((1-\dfrac{\sqrt{2} }{2} ,\dfrac{3\pi }{4} ) and (1+\dfrac{\sqrt{2} }{2} ,\dfrac{7\pi }{4} )\)

Given:

Equations: r = 1 + cos θ                                                                      ...(1)

                  r = 1 − sin  θ                                                                      ...(2)                                                                

Where, r ≥ 0, 0 ≤ θ < 2π

We need to determined the point of intersection of the graphs of the equations.

To obtain the points of intersection, Equate the two equations above as follows;

r = 1 + cos θ = 1 - sin θ

=> 1 + cos θ = 1 - sin θ

Solve further for θ. We get,

1 + cos θ = 1 - sinθ

cos θ  = - sinθ

Now dividing both sides by - cos θ and solve it further,

\(\dfrac{cos\;\theta}{-cos\;\theta} =\dfrac{-sin\;\theta}{-cos\;\theta}\\\\tan\;\theta=-1\\\\\theta=tan^{-1}(1)\\\\\theta=45^{0}=\dfrac{-\pi }{4}\)

To get the 2nd quadrant value of θ, add π ( = 180°) to the value of θ. i.e

\(\dfrac{-\pi }{4} +\pi =\dfrac{3\pi }{4} \\\)

Similarly, to get the fourth quadrant value of θ, add 2π ( = 360° ) to the value of θ. i.e

\(\dfrac{-\pi }{4} +2\pi =\dfrac{7\pi }{4} \\\)

Therefore, the values of θ are 3π / 4 and 7π / 4.

Now substitute these values into equations (i) and (ii) as follows;

\(When \;\theta=\dfrac{3\pi }{4},\)

\(r = 1 + cos\;\theta = 1 + cos \dfrac{3\pi }{4} =1+\dfrac{-\sqrt{2} }{2} =1-\dfrac{\sqrt{2} }{2}\)

\(r = 1 + sin\;\theta = 1 - sin \dfrac{3\pi }{4} =1-\dfrac{\sqrt{2} }{2} =1-\dfrac{\sqrt{2} }{2}\)

\(When\; \theta=\dfrac{7\pi }{4}\)

\(r = 1 + cos\;\theta = 1 + cos \dfrac{7\pi }{4} =1+\dfrac{\sqrt{2} }{2} =1+\dfrac{\sqrt{2} }{2}\)

\(r = 1 + sin\;\theta = 1 - sin \dfrac{7\pi }{4} =1-\dfrac{-\sqrt{2} }{2} =1+\dfrac{\sqrt{2} }{2}\)

Represent the results  above in polar coordinates of the form (r, θ). i.e

\((1-\dfrac{\sqrt{2} }{2} ,\dfrac{3\pi }{4} ) and (1+\dfrac{\sqrt{2} }{2} ,\dfrac{7\pi }{4} )\)

Therefore, at the pole where r = 0, is also one of the points of intersection.  

Therefore, the points of intersection of the graphs of the equations at 0 ≤ θ < 2π are:

\((1-\dfrac{\sqrt{2} }{2} ,\dfrac{3\pi }{4} ) and (1+\dfrac{\sqrt{2} }{2} ,\dfrac{7\pi }{4} )\)

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Answer:

(a) therefore the equation is

a(t) = -1000t + 32800

Step-by-step explanation:

(a) a(t) = mt + b

(8, 24800). ( 20, 12800)

(t, a). ( t¹ , a¹)

slope or gradient, m = a¹ - t¹

a - t

m = 12800 - 24800

20 - 8

m = -12000

12

m = -1000

The slope is -1000

let's use (8, 24800) to find b.

24800 = -1000 (8) + b

24800 = -8000 + b

b= 24800 + 8000

b = 32800

therefore the equation is

a(t) = -1000t + 32800

(b) Since the slope is negative (-1000), it means that the altitude is decreasing at a constant rate of 1000 feet per minute. The negative sign indicates the descent, as the altitude is decreasing over time.

Therefore, the slope tells us that the plane is descending at a constant rate of 1000 feet per minute.

(c) The value 32800 tells us the initial altitude of the plane before descending. It indicates the starting point or the initial position of the aircraft above the ground level.

Therefore, the value 32800 represents the initial altitude of the plane before it began descending.

A square and a traingle are present in a large rectangle with given dimensions in the figure.

\(\qquad\displaystyle \tt \dashrightarrow \: 12 \times 8\)

\(\qquad\displaystyle \tt \dashrightarrow \: 96 \: \: unit {}^{2} \)

\(\qquad\displaystyle \tt \dashrightarrow \: 4 \times 4\)

\(\qquad\displaystyle \tt \dashrightarrow \: 16 \: \: unit {}^{2} \)

\(\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{2} \times 4 \times 5\)

\(\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{2} \times 4 \times 5\)

\(\qquad\displaystyle \tt \dashrightarrow \: 2 \times 5\)

\(\qquad\displaystyle \tt \dashrightarrow \: 10 \: \: unit {}^{2} \)

[ area of square / total area ]

\(\qquad\displaystyle \tt \dashrightarrow \: p(inside \: \: the \: \: square) = \frac{16}{96} \)

\(\qquad\displaystyle \tt \dashrightarrow \: p(inside \: \: the \: \: square) = \frac{1}{6} \)

[ total area except area of triangle / total area ]

\(\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{96 - 10}{96} \)

\(\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{86 }{96} \)

\(\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{43}{48} \)

Answer:

A

Step-by-step explanation:

I think it's A hope that helps

Answer:

no

Step-by-step explanation:

Answer:

a) The probability that the mean printing speed of the sample is greater than 17.55 ppm = 0.4247

b) The probability that more than 48.6% of the sampled printers operate at the advertised speed = 0.4197

Step-by-step explanation:

The central limit theorem explains that for an independent random sample, the mean of the sampling distribution is approximately equal to the population mean and the standard deviation of the distribution of sample is given as

σₓ = (σ/√n)

where σ = population standard deviation

n = sample size

So,

Mean of the distribution of samples = population mean

μₓ = μ = 17.35 ppm

σₓ = (σ/√n) = (3.25/√10) = 1.028 ppm

a) The probability that the mean printing speed of the sample is greater than 17.55 ppm.

P(x > 17 55)

We first normalize 17.55 ppm

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (x - μ)/σ = (17.55 - 17.35)/1.028 = 0.19

To determine the required probability

P(x > 17.55) = P(z > 0.19)

We'll use data from the normal probability table for these probabilities

P(x > 17.55) = P(z > 0.19) = 1 - P(z ≤ 0.19)

= 1 - 0.57535 = 0.42465 = 0.4247

b) The probability that more than 48.6% of the sampled printers operate at the advertised speed

We first find the probability that one randomly selected printer operates at the advertised speed.

Mean = 17.35 ppm

Standard deviation = 3.25 ppm

Advertised speed = 18 ppm

Required probability = P(x ≥ 18)

We standardize 18 ppm

z = (x - μ)/σ = (18 - 17.35)/3.25 = 0.20

To determine the required probability

P(x ≥ 18) = P(z ≥ 0.20)

We'll use data from the normal probability table for these probabilities

P(x ≥ 18) = P(z ≥ 0.20) = 1 - P(z < 0.20)

= 1 - 0.57926 = 0.42074

48.6% of the sample = 48.6% × 10 = 4.86

Greater than 4.86 printers out of 10 includes 5 upwards.

Probability that one printer operates at advertised speed = 0.42074

Probability that one printer does not operate at advertised speed = 1 - 0.42074 = 0.57926

probability that more than 48.6% of the sampled printers operate at the advertised speed will be obtained using binomial distribution formula since a binomial experiment is one in which the probability of success doesn't change with every run or number of trials. It usually consists of a number of runs/trials with only two possible outcomes, a success or a failure. The outcome of each trial/run of a binomial experiment is independent of one another.

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = 10

x = Number of successes required = greater than 4.86, that is, 5, 6, 7, 8, 9 and 10

p = probability of success = 0.42074

q = probability of failure = 0.57926

P(X > 4.86) = P(X ≥ 5) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) = 0.4196798909 = 0.4197

Hope this Helps!!!

Answer:

3/2

Step-by-step explanation:

K = Y/X

Least to greatest: 20,564 22,755 2,3805

The compound inequality that represents the range of the actual weight of the object is 125.4 ≤ w ≤ 126.2.

Part A: The absolute value inequality that describes the range of the actual weight of the object is:

|w - 125.8| ≤ 0.4

Part B: To solve the absolute value inequality, we can break it down into two separate inequalities:

w - 125.8 ≤ 0.4 and - (w - 125.8) ≤ 0.4

Solving the first inequality:

w - 125.8 ≤ 0.4

Add 125.8 to both sides:

w ≤ 126.2

Solving the second inequality:

-(w - 125.8) ≤ 0.4

Multiply by -1 and distribute the negative sign:

-w + 125.8 ≤ 0.4

Subtract 125.8 from both sides:

-w ≤ -125.4

Divide by -1 (note that the inequality direction flips):

w ≥ 125.4

Combining the solutions, we have:

125.4 ≤ w ≤ 126.2

The object is 125.4 ≤ w ≤ 126.2.

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The second pair of points representing the solution set of the system of equations is (-6, 29).

To find the second pair of points representing the solution set of the system of equations, we need to substitute the x-coordinate of the second point into one of the equations and solve for y.

Given the system of equations:

y = x^2 - 2x - 19

y + 4x = 5

Substituting the x-coordinate of the second point (-6) into equation 2:

y + 4(-6) = 5

y - 24 = 5

y = 5 + 24

y = 29

Therefore, the second pair of points representing the solution set of the system of equations is (-6, 29).

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Question

Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.

y = x2 − 2x − 19

y + 4x = 5

The pair of points representing the solution set of this system of equations is (-6, 29) and

_________.

3. The volume of the new gumball machine is of 861.75 in³.

4. This is a real solution, as it is a positive number.

The volume of the old machine was of:

5575 in.³

(as stated in the problem)

The volume of the new machine is obtained as follows:

532 in.³ less than 1/4 of the volume of the globe of the giant gumball machine.

One fourth of the volume of the old machine is obtained as follows:

1/4 x 5575 in³ = 1393.75 in³.

532 in.³ less than that is obtained as follows:

1393.75 - 532 = 861.75 in³.

An extraneous volume is a negative value to the volume, as the volume cannot be negative. Since the number is positive, it represents a real solution.

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Answer:

w = - 8

Step-by-step explanation:

-Simplify both sides of the equation

-Distribute

-Combine like terms

-Add 22 to both sides

-Divide both sides by -3

The correct definition for sec(0) is 1.

We have,

The secant function is defined as the reciprocal of the cosine function so for any angle θ.

sec(θ) = 1 / cos(θ)

In this case,

We are looking for the value of sec(0).

The cosine of 0 radians is equal to 1, since 0 radians is the angle at which the x-axis intersects the unit circle, and at that point, the x-coordinate (which represents the cosine) is equal to 1.

So,

cos(0) = 1

Now,

Using the definition of secant, we can find the value of sec(0) by taking the reciprocal of cos(0):

sec(0) = 1 / cos(0)

And,

cos(0) = 1

sec(0) = 1 / 1

Simplifying the expression.

sec(0) = 1

Therefore,

The correct definition for sec(0) is 1.

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Answer: x = 138

Step-by-step explanation:

\(108+117+x=3 * 121\\3 * 121 = 363\\Rearrange unknown terms to the left side of the equation\\x=(363-108)-117\\\\255-117=138\\x=138\)

Answer:

V = 2143.57 cm^3

Step-by-step explanation:

The volume of a sphere is

V = 4/3 pi r^3

The diameter is 16 so the radius is 1/2 of the diameter or 8

V = 4/3 ( 3.14) (8)^3

V =2143.57333

Rounding to the nearest hundredth

V = 2143.57 cm^3

Answer:

2143.57 cm^3

Step-by-step explanation:

V = 4/3 * 3.14 * r^3

r = 1/2 * 16 = 8

So V = 4/3 * 3.14 * 8^3

= 2143.57 cm^3.

Step-by-step explanation:

y <= x/2 + 2

the maximum is to have 2 boondins and add 1/2 boondin every day.

but every time Mimstoon did not use the max. possible, the resulting total sum of boodins stays smaller that the maximum, and is therefore a valid data point.

(6, 7) is not in the inequality relationship.

because the max. boondins after 6 days is 2 (from the beginning) and 1/2 every day = 2 + 1/2 × 6 = 2 + 3 = 5.

but the data point shows 7 as y value (which is larger than the allowed max. of 5).

therefore the inequality is false for this data point, and therefore the data point is not in the inequality relationship.

Answer:

D

Step-by-step explanation:

D option is correct xy=-10

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Pretty with this. this is worth 20

The linear function y = 3 · w - 1 represents the number of sea shells found in each week.

The speed of the driven gear is 180 rounds per minute.

In the first problem we have an example of linear progression, in which the number of sea shells is increased linearly every week. After a quick analysis, we conclude that the linear function y = 3 · w - 1, a kind of direct relationship.

In the second problem, we must an inverse relationship to determine the speed of the driven gear. Please notice that the speed of the gear is inversely proportional to the number of teeths. Then, we proceed to calculate the speed:

\(\frac{v_{1}}{v_{2}} = \frac{N_{2}}{N_{1}}\)

If we know that \(v_{2} = 60\,rpm\), \(N_{2} = 60\) and \(N_{1} = 20\), then the speed of the driven gear is:

\(v_{1} = v_{2}\times \frac{N_{2}}{N_{1}}\)

\(v_{1} = 60\,rpm \times \frac{60}{20}\)

\(v_{1} = 180\,rpm\)

The speed of the driven gear is 180 rounds per minute.

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On solving the provided question, we can say that Therefore, the function has an average rate of change of -1.

The subject of mathematics includes quantities and their variations, equations and related structures, shapes and their locations, and places where they can be found. The term "function" refers to the relationship between a set of inputs, each of which has an associated output. A connection between inputs and outputs in which each input leads to a single, distinct result is known as a function. Each function is given a domain and a codomain, or scope. Usually, f is used to denote functions (x). input is an x. There are four main types of functions accessible. based on the following factors: on functions, one-to-one functions, many-to-one functions, inside functions, and on functions.

The function has an average rate of change of -1.

R = 4-5/1-0

R = -1

Therefore, the function has an average rate of change of -1.

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Answer:

Discount = $24

Step-by-step explanation:

find the discount by multiplying the percent off and the cost.
40% × 60 = 0.4 × 60 = 24

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Answer:

1/4

Step-by-step explanation:

The probability of tossing tails on a fair coin is 1/2. The probability of rolling a multiple of 2 on a fair six-sided number cube is 1/2 (since there are three multiples of 2: 2, 4 and 6). Since these two events are independent, the probability of both events happening is the product of their individual probabilities: (1/2) * (1/2) = 1/4.

The average response time for the ambulance company is 13 minutes, with 95% of response times falling between 10 and 16 minutes.

The response times for a specific ambulance company follow a normal distribution with a mean of 13 minutes. This means that the majority of response times will cluster around the average of 13 minutes.

Furthermore, we know that 95% of the response times fall within the range of 10 to 16 minutes. This suggests that the response times are relatively consistent, with only a small percentage of outliers falling outside this range.

In summary, the average response time for this ambulance company is 13 minutes, and 95% of their response times fall between 10 and 16 minutes.

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Answer: $16137.07

Step-by-step explanation:

A = p(1+r/n)^nt

A= 4600(1+0.16/4)^4x8

A= 16137.07

Answer:

3 liquid gallons

Step-by-step explanation:

3 + 4 + 5 = 12

12 ÷ 4 = 3 liquid gallons

By analyzing the graph and evaluating the objective function at each vertex of the feasible region, we can find the optimal solution, which is the vertex that maximizes the objective function z = 3x + y.

To find the optimal solution of the given problem using the graphical method, we need to plot the feasible region determined by the given constraints and then identify the point within that region that maximizes the objective function.

Let's start by graphing the constraints:

1. Plot the line 2x - y = 5. To do this, find two points on the line by setting x = 0 and solving for y, and setting y = 0 and solving for x. Connect the two points to draw the line.

2. Plot the line -x + 3y = 6 using a similar process.

3. The x-axis and y-axis represent the constraints x ≥ 0 and y ≥ 0, respectively.

Next, identify the feasible region, which is the region where all the constraints are satisfied. This region will be the intersection of the shaded regions determined by each constraint.

Finally, we need to identify the point within the feasible region that maximizes the objective function z = 3x + y. The optimal solution will be the vertex of the feasible region that gives the highest value for the objective function. This can be determined by evaluating the objective function at each vertex and comparing the values.

Note: Without a specific graph or additional information, it is not possible to provide the precise coordinates of the optimal solution in this case.

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