A manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 430 gram setting. It is believed that the machine is underfilling the bags. A 17 bag sample had a mean of 423 grams with a variance of 676. A level of significance of 0.01 will be used. Assume the population distribution is approximately normal. State the null and alternative hypotheses

Answer:

20m squared

Step-by-step explanation:

The formula for working the area os a triangle is

base x height/2

5x8=40

40/2= 20

hope it helps

Answer:

If a family chosen at random bought a car, we need to find the probability that the family had not previously indicated an intention to buy a car = P(I'|B) = 0.3362

Step-by-step explanation:

Let the event that a family that intends to buy a car be I

Let the event that a family does not intend to buy a car be I'

Let the event that a family buys a car in those 3 months be B

Let the event that a family does not buy a car in those 3 months be B'

Given,

P(B|I) = 0.70

P(B|I') = 0.10

P(I) = 0.22

P(I') = 1 - P(I) = 1 - 0.22 = 0.78

If a family chosen at random bought a car, we need to find the probability that the family had not previously indicated an intention to buy a car = P(I'|B)

The conditional probability P(A|B), is given as

P(A|B) = P(A n B) ÷ P(B)

So,

P(B|I) = P(B n I) ÷ P(I)

P(B n I) = P(B|I) × P(I) = 0.70 × 0.22 = 0.154

P(B|I') = P(B n I') ÷ P(I')

P(B n I') = P(B|I') × P(I') = 0.10 × 0.78 = 0.078

P(B) = P(B n I) + P(B n I') = 0.154 + 0.078 = 0.232

P(B') = 1 - 0.232 = 0.768

P(I'|B) = P(B n I') ÷ P(B)

= 0.078 ÷ 0.232 = 0.3362

Hope this Helps!!!

Using Bayes' theorem, the probability that a randomly chosen family bought a car without previously indicating the intention is 33.62%.

Calculating the Probability of a Randomly Selected Family Buying a Car Without Prior Intent

To find the probability that a family chosen at random bought a car without previously indicating an intention to buy a car, we need to use conditional probability and Bayes' theorem.

Given the survey results, 70% of families who intended to buy a new car did so within 3 months, and 10% of families without prior intent also bought a car.

Let I be the event that a family indicated an intention to buy a car, and N be the event that a family did not indicate an intention.

We're given that P(I) = 0.22 and P(N) = 0.78 (since there are only two options, either they intended or did not, which sums to 1).

Let C be the event that a family bought a car. We want to find P(N|C), which is the probability a family had not previously indicated the intention to buy a car given that they bought a car.

We use Bayes' theorem:

P(N|C) = [P(C|N) × P(N)] / [P(C|I) × P(I) + P(C|N) × P(N)]

Substitute the values we know:

P(N|C) = [(0.10) ×(0.78)] / [(0.70) × (0.22) + (0.10) × (0.78)]

Calculate the probability:

P(N|C) = (0.078) / (0.154 + 0.078)

P(N|C) = 0.078 / 0.232

P(N|C) = 0.3362 or 33.62%

Therefore, there's a 33.62% chance that a family chosen at random bought a car without having indicated an intention

Answer:

19 11/16 or 315/16

Steps:

Turn the fractions into an improper fraction and then multiply straight across

5 1/4 = 21/4

3 3/4 = 15/4

(21/4)*(15/4)= 315/16= 19 11/16

Yes you got it right :)

Answer:

Step-by-step explanation:

Area = Length times Width

  5 1/4  times 3 3/4

 5 x 4 + 1 = 21

21/4

3 x 4 +3 = 15

15/4

21/4 x 15/4 = 315 / 16 or 19 11/16 ft^2

Answer:

a. 113.72

b. 115

c. 107, 120

d. 100

Step-by-step explanation:

Hypothesis is seen as an assumption, an idea that is proposed for the sake of argument so that it can be tested to see if it might be true. In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review.

Sampling distribition can be seen as the distribution of frequencies of a range of different outcomes that could possibly occur for a statistic of a population.

Please go to attachment for the detailed analysis.

Answer: 1.92

Step-by-step explanation:

24.00 x 0.08

1.92

$1.92

Answer: y=3x

Step-by-step explanation:

Answer:

–3x + y = 0

Step-by-step explanation:

line through (0,0) always has zero constant, divide by 8 for simplicity we get –3x + y = 0

Answer:

The correct option is (d).

Step-by-step explanation:

The (1 - α)% confidence interval for the difference between two means with same sample size is:

[tex]CI=(\bar x_{1}-\bar x_{2})\pm CV\times SD\times \sqrt{\frac{2}{n}}[/tex]

The width of the interval is:

[tex]\text{Width}=2\times CV\times SD\times \sqrt{\frac{2}{n}}[/tex]

From the formula of the width of the confidence interval it can be seen that the sample size is inversely related to the width.

That is, if the sample size is increased the width of the interval will be decreased and if the sample size is decreased the width of the interval will be increased.

It is provided that two confidence intervals are constructed for the difference between the means of two populations R and J.

One One confidence interval, will be constructed using samples of size 400 from each of R and J.

And the other confidence interval, will be constructed using samples of size 100 from each of R and J.

Determine the formula of width for both sample sizes as follows:

[tex]\text{Width}_{1}=2\times CV\times SD\times \sqrt{\frac{2}{400}}\\[/tex]

           [tex]=2\times CV\times SD\times \frac{\sqrt{2}}{20}[/tex]

[tex]\text{Width}_{2}=2\times CV\times SD\times \sqrt{\frac{2}{100}}\\[/tex]

           [tex]=2\times CV\times SD\times \frac{\sqrt{2}}{10}[/tex]

So, the width of I₄₀₀ is half times the width of I₁₀₀.

The correct option is (d).

Answer:

D

Step-by-step explanation:

I got 18/18

Answer:

1/4 < 3/8

Step-by-step explanation:

We need to get a common denominator to compare.  We will use a common denominator of 8

1/4 *2/2    or 3/8

2/8 or 3/8

since the denominators are the same

2<3

so 1/4 < 3/8

3/8 is greater than 1/4.

To determine which fraction is greater between 1/4 and 3/8, we can compare them by finding a common denominator and then comparing the numerators.

To find a common denominator, we need to determine the least common multiple (LCM) of the denominators, which in this case are 4 and 8. The LCM of 4 and 8 is 8.

Next, we need to convert both fractions to have a denominator of 8:

1/4 = (1/4) x (2/2) = 2/8

3/8 = 3/8

Now that both fractions have a common denominator of 8, we can compare the numerators:

2/8 vs. 3/8

Since the denominator is the same, we can see that the fraction with the greater numerator, 3/8, is greater than the fraction with the smaller numerator, 2/8.

Therefore, 3/8 is greater than 1/4.

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Answer:

A

Step-by-step explanation:

Answer:

Yes

Step-by-step explanation:

−4.9t² + 18.24t + 0.8 = −1.43t + 4.26

-4.9t² + 19.67t - 3.46 = 0

4.9t² - 19.67t + 3.46 = 0

Using quadratic formula:

t = [19.67 +/- sqrt(319.0929l)]/9.8

t = (19.67 +/- 17.8631716109)/9.8

t = 3.8299154705, 0.1843702438

Since the two can be at the same height at the same time, the blocker can knock down the punt

Yes, the blocker can block down the punt.

To determine whether the blocker can knock down the punt, compare the heights of the football and the blocker’s hands at the same time t. The equations for the height of the football h(t) and the height of the blocker’s hands g(t) are given as follows:

[tex]h(t)=-4.9t^2+18.24t+0.8[/tex]

[tex]g(t)=-1.43t+4.26[/tex]

For the blocker to knock down the punt, the heights must be equal at some time t. Therefore, solve for t when h(t) = g(t):

[tex]-4.9t^2+18.24t+0.8=-1.43t+4.26[/tex]

First, move all terms to one side of the equation to set it to zero:

[tex]-4.9t^2+18.24t+0.8+1.43t-4.26=0[/tex]

Combine like terms:

[tex]-4.9t^2+19.67t-3.46=0[/tex]

This is a quadratic equation of the form [tex]ax^2+bx+c[/tex], where:

a=−4.9, b=19.67, c=−3.46

We can solve this quadratic equation using the quadratic formula:

[tex]t=\frac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]

Substitute the values of a, b, and c:

[tex]t = \frac{-19.67\pm\sqrt{(19.67)^2-4(-4.9)(-3.46)} }{2(-4.9)}=\frac{-19.67 \pm \sqrt{386.909 - 67.816} }{-9.8}=\frac{-19.67\pm\sqrt{319.903} }{-9.8}[/tex]

[tex]t = \frac{-19.67\pm 17.89 }{-9.8}[/tex]

Now solve for both possible values of t:

[tex]t_1 = \frac{-19.67+17.89}{-9.8} = \frac{-1.78}{-9.8} \approx 0.18\ seconds[/tex]

[tex]t_2 = \frac{-19.67-17.89}{-9.8} = \frac{-37.56}{-9.8} \approx 3.83\ seconds[/tex]

Next, check if these times are within the reasonable range for a football to stay in the air. Since 0.186 seconds and 3.83 seconds are within a typical range for a football's hang time, let's proceed to compare the heights at these times.

For t = 0.186 seconds:

[tex]h(0.186)=-4.9(0.18)^2+18.24(0.18)+0.8[/tex]

[tex]h(0.18) = -4.9(0.0324)+3.2832+0.8[/tex]

[tex]h(0.18)= -0.15876+3.2832+0.8 = 3.92444\ meters[/tex]

[tex]g(0.18)=-1.43(0.18)+4.26[/tex]

[tex]g(0.18) = -0.2574+4.26 = 4.0026[/tex]

For t = 3.83 seconds:

[tex]h(3.83)=-4.9(3.83)^2+18.24(3.83)+0.8[/tex]

[tex]h(3.83)=-4.9(14.6689)+69.8592+0.8[/tex]

[tex]h(3.83)=-71.8781+69.8592+0.8 = -2.0189+0.8=-1.2189\ meters[/tex]

[tex]g(3.83)=-1.43(3.83)+4.26[/tex]

[tex]g(3.83)=-5.4769+4.26[/tex]

[tex]g(3.83)=-1.2169\ meters[/tex]

Since at t = 0.186 seconds, the heights h(0.186) and g(0.186) are nearly equal and are positive, the blocker can knock down the punt at t around 0.186 seconds.

Answer:

Radius of outer circle = 6.5 ft

Circumference outer circle = 40.9 ft

Step-by-step explanation:

We can find the radius of the inner circle

C = 2 * pi r

22 = 2 * pi *r

22 /2pi = 2pi r/2pi

11/pi = r

11/(22/7) =r

3.5 =r

Add 3 to get the radius of the outer circle

The radius of the outer circle is 3+3.5 = 6.5 ft

We can find the circumference of the outer circle by

C = 2*pi*r

C = 2 * 22/7 *6.5

C=40.85714286

Rounding to the nearest tenth

C = 40.9 ft

Answer:

18.9 ft

40.9 ft

Step-by-step explanation:

Circumference of the inner circle:

22 = 2 pi × r

22 = 2 × 22/7 × r

r = 7/2 = 3.5

Distance between the circles is the difference between the radii

Outer circle radius: 3.5 + 3 = 6.5

Circumference of the outer circle is:

2 × 22/7 × 6.5

40.85614286

To the nearest tenth: 40.9

Difference between the circumferences:

40.9 - 22

18.9 ft

Answer:

represents the combinations of K and L that cost the firm the same amount of money.

Step-by-step explanation:

Isocost is a graph representing factor inputs ( labour, capital ) ; which costs firm the same level of total production expenditure.

The curve is analogous to consumer's budget line - product combinations costing same to consumers. So, it is likely a straight line downward sloping curve also. Such because : factors are inversely related, given same total cost; and the slope is constant = price ratios of the two factor inputs.

An isocost line represents combinations of capital (K) and labor (L) that cost the same total amount for a firm. Option 2 correctly defines an isocost line. The line's slope is determined by the prices of labor and capital.

An Isocost Line in Economics

An isocost line is a graphical representation in economics that shows all the combinations of inputs that cost a firm the same total amount. When referring to factors of production such as capital (K) and labor (L), the isocost line equation could be expressed as rK + wL = constant, where 'r' represents the cost of capital and 'w' represents the wage or cost of labor. If we are looking at the firm's input choices to minimize cost for a given level of output, the isocost line will have a negative slope that represents the trade-off between the quantities of capital and labor the firm can use subject to its budget constraint.

The correct option to describe an isocost line from the given choices would be 2) represents the combinations of K and L that cost the firm the same amount of money. This means that all input combinations lying on the same isocost line have the same total cost (TC). Also, the slope of the isocost line is determined by the ratio of the prices of the factors, i.e., -w/r.

In economic analysis, firms are often visualized as combining inputs of labor and capital in the most cost-efficient manner to produce a certain level of output, as shown by the isoquant curves. By finding the point where an isocost line is just tangent to an isoquant, a firm achieves the least-cost combination of labor and capital for producing the given quantity of output.

Answer:

a

Step-by-step explanation:

Final answer:

To calculate the interquartile range (IQR), identify the median, lower quartile (Q1), and upper quartile (Q3) of the data set, then find the difference between Q3 and Q1, giving you the IQR of the middle 50% of the data.

Explanation:

To find the interquartile range (IQR) of a data set, you first need to find the median, which divides the data into two halves. Then, determine the median for the lower half (Q1) and the upper half (Q3). Finally, subtract Q1 from Q3 to find the IQR. In this case, for the data set provided, the IQR would be calculated as follows:

Arrange the data in ascending order: 0, 0, 1/4, 1/2, 1/2, 5/4, 1, 1, 1, 2, 2.

Find the median (Q2), which is the middle value: 1/2.

Calculate the medians of the lower and upper halves: Q1 = 1/4 and Q3 = 1.

Subtract Q1 from Q3 to find the IQR: IQR = 1 - 1/4 = 3/4.

Answer:

a=-2

Step-by-step explanation:

Let me know if you need the steps tho.

Answer:

The answer will be 1/12 of

Step-by-step explanation:

Answer:

1/64

Step-by-step explanation:

Each of the questions has 4 choices, making the chance to get the correct answer 1/4. So, to get 3 questions correct, you can use 1/4^3 to find the probability. So, the answer is 1/64.

The probability of exactly 3 questions are  correct is, [tex]\frac{1}{64}[/tex]

In test every question has four choices with one correct answer.

So that, the probability of one question is correct  [tex]=\frac{1}{4}[/tex]

the probability of exactly 3 questions are  correct is,

                  [tex]P(E)=\frac{1}{4}*\frac{1}{4}*\frac{1}{4}\\ \\ P(E)=\frac{1}{64}[/tex]

The probability of exactly 3 questions are  correct is, [tex]\frac{1}{64}[/tex]

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Answer:

1. The vertex of the function is at (1, -25).

5. The graph is negative on the entire interval -4 < x < 6.

Answer:

A)

Step-by-step explanation:

on edge

Answer:

This is a hypothesis test for a proportion.

There is not enough evidence to support the claim that the true proportion of people in this town suffering from depression or a depressive illness is lower than the percent in the general adult American population (P-value=0.248).

Step-by-step explanation:

This is a hypothesis test for a proportion.

The claim is that the true proportion of people in this town suffering from depression or a depressive illness is lower than the percent in the general adult American population.

Then, the null and alternative hypothesis are:

[tex]H_0: \pi=0.095\\\\H_a:\pi<0.095[/tex]

The significance level is 0.05.

The sample has a size n=100.

The sample proportion is p=0.07.

[tex]p=X/n=7/100=0.07[/tex]

The standard error of the proportion is:

[tex]\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.095*0.905}{100}}\\\\\\ \sigma_p=\sqrt{0.001}=0.029[/tex]

Then, we can calculate the z-statistic as:

[tex]z=\dfrac{p-\pi-0.5/n}{\sigma_p}=\dfrac{0.07-0.095+0.5/100}{0.029}=\dfrac{-0.02}{0.029}=-0.682[/tex]

This test is a left-tailed test, so the P-value for this test is calculated as:

[tex]P-value=P(z<-0.682)=0.248[/tex]

As the P-value (0.248) is bigger than the significance level (0.05), the effect is  not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that the true proportion of people in this town suffering from depression or a depressive illness is lower than the percent in the general adult American population.

This is a test of proportion to determine if the true proportion of people in the town suffering from depression or a depressive illness is lower than the general adult American population.

This is a test of proportion since we are comparing the proportion of people in the town suffering from depression or a depressive illness to the percentage in the general adult American population.

The null hypothesis (H0) states that the true proportion of people in the town suffering from depression or a depressive illness is not lower than the percentage in the general adult American population, while the alternative hypothesis (Ha) states that the true proportion is lower than the general population.

To conduct a hypothesis test, we can use a one-sample z-test to compare the observed proportion of people suffering from depression or a depressive illness in the town to the expected proportion based on the general adult American population.

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Answer:

Ella has the greatest return in the current year.

Step-by-step explanation:

Debby would receive $0.80 for each of her 2000 common stock in the oil company,hence Debby's return on investment in the current year is $1600($0.80*2000)

Besides,Ella's return on the stock investment in the current year is computed thus:

Ella's return= 5%*1000*$50=$2,500

In addition,Unique's dollar return on the investment is computed as follows:

Unique's return on  investment=4%*2000*$20=$1,600

From the above computations,Ella seems to have the highest return in the current year of $2,500 whereas the two others managed to have $1600 return each

Answer:

a) [tex]P(X \leq 3) = 0.99328[/tex]

b) 0.6957

Step-by-step explanation:

Let X represent the number of 4's when  n = 5 independent spins

each has a probability of 0.2 (i.e p = 0.2)

This notation is represented as:

X [tex]\approx[/tex] Binomial (n = 5, p = 0.2)

Probability of  [tex]x[/tex]  number of 4's is:

[tex]P(X=x)= (\left \ n \atop x \right) p^x (1-p)^{(n-x)}[/tex]

here; [tex](\left \ n \atop x \right)[/tex] is the combinatorial expression

[tex](\left \ n \atop x \right)[/tex] = [tex]\frac{n!}{x!(n-x)!}[/tex]

[tex]P(X \leq3), n =5 , p = 0.2[/tex]

[tex]P(X \leq3) = 1-P(X > 3)[/tex]

So; let's first find:

[tex]P(X > 3)[/tex]

[tex]= P(3 <X \leq 5) \\ \\ = P(4 <X \leq 5) \\ \\ = P (X = 4, 5) \\ \\ = P (X=4)+P(X = 5 ) \ \ \ (disjoint \ events)[/tex]

[tex]P(X = 4) =( \left \ {{5} \atop {4}} \right. ) (0.2)^4 (1-0.2)^1 \\ \\ P(X = 4) = 5(0.2)^4(0.8)^1 \\ \\ P(X = 4) = 0.0064[/tex]

[tex]P(X = 5) =( \left \ {{5} \atop {5}} \right. ) (0.2)^5 (1-0.2)^0 \\ \\ P(X = 5) = 5(0.2)^5(0.8)^0 \\ \\ P(X = 5) = 0.00032[/tex]

[tex]P (X=4)+P(X = 5 ) \\ \\ = 0.0064 + 0.00032 = 0.006720 \\ \\ \approx 0.007[/tex]

[tex]P(X > 3 ) = 0.00672 \\ \\ P(X \leq 3) = 1- P(X > = 3 ) \\ \\ =1 - 0.00672 \\ \\ = 0.99328[/tex]

[tex]P(X \leq 3) = 0.99328[/tex]

b)

Given that:

The ratio of boys to girls at birth in  Singapore is quite high at 1.09:1

What proportion of Singapore families with exactly 6 children will have at least 3 boys?

Probability of having a boy = [tex]\frac{1.09}{1+1.09}[/tex] = 0.5215

Binomial Problem with n = 6

P(3<= x <=6) = 1 - P(0<= x <=2)

= 1 - binomial (6,0.5215,2)

= 0.6957

Answer:

  7/15

Step-by-step explanation:

We assume you want xy+x. Put the numbers where the variables are and do the arithmetic.

  (1/3)(2/5) +(1/3) = 2/15 + 5/15 = 7/15

Answer:

  Recipe 3

Step-by-step explanation:

Honey to Oats ratios for the four recipes are ...

  1: 5/2 = 2.5

  2: 6/3 = 2.0

  3: 3/1 = 3.0

  4: 8/4 = 2.0

The greatest ratio of Honey to Oats is found in Recipe 3, where it is 3:1.

Recipe 3 is the answer.

just took the test.

La mejor forma para calcular el costo que Andrea pagará por los minutos de teléfono el lunes depende del número total de minutos hablados. Primero, multiplicar los primeros 20 minutos al precio de $30 cada uno. Después, si hubiera minutos adicionales, multiplicarlos al precio de $20 cada uno y sumar ambos montos.

Para calcular cuánto paga Andrea por los minutos de teléfono el lunes, debemos tener en cuenta la tarifa descrita: los primeros 20 minutos se cobran a $30 cada uno, y cualquier minuto adicional es a $20. No tenemos la información sobre cuántos minutos usó en total Andrea, pero podemos evaluar las opciones dadas.

La opción B parece la más adecuada si consideramos que Andrea usó 10 minutos en una línea y 25 minutos en la otra, donde los primeros 20 minutos de ambas líneas suman 30 minutos, multiplicados por $30, y los 5 minutos restantes se multiplicarían por $20. Sin embargo, si esta fue la distribución real de los minutos debería sumarse los primeros 20 minutos de cada línea a $30 cada uno, lo que indica que tal vez la opción C seria la correcta. La opción A y D son incorrectas porque no toman en cuenta la tarifa diferenciada.

Answer:

b.75

Step-by-step explanation:

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:

Mean of the population is 75.

By the Central Limit Theorem,

The mean of the sample, [tex]\bar{x}[/tex], is expected to be also 75.

So the correct answer is:

b.75

This question is based on the concept of statistics.Therefore, the expected value of  mean is 75. Hence, the correct option is (b) 75.

Given:

Mean = 75

Standard deviation = 8

Random sample size = 800

According to the question,

By using the central limit theorem states that,

This theorem states that, the distribution of sample means approximate normal distribution as the sample size gets larger.

Hence, for a skewed variable, the central limit theorem can also be applied, as long as n is at least 30.

By the above theorem, the mean of the sample, is expected to be also 75.

Therefore, the expected value of  mean is 75. Hence, the correct option is (b) 75.

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Answer:

t= 1/8, pi/8, 2pi/8,3pi/8

Step-by-step explanation:

Given

m=(8/32) lb s^2/ft

K=8/(6/12)=16 lb/ft

Use the following equation and plug in values

mu''+ku=f(t)

1/4u''+16u=8sin8t

u''+64u=32sin8t

This equation corresponds to the following homogeneous equation

u''+64u=0

r=+/-8i

uc(t)=c1cos8t+c2sin8t

Now find the particular solution

u(t)=Atcos8t+Btsin8t

u'(t)=-8Atsin8t+Acos8t+B8tcos8t+Bsin8t

u''(t)=-8tAsin8t-64Atcos8t-8Asin8t+B8cos8t-64Btsin8t+8Bcos8t

Substitute these values into the original equation and solve for Aand B

A=-2 B=0

the particular solution is u(t)=-2tcos8t

the general solution is u=u1(t)+u(t)

u=c1cos8t+c2sin8t-2tcos8t

Use the initial conditions to solve for c1 andc2

c1+0=(1/4)   8c2-2=0

c1=(-1/4) c2=(1/4)

u=(1/4)[cos8t+sin8t-8tcos8t]

To solve the next step differentiate u

u'=-2sin8t+2cos8t-2cos8t+16tsin8t

= -2sin8t+16sin8t

= 2sin8t(8t-1)

Velocity=2sin8t(8t-1)

Set this equation equal to zero to solve for zero velocity

8t-1=0 t=1/8

t= 1/8, pi/8, 2pi/8,3pi/8

The number of permutations of 3 objects taken 2 at a time without repetition is 3. The permutations are ml, mn, lm, ln, nm, nl.

The number of permutations of three objects taken two at a time without repetition is given by the formula 3P2 = 3!/(3-2)! = 3!/1! = 3.

The permutations of three objects (m, l, and n) taken two at a time without repetition are:

Without additional information, it's not possible to calculate the exact probability that Artemisia's conclusion is correct using conditional probability and Bayes' theorem.

To calculate the probability that Artemisia's conclusion about her new phone number being correct, we need to use the concept of conditional probability. Since she is 50% sure that the number is correct, and given the probability of any seven-digit phone number being busy is 1%, we need to consider both pieces of information. We can use Bayes' theorem to update the probability of Artemisia's belief in light of the new evidence (getting a busy signal).

However, we need additional information to accurately calculate this. Specifically, we would need to know the probability that Artemisia would get a busy signal if the number was incorrect. Without this information, we cannot provide a definitive answer to the student's question.

Answer:

[tex]P=\frac{1}{132}[/tex]

Step-by-step explanation:

If there are twelve contestants, the number of ways in which they can select the first and second turn can be calculated using the rule of multiplication as:

         12               *                11            =  132

     1st turn                     2nd turn

Because, they are going to have 12 contestants for the first turn and then they are going to have 11 contestants for the second.

On the other hand, 1 of these options is that you go first and your friend goes second, so the probability that this happens is equal to:

[tex]P=\frac{1}{132}[/tex]