CD Express offers 4 CDs for $60. Music
Place offers 6 CDs for $75. Which store
offers the better buy?

Step-by-step explanation:

[tex]F=ma\\\\F=(300kg)(4m/s^2)\\F=1200N[/tex]

Answer:

[tex](a)v(t)=-\frac{\pi }{3}sin(\frac{\pi t}{3})[/tex]

(b)-0.91 ft/s

Step-by-step explanation:

Given the position function  s = f(t) where f(t) = cos(πt/3), 0 ≤ t ≤ 6

(a)The velocity at time t in ft/s is the derivative of the position vector.

[tex]If\: f(t)=cos(\frac{\pi t}{3})\\f'(t)=-\frac{\pi }{3}sin(\frac{\pi t}{3})\\v(t)=-\frac{\pi }{3}sin(\frac{\pi t}{3})[/tex]

(b)Velocity after 2 seconds

When t=2

[tex]v(2)=-\frac{\pi }{3}sin(\frac{\pi *2}{3})\\=-0.91 ft/s[/tex]

The particle moves 0.91 ft/s in the opposite direction.

The velocity v(t) of a particle moving with function s=f(t)=cos(πt/3) is given by v(t) = -(π/3)sin(πt/3). When t=2 seconds, the velocity of the particle is approximately -1.81 ft/s.

To find the velocity, v(t), at time t for the particle you need to find the derivative of s = f(t) = cos(πt/3) with respect to time, t. Using the chain rule, the derivative will be v(t) = -sin(πt/3) * (π/3), which simplifies to v(t) = -(π/3)sin(πt/3). This formula will provide the velocity of the particle at any time, t, within the given range.

To find the velocity of the particle after 2 seconds, substitute t = 2 into the velocity function. So, v(2) = -(π/3)sin(π*2/3). This simplifies to approximately -1.81 ft/s, when rounded to two decimal places. Therefore, the velocity of the particle at 2 seconds is -1.81 ft/s.

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Answer:

[-3, ∞)

Step-by-step explanation:

There are many ways to find the range but I will use the method I find the easiest.

First, find the derivative of the function.

f(x) = x² - 10x + 22

f'(x) = 2x - 10

Once you find the derivative, set the derivative equal to 0.

2x - 10 = 0

Solve for x.

2x = 10

x = 5

Great, you have the x value but we need the y value. To find it, plug the x value of 5 back into the original equation.

f(x) = x² - 10x + 22

f(5) = 5² - 10(5) + 22

      = 25 - 50 +22

      = -3

Since the function is that of a parabola, the value of x is the vertex and the y values continue going up to ∞.

This means the range is : [-3, ∞)

Another easy way is just graphing the function and then looking at the range. (I attached a graph of the function below).

Hope this helped!

Answer:

The correct answer is B

Step-by-step explanation:

Answer:

y = -3x+8

Step-by-step explanation:

We can use the slope intercept form of an equation

y = mx+b  where m is the slope and b is the y intercept

y = -3x+b

Substitute the point into the equation

-1 = -3(3)+b

-1 = -9+b

Add 9 to each side

-1+9 = -9+9+b

8 =b

y = -3x+8

Answer:

1:C

2:B

3:A

4:B

5:D

6:D

7:A

8:C

9:C

10:B

Step-by-step explanation:

Answer:

There is enough evidence, at a significance level of 0.05, that the population mean age of the Bluestone formations is different from 2950 BCE.

Step-by-step explanation:

We have a sample and we want to perform a hypothesis test on the mean.

The null hypothesis is the Corbin's declared age (2950 BCE). The alternative hypothesis states that the age differ from that value.

They can be expressed as:

[tex]\H_0:\mu=2950\\\\H_a:\mu\neq2950[/tex]

The significance level is 0.05.

The sample has a size of n=3, a mean of 2193.3 BCE and a standard deviation of 104.1 years.

As the standard deviation is estimated from the sample, we have to calculate the t-statistic.

[tex]t=\dfrac{\bar x-\mu}{s/\sqrt{n}}=\dfrac{2193.3-2950}{104.1/\sqrt{3}}=\dfrac{-756.7}{60.1}=-12.59[/tex]

The degrees of freedom for this test are:

[tex]df=n-1=3-1=2[/tex]

The critical value for a two side test with level of significance α=0.05 and 2 degrees of freedom is t=±4.271.

As the statistic t=-12.59 lies outside of the acceptance region, the null hypothesis is rejected.

There is enough evidence, at a significance level of 0.05, that the population mean age of the Bluestone formations is different from 2950 BCE.

Answer:

There is no enough evidence to support the claim that the proportion of US teens that have some level of hearing loss differs from 20%.

P-value=0.12

Step-by-step explanation:

We have to perform a test of hypothesis on the proportion.

The claim is that the proportion of US teens that have some level of hearing loss differs from 20%.

Then, the null and alternative hypothesis are:

[tex]H_0: \pi=0.20\\\\H_a:\pi\neq0.20[/tex]

The significance level is assumed to be 0.05.

The sample, of size n=1981, has 369 positive cases. Then, the proportion is:

[tex]p=X/n=369/1981=0.186[/tex]

The standard error of the proportion is:

[tex]\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.2*0.8}{1981}}=\sqrt{ 0.000081 }= 0.009[/tex]

Now, we can calculate the statistic z:

[tex]z=\dfrac{p-\pi+0.5/n}{\sigma_p}=\dfrac{0.186-0.20+0.5/1981}{0.009}=\dfrac{-0.014}{0.009}=-1.556[/tex]

The P-value for this two-tailed test is:

[tex]P-value=2*P(z<-1.556)=0.12[/tex]

The P-value is below the significance level, so the effect is not significant. The null hypothesis failed to be rejected.

There is no enough evidence to support the claim that the proportion of US teens that have some level of hearing loss differs from 20%.

Since -0.700 falls within this range (-1.96 to 1.96), you fail to reject the null hypothesis (H0). This means that there is not enough evidence to conclude that the true proportion of U.S. teens with hearing loss is different from 0.20.

To investigate whether the headline "1 in 5 U.S. teens has hearing loss, study says" is appropriate or misleading, you can conduct a hypothesis test based on the given hypotheses:

Null Hypothesis (H0): π = 0.20 (The true proportion of U.S. teens with hearing loss is 0.20, or 20%.)

Alternative Hypothesis (H1): π ≠ 0.20 (The true proportion of U.S. teens with hearing loss is not equal to 0.20.)

Here, π represents the population proportion of U.S. teenagers with hearing loss.

To test these hypotheses, you can perform a hypothesis test for a population proportion using a significance level (alpha), such as 0.05 (5%). You can use the z-test for proportions to determine whether the observed proportion of hearing loss in the sample significantly differs from the claimed proportion of 0.20.

The test statistic for the z-test for proportions is calculated as:

z= (p−π)/√(π(1−π)/n)

​Where:

p is the sample proportion (369 out of 1981 in this case).

π is the hypothesized population proportion (0.20).

n is the sample size (1981).

Calculate the sample proportion:

p= 369/1981​ ≈0.186

Now, calculate the test statistic

z= (0.186−0.20)/√(0.20(1−0.20)/1981)

​Calculate the standard

SE=√(0.20(1−0.20)/1981)≈0.020

Now, calculate

z≈ (0.186−0.20)/0.020​ ≈−0.700

Now, you can find the critical values for a two-tailed test at a 95% confidence level (alpha = 0.05). You can use a standard normal distribution table or calculator to find the critical z-values. For a two-tailed test with alpha = 0.05, the critical z-values are approximately -1.96 and 1.96.

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Answer:

The GPA for this transcript is:

GPA = 2.73

Step-by-step explanation:

A=4, B=3, C=2, D=1, F=0

Sociology: 3cr: A, Biology: 4cr: C, Music 1cr: B, Math 4cr:B, English 3cr: C

Total number of credit hour = 3 + 4 + 1 + 4 + 3 = 15

Product for each course = the number of credit hours for a course * the number assigned to each grade

Sociology = 3 * 4 = 12

Biology = 4 * 2 = 8

Music = 1 * 3 = 3

Math = 4 * 3 = 12

English = 3 * 2 = 6

Total product = 12 + 8 + 3 + 12 + 6 = 41

GPA = Sum of product / Total credit hour

GPA = 41 / 15 = 2.7333333

GPA = 2.73

Answer:

The results is 2.73.

Step-by-step explanation:

First let's make the calculations for each course;

For Sociology, an A (which is 4) for 3 credits equals to 12.

For Biology, a C (which is 2) for 4 credits equals to 8.

For Music, a B (which is 3) for 1 credit equals to 3.

For Math, a B (which is 3) for 4 credits equals to 12.

For English, a C (which is 2) for 3 credits equals to 6.

If we sum them all up, we find the results to be 41 and the total credits to be 15 for the 5 courses.

Lastly we should divide 41/15 which will be equal to 2.73 which is the GPA.

I hope this answer helps.

Answer:64 pi units

What is the ratio of the arc length to the circumference?Answer: 1/4

What is the measure of central angle XOY?Answer: 90 degrees

Answer:

1. 64 pi units

2. 1/4

3. 90°

Step-by-step explanation: edge

Answer:

its a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

Formula:

[tex]a^{2} + b^{2} = c^{2}[/tex]

Answer:

-1

Step-by-step explanation:

The slope of the line can be found by

m = (y2-y1)/(x2-x1)

    = (-2-1)/(1--2)

     =-3 /(1+2)

      =-3/3

       -1

4x - 2y = 7

x + 2y =  3

5x = 10

x=2 and y=0

Answer:

{x,y} = {2,1/2}

Step-by-step explanation:

Solve by Substitution :

1. Solve equation [2] for the variable  x  

 [2]    x = -2y + 3

2. Plug this in for variable  x  in equation [1]

  [1]    4•(-2y+3) - 2y = 7

  [1]     - 10y = -5

3.Solve equation [1] for the variable  y  

  [1]    10y = 5

  [1]    y = 1/2

By now we know this much :

   x = -2y+3

   y = 1/2

4.Use the  y  value to solve for  x  

   x = -2(1/2)+3 = 2

Solution :

{x,y} = {2,1/2}

Answer:

C,H,I..............

Answer:

m = b(t)

t= the price per box

Answer:

m = ?b

Step-by-step explanation:

m = total

b = number of boxes sold

? = price of chocolate

Answer:

B

Step-by-step explanation:

Option B is correct. A quadratic function can have symmetry about the y-axis.

A quadratic equation is a second-order polynomial equation in a single variable x , ax²+bx+c=0. with a ≠ 0 .

A quadratic function can have symmetry about the axis of symmetry, which is a vertical line that passes through the vertex of the parabola. The axis of symmetry is given by the equation x = -b/(2a)

where a and b are the coefficients of the quadratic function ax² + bx + c.

If a quadratic function has symmetry about the x-axis, then its equation is of the form y = ax² + c, where a and c are constants.

If it has symmetry about the y-axis, then its equation is of the form y = ax², where a is a constant.

Therefore, Option B is correct. A quadratic function can have symmetry about the y-axis.

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Answer:

This is a weird assignment

Step-by-step explanation:

1)    7.5 minute per mile

10 convert hours to min (900 min) and then simply the unit of 6771 miles by 900 min. The objective is to know how far you travel in one minute.

3) Chose any country in South America:

Brazil, Argentina, Peru, Columbia, Venezuela, Bolivia, etc.

Step-by-step explanation:

[tex](7m + 8)(4m + 1) \\ = 7m(4m + 1) + 8(4m + 1) \\ = 28 {m}^{2} + 7m + 32m + 8 \\ \purple { \bold{= 28 {m}^{2} + 39m + 8}}[/tex]

Final answer:

The length of a 180° arc in a circle with a radius of 9 miles is 9π miles, which is half of the circle's total circumference.

Explanation:

The radius of a circle is the distance from the center to any point on the circle, and the arc length is the distance measured along the circumference of the circle that corresponds to a particular angle. In our case, to calculate the length of a 180° arc, we first need the circumference of the circle, which can be found using the formula 2πr (where r is the radius), and then we find the proportion of the circumference that corresponds to a 180° angle, or half a circle.

The circumference of a circle with a radius of 9 miles is given by:
Circumference = 2π × 9 miles = 18π miles.
Since 180° is half of a full 360° rotation, the arc length for 180° will be half of the circumference:
Arc Length for 180° = ½ × 18π miles = 9π miles.

Given:

The base of the triangle = 8.6 yd

The height of the triangle = 10.9 yd

To find the area of the triangle.

Formula

The area of a triangle with b as base and h as height is

[tex]A=\frac{1}{2}bh[/tex]

Now,

Taking, b= 8.6 and h = 10.9 we get,

[tex]A=\frac{1}{2}(8.6)(10.9)[/tex] sq yd

or, [tex]A= 46.87[/tex] sq yd

Hence,

The area of the given triangle is 46.87 sq yd.

Answer:

46.87 yd^2

Step-by-step explanation:

The area of the triangle is given by

A = 1/2 bh

A = 1/2 (8.6)(10.9)

A =46.87 yd^2

Answer:

Type II error

Step-by-step explanation:

Type 1 error occurs when:

We reject a True Null Hypothesis

Type 2 error occurs when:

We fail to reject a wrong Null Hypothesis.

The given hypothesis are:

[tex]H_{o}: \mu=20\\\\ H_{a}:\mu>20[/tex]

Level of significance = α = 0.05

The calculated z test statistic = z = 1.47

In order to make a decision we first need to convert z = 1.47 to its equivalent p-value. From the z-table the p value for z score being greater than 1.47 comes out to be:

p-value = 0.0708

Since, p-value is greater than the level of significance, we fail to reject the Null Hypothesis.

It is given that the true value of μ is 25. If the true value of μ is 25, then the Null hypothesis was false. But from the test we performed, we failed to reject the Null Hypothesis.

Since, we failed to reject a False Null Hypothesis, the conclusion resulted in a Type II error.

Answer:

P (Type I Error) = P (Revokes the license | Restaurant is not in gross violation)

Step-by-step explanation:

A type I error occurs when we reject a true null hypothesis (H₀). It is the probability of rejecting the null hypothesis when the null hypothesis is true.

The type I error is also known as the level of significance. It is denoted by α .

P (Type I Error) = P (Rejecting H₀ | H₀ is true) = α.

In this case, the food inspector uses a hypothesis testing framework to evaluate whether regulations are not being met.

He decides the the restaurant's license to serve food will be revoked if the restaurant is in gross violation.

So the hypothesis is defined as:

H₀: The restaurant is not in gross violation.

Hₐ: The restaurant is in gross violation.

The type I error will be committed by the food inspector if he concludes that the the restaurant is in gross violation and revokes their license, when in fact the restaurant is not in gross violation.

α = P (Revokes the license | Restaurant is not in gross violation)

A Type 1 Error in this context would be when the food safety inspector incorrectly identifies the restaurant as violating sanitation regulations when in reality, it was following all necessary practices. It’s the wrongful rejection of the null hypothesis.

In statistical hypothesis testing, a Type 1 Error occurs when a true null hypothesis is rejected. In the context of the food safety inspector's investigation, the null hypothesis would likely be that the restaurant is abiding by all required sanitation practices. Therefore, a Type 1 Error would be if the food safety inspector concludes that the restaurant is in serious violation of sanitation regulations and revokes its license, but in actuality, the restaurant was not violating any regulations, i.e., the inspector incorrectly identified the restaurant as unclean and unhealthy.

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Answer:

The minimum distance of   √((195-19√33)/8)  occurs at  ((-1+√33)/4; (-1+√33)/4; (17-√33)/4)  and the maximum distance of  √((195+19√33)/8)  occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)

Step-by-step explanation:

Here, the two constraints are

g (x, y, z) = x + y + 2z − 8  

and  

h (x, y, z) = x ² + y² − z.

Any critical  point that we find during the Lagrange multiplier process will satisfy both of these constraints, so we  actually don’t need to find an explicit equation for the ellipse that is their intersection.

Suppose that (x, y, z) is any point that satisfies both of the constraints (and hence is on the ellipse.)

Then the distance from (x, y, z) to the origin is given by

√((x − 0)² + (y − 0)² + (z − 0)² ).

This expression (and its partial derivatives) would be cumbersome to work with, so we will find the the extrema  of the square of the distance. Thus, our objective function is

f(x, y, z) = x ² + y ² + z ²

and

∇f = (2x, 2y, 2z )

λ∇g = (λ, λ, 2λ)

µ∇h = (2µx, 2µy, −µ)

Thus the system we need to solve for (x, y, z) is

                           2x = λ + 2µx                         (1)

                           2y = λ + 2µy                       (2)

                           2z = 2λ − µ                          (3)

                           x + y + 2z = 8                      (4)

                           x ² + y ² − z = 0                     (5)

Subtracting (2) from (1) and factoring gives

                     2 (x − y) = 2µ (x − y)

so µ = 1  whenever x ≠ y. Substituting µ = 1 into (1) gives us λ = 0 and substituting µ = 1 and λ = 0  into (3) gives us  2z = −1  and thus z = − 1 /2 . Subtituting z = − 1 /2  into (4) and (5) gives us

                            x + y − 9 = 0

                         x ² + y ² +  1 /2  = 0

however, x ² + y ² +  1 /2  = 0  has no solution. Thus we must have x = y.

Since we now know x = y, (4) and (5) become

2x + 2z = 8

2x  ² − z = 0

so

z = 4 − x

z = 2x²

Combining these together gives us  2x²  = 4 − x , so

2x²  + x − 4 = 0 which has solutions

x =  (-1+√33)/4

and

x = -(1+√33)/4.

Further substitution yeilds the critical points  

((-1+√33)/4; (-1+√33)/4; (17-√33)/4)   and

(-(1+√33)/4; - (1+√33)/4; (17+√33)/4).

Substituting these into our  objective function gives us

f((-1+√33)/4; (-1+√33)/4; (17-√33)/4) = (195-19√33)/8

f(-(1+√33)/4; - (1+√33)/4; (17+√33)/4) = (195+19√33)/8

Thus minimum distance of   √((195-19√33)/8)  occurs at  ((-1+√33)/4; (-1+√33)/4; (17-√33)/4)  and the maximum distance of  √((195+19√33)/8)  occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)

In this question, we have 2 constraints:

The plane                        g ( x , y , z ) = x + y + 2 z - 8

The paraboloid             h ( x , y , z ) = x² + y² - z

We need to apply Lagrange Multipliers to answer it

The solution are:

The nearest point P = ( -9.06/4   ,   -9.06 /4  ,  6.23 )

The farthest point Q  (  (7.06) /4  ,  (7.06) /4 ,  10.26)

The Objective Function (F) is the distance between the ellipse and the origin, In this case,  we don´t need to know the equation of the ellipse

The Objective Function is

F = √ x² + y² + z²    and as this function has the same critical points that

F  = x² + y² + z²  we will use this one

Then:

     δF/δx  = 2×x               δF/δy  = 2×y                 δF/δz = 2×z

   λ ×δg/δx =  λ                  λ δg/δy = λ                  λ δg/δz = 2× λ    

   μ× δh/δx = 2× μ×x         μ× δh/δy = 2× μ×y        μ× δh/δz= - μ

Therefore we get our five equations.

2×x  =  λ   +  2× μ×x      (1)

2×y  =  λ   +  2× μ×y      (2)

2×z  = 2× λ  -  μ             (3)

x + y + 2 z - 8 = 0          (4)

x² + y² - z = 0                (5)

Subtracting equation (2) from equation (1)

2×x  - 2×y  = 2× μ×x -  2× μ×y

( x - y ) =  μ × ( x - y )       then   μ = 1   and by substitution in eq. (2)

2×y  =  λ   +  2×y        then       λ  = 0

From eq. (3)

2×z  = - 1                                   z = -1/2

By subtitution in eq. (4) and (5)

x  +  y  - 1 - 8 = 0     ⇒   x  +  y  = 9

x² + y² + 1/2  = 0        this equation has no solution.

If we make  x = y

Equation (4) and (5) become

2× x + 2× z = 8

2×x² - z = 0        ⇒         z = 2×x²

2× x + 4×x² = 8     ⇒   2×x² + x - 8 = 0

Solving for x          x₁,₂ = ( -1 ± √ 1 + 64 ) / 4

x₁,₂ = ( -1 ± √65 ) 4

x₁ = (-1 + √65) /4            x₂ = ( -1 - √65) /4          √ 65  = 8.06

x₁ = 1.765            x₂ = - 2.265

And  z = 2×x²      ⇒      z₁ =  6.23          z₂  =

And critical points are:

P ( x₁  y₁  z₁ )      (  (7.06) /4  ,  (7.06) /4 ,  6.23 )

Q ( x₂  y₂ z₂ )      ( -9.06/4   ,   -9.06 /4  ,  10.26 )

And by simple  inspeccion we see That

minimum distance is the point P = ( -9.06/4   ,   -9.06 /4  ,  6.23 )

the point Q  (  (7.06) /4  ,  (7.06) /4 ,  10.26) is the farthest point

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Answer:

B on edge 2020

Step-by-step explanation:

Applying [tex]\[M_f = M_n \cdot M_{n-1} \cdot ... \cdot M_3 \cdot M_2 \cdot M_1\][/tex] final transformation matrix to the pre-image GHJK [tex](\(P\))[/tex], we get the image G'H''J''K''.

To determine the final transformation that maps the pre-image GHJK to the image G'H''J''K'', we need to break down the transformations and apply them in the correct order.

Let's assume there are several transformations involved, such as translations, rotations, reflections, or dilations. Each transformation can be represented by a matrix or a set of rules.

Let's denote the initial pre-image GHJK as [tex]\(P\).[/tex] The series of transformations can be represented as [tex]\(T_1 \cdot T_2 \cdot T_3 \cdot ... \cdot T_n\)[/tex], where [tex]\(T_1\) to \(T_n\)[/tex] are individual transformations.

To find the final transformation, we need to multiply the matrices representing these transformations in the reverse order. If [tex]\(M_1, M_2, M_3, ..., M_n\)[/tex] are the matrices representing [tex]\(T_1, T_2, T_3, ..., T_n\)[/tex]respectively, the final transformation matrix [tex]\(M_f\)[/tex] would be:

[tex]\[M_f = M_n \cdot M_{n-1} \cdot ... \cdot M_3 \cdot M_2 \cdot M_1\][/tex]

Applying this final transformation matrix to the pre-image GHJK [tex](\(P\))[/tex], we get the image G'H''J''K''.

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Answer:

x = 1 or x = 3/2

Step-by-step explanation:

Solve for x:

(4 x^2 - 10 x + 6)/(4 x + 2) = 0

Multiply both sides by 4 x + 2:

4 x^2 - 10 x + 6 = 0

The left hand side factors into a product with three terms:

2 (x - 1) (2 x - 3) = 0

Divide both sides by 2:

(x - 1) (2 x - 3) = 0

Split into two equations:

x - 1 = 0 or 2 x - 3 = 0

Add 1 to both sides:

x = 1 or 2 x - 3 = 0

Add 3 to both sides:

x = 1 or 2 x = 3

Divide both sides by 2:

Answer: x = 1 or x = 3/2

Answer:

7 inches hope this helps

Step-by-step explanation:

Answer:

7 inches

Step-by-step explanation:

12 inches is a foot

Answer:

33.12 rad/s

Step-by-step explanation:

We are given that

Diameter=d=23.3 in

Radius,=[tex]r=\frac{d}{2}=\frac{23.3}{2}=11.65 in=11.65\times 0.0254= 0.29591 m[/tex]

1 in=0.0254 m

We have to find the angular speed of drum would have to spin.

Force=[tex]mg[/tex]

Centripetal force=[tex]m\omega^2 r[/tex]

[tex]mg=m\omega^2 r[/tex]

[tex]\omega^2=\frac{g}{r}[/tex]

[tex]\omega=\sqrt{\frac{g}{r}}[/tex]

Where [tex]g=9.8m/s^2[/tex]

[tex]\omega=\sqrt{\frac{9.8}{0.29591}[/tex]

[tex]\omega=33.12 rad/s[/tex]

Answer:

English plz

Step-by -step explanation:

what dose this say

El condenado sobrevivió al destruir una de las 'papeletas de muerte' sin leerla, y utilizar la ley del Sultán a su favor para afirmar que su 'papeleta de muerte' destruida era la 'papeleta de vida'.

El reo aseguró su supervivencia actuando de manera astuta. Sabiendo que ambas papeletas tenían la palabra muerte, decidió escoger una y, sin leerla, la destruyó por completo. Entonces solicitó que se leyera la papeleta restante, si en la papeleta restante dice muerte, entonces es obvio que la papeleta que destruyó debía tener la sentencia de vida, ya que según el Sultán, el Gran Visir hizo dos papeletas diferentes. De esta manera, aunque el Gran Visir deseaba que obtuviera la sentencia de muerte, el reo se salvó por su astucia sin necesidad de revelar el complot del Visir.

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Answer:

a) 0.011 = 1.1% probability that the sample mean hardness for a random sample of 17 pins is at least 51

b) 0.0001 = 0.1% probability that the sample mean hardness for a random sample of 45 pins is at least 51

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

[tex]\mu = 50, \sigma = 1.8[/tex]

(a) If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 17 pins is at least 51?

Here [tex]n = 17, s = \frac{1.8}{\sqrt{17}} = 0.4366[/tex]

This probability is 1 subtracted by the pvalue of Z when X = 51. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{51 - 50}{0.4366}[/tex]

[tex]Z = 2.29[/tex]

[tex]Z = 2.29[/tex] has a pvalue of 0.9890

1 - 0.989 = 0.011

0.011 = 1.1% probability that the sample mean hardness for a random sample of 17 pins is at least 51

(b) What is the (approximate) probability that the sample mean hardness for a random sample of 45 pins is at least 51?

Here [tex]n = 17, s = \frac{1.8}{\sqrt{45}} = 0.2683[/tex]

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{51 - 50}{0.0.2683}[/tex]

[tex]Z = 3.73[/tex]

[tex]Z = 3.73[/tex] has a pvalue of 0.9999

1 - 0.9999 = 0.0001

0.0001 = 0.1% probability that the sample mean hardness for a random sample of 45 pins is at least 51

Final answer:

To find the probability that the sample mean hardness for a random sample of 17 pins is at least 51, we can convert it to a standard normal distribution and use the z-score formula. The probability is approximately 0.011. For a sample size of 45 pins, the probability is approximately 0.001.

Explanation:

To find the probability that the sample mean hardness for a random sample of 17 pins is at least 51, we can convert it to a standard normal distribution and use the z-score formula. The formula for the z-score is:

z = (x - μ) / (σ / sqrt(n))

where x is the value we are interested in (51), μ is the mean (50), σ is the standard deviation (1.8), and n is the sample size (17).

Plugging in the values, we get:

z = (51 - 50) / (1.8 / sqrt(17))

Calculating this, we find that the z-score is approximately 1.044. Looking up this z-score in the z-table, we find that the probability is 0.853. However, we are interested in the probability that the hardness is at least 51, which means we need to find the area to the right of the z-score. So, we subtract the probability from 1:

Probability = 1 - 0.853 = 0.147, or approximately 0.011 when rounded to four decimal places.

Therefore, the probability that the sample mean hardness for a random sample of 17 pins is at least 51 is approximately 0.011.

To find the probability that the sample mean hardness for a random sample of 45 pins is at least 51, we follow the same process. The only difference is that the sample size is now 45 instead of 17. Plugging in the values into the z-score formula, we find that the z-score is approximately 3.106. Looking up this z-score in the z-table, we find that the probability is 0.999. Subtracting this probability from 1, we get:

Probability = 1 - 0.999 = 0.001.

Therefore, the probability that the sample mean hardness for a random sample of 45 pins is at least 51 is approximately 0.001.

Answer:

The null hypothesis is not rejected.

There is no enough evidence to support the claim that the CO level is lower  in non-smoking working areas compared to smoking work areas.

P-value = 0.07.

Step-by-step explanation:

We have to perform a test on the difference of means.

The claim that we want to test is that CO is less present in no-smoking work areas.

Then, the null and alternative hypothesis are:

[tex]H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2 > 0[/tex]

being μ1: mean CO level in smoking work areas, and μ2: mean CO level in no-smoking work areas.

The significance level is assumed to be 0.05.

Smoking areas sample

Sample size n1=40.

Sample mean M1=11.6

Sample standard deviation s1=7.3

No-smoking areas sample

Sample size n2=40

Sample mean M2=6.9

Sample standard deviation s2=2.7

First, we calculate the difference between means:

[tex]M_d=M_1-M_2=11.6-7.3=4.3[/tex]

Second, we calculate the standard error for the difference between means:

[tex]s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{7.3^2}{40}+\dfrac{2.7^2}{40}}=\sqrt{\dfrac{53.29+7.29}{40}}=\sqrt{\dfrac{60.58}{40}}\\\\\\s_{M_d}=\sqrt{1.5145}=1.23[/tex]

Now, we can calculate the t-statistic:

[tex]t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{4.3-0}{1.23}=3.5[/tex]

The degrees of freedom are calculated with the Welch–Satterthwaite equation:

[tex]df=\dfrac{(\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2})^2}{\dfrac{s_1^4}{n_1(n_1-1)}+\dfrac{s_2^4}{n_2(n_2-1)}} \\\\\\\\df=\dfrac{(\dfrac{7.3^2}{40}+\dfrac{2.7^2}{40})^2}{\dfrac{7.3^4}{40(39)}+\dfrac{2.7^4}{40(39)}} =\dfrac{(\dfrac{53.29}{40}+\dfrac{7.29}{40})^2}{\dfrac{2839.82}{1560}+\dfrac{53.14}{1560}} \\\\\\\\df=\dfrac{1.5145^2}{1.8545}=\dfrac{2.2937}{1.8545}=1.237[/tex]

The P-value for this right tail test, with 1.237 degrees of freedom and t=3.5 is:

[tex]P-value=P(t>3.5)=0.07[/tex]

The P-value is bigger than the significance level, so the effect is not significant. The null hypothesis is not rejected.

There is no enough evidence to support the claim that the CO level is lower  in non-smoking working areas compared to smoking work areas.

The p-value, which indicates the likelihood that the difference in CO levels in the work areas is due to chance, can be computed from the mean CO levels and the standard deviations using a t-test for unequal variance. The computation requires several steps, including calculating the degrees of freedom and the t-statistic.

To conduct the t-test for unequal variance, we need to follow several steps. Below are the necessary steps:

In this scenario, the mean CO levels and standard deviations in work areas where smoking was permitted and not permitted are given. By plugging these into the formulas, we can find the t-value and then use the t-distribution to find the corresponding p-value.

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Answer:

Step-by-step explanation:

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