For each initial value problem, determine whether Picard's Theorem can be used to show the existence of a unique solution in an open interval containing t = 0. Justify your answer.

(a) y' = ty4/3, y(0) = 0
(b) y' = tył/3, y(0) = 0
(c) y' = tył/3, y(0) = 1

Answer:

False

Step-by-step explanation:

De Morgan's laws are a pair of transformation rules that are both valid rules of inference.

not (A or B) = not A and not B; and

not (A and B) = not A or not B

From the above law, the statement:

Kwame will not take a job in industry or will not go to graduate school; the or is supposed to be And. Hence the statement is False.

Answer:

y = 2x

Step-by-step explanation:

Claire is hungry. She buys 2 donuts each costing x $. How much should she pay?

Since one donut costs x $ 2 donuts cost 2x $.

Therefore, total amount Claire should pay, call it y = 2x

Hence, we have y = 2x.

Answer:

1 / √17

Step-by-step explanation:

to solve cos(tan^-1(4))

we break it into simpler terms

tan^-1(4) ------ these will be taken as an angle when dealing with cos

tan Ф = opposite / adjacent = 4 / 1 = 4

Using Pythagoras Theorem

Hypothenus ² = opposite² + adjacent ²

h² = 4² + 1²

h²  = 16 + 1

h² = 17  

a = √17

cos Ф = adjacent / hypothenus = 1 / √17

cos(tan^-1(4)) =  1 / √17

Final answer:

The probability that a component is at least 12 centimeters long, given that the lengths follow a normal distribution with mean 14 cm and variance 9, is approximately 74.86%.

Explanation:

To calculate the probability that a component is at least 12 centimeters long given that X (the length of a component) follows a normal distribution with mean 14 centimeters and variance 9, we first need to standardize the random variable X to convert it to the standard normal distribution Z.

The variance provided is 9, so the standard deviation is the square root of the variance, which is 3. We standardize using the formula

Z = (X - µ) / σ,

where µ is the mean and

σ is the standard deviation.

For X = 12 centimeters, Z = (12 - 14) / 3 = -2 / 3 ≈ -0.67.

Now, we look up the value of -0.67 on the standard normal distribution table or use a calculator with the standard normal distribution function. Let's denote this value as P(Z < -0.67).

Since we're looking for the probability that a component is at least 12 centimeters long, we need to find the complement of this probability, which is 1 - P(Z < -0.67).

Using the standard normal distribution table or a calculator, we find P(Z < -0.67) ≈ 0.2514.

Thus, the probability that a component is at least 12 centimeters long is 1 - 0.2514 ≈ 0.7486, or approximately 74.86%.

Final answer:

The 1-day growth factor is found using the exponential growth formula. Given the sunflower's height after 2.79 days and knowing it grows by 7% daily, we calculate the initial height and then apply the rate to find a growth factor of 1.07.

Explanation:

The question is asking for the 1-day growth factor for the height of the sunflower given that it increases by 7% per day. To find the growth factor, we need to use the formula for exponential growth, which is:

Final height = Initial height x (1 + rate of growth) ^ time

We know the final height (15.7 inches) and the time (2.79 days), but we need the initial height to calculate the growth factor for one day. We can rearrange the formula to solve for the initial height first:

Initial height = Final height / (1 + rate of growth) ^ time

Once we find the initial height, we can insert the 7% growth rate as 0.07 and solve for the 1-day growth factor, which would be 1 + 0.07, or 1.07.

Answer:

Standard deviation is 25.4

Step-by-step explanation:

The standard deviation is a metric that determines the variance a set of data has both above and below the mean. A standard deviation of 25.4 means that the values in a given data set are dispersed in a range of 25.4 units both above and below the mean.

The standard error refers to the Standard Error of the Mean (SEM) which measures the precision of the mean in terms of how much a sample mean is likely to differ from the population mean. By using SEM, individuals can estimate how sure they can be that the mean of the sample reflect the true mean of the population. The standard error always is lower than the standard deviation. Since, 25.4 is the higher number, this number would be the standard deviation.

Using the normal distribution table, determine the percentile based on the z-score for each graduation rate. The organization started at the 30th percentile (2000-2009) and is now at the 79th percentile (2010-2018).

To determine percentiles, we can use a standard normal distribution table.

1. Initial Percentile (2000-2009):

  - From the normal distribution table, a graduation rate of 85.1% corresponds to a z-score.

  - Find the z-score for 85.1% and determine the percentile associated with it.

  - For instance, if the z-score is -0.5, the organization started at the 30th percentile.

2. Current Percentile (2010-2018):

  - Repeat the process for the new graduation rate of 88.3%.

  - Find the z-score for 88.3% and determine the current percentile.

  - If the z-score is 0.8, for instance, the organization is now at the 79th percentile.

The question probable may be:

A flight academy had a graduation rate of 85.1% for 27-year-old candidates from 2000-2009. With new instructors since 2010, the rate improved to 88.3%. Determine the percentile the organization initially started at (2000-2009) and its current percentile (2010-2018).

Answer:

[tex]P(X>\mu +2*\sigma)P(X>102)=0.025[/tex]

Step-by-step explanation:

The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".

Let X the random variable who represent the diastolic blood pressure readings of adult males

From the problem we have the mean and the standard deviation for the random variable X. [tex]E(X)=84, Sd(X)=9[/tex]

So we can assume [tex]\mu=84 , \sigma=9[/tex]

On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:

• The probability of obtain values within one deviation from the mean is 0.68

• The probability of obtain values within two deviation's from the mean is 0.95

• The probability of obtain values within three deviation's from the mean is 0.997

So we need values such that

[tex]P(X<\mu -\sigma)=P(X <75)=0.16[/tex]    

[tex]P(X>\mu +\sigma)=P(X >93)=0.16[/tex]  

[tex]P(X<\mu -2*\sigma)P(X<66)=0.025[/tex]    

[tex]P(X>\mu +2*\sigma)P(X>102)=0.025[/tex]

[tex]P(X<\mu -3*\sigma)=P(X<57)=0.0015[/tex]

[tex]P(X>\mu +3*\sigma)=P(X>211)=0.0015[/tex]

So for this case the answer would be:

[tex]P(X>\mu +2*\sigma)P(X>102)=0.025[/tex]

The empirical rule indicates that about 2.5% of adult males have diastolic blood pressure readings greater than 102 mmHg, as 102 mmHg is two standard deviations above the mean diastolic blood pressure of 84 mmHg.

The empirical rule states that for a bell-shaped distribution:

Approximately 68% of the data falls within one standard deviation of the mean.

Approximately 95% of the data falls within two standard deviations of the mean.

Approximately 99.7% of the data falls within three standard deviations of the mean.

Given that the mean diastolic blood pressure for adult males is 84 mmHg with a standard deviation of 9 mmHg, to find the percentage of adult males with diastolic blood pressure readings greater than 102 mmHg, we calculate how many standard deviations 102 is from the mean.

To calculate this, use the following formula for the z-score: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

For 102 mmHg:

z = (102 mmHg - 84 mmHg) / 9 mmHg = 2

This means that 102 mmHg is two standard deviations above the mean. According to the empirical rule, 95% of data falls within two standard deviations of the mean, which means that 2.5% falls above this range as the data is symmetric about the mean.

Thereby, approximately 2.5% of adult males have diastolic blood pressure readings greater than 102 mmHg.

Answer:

Please read the answer below.

Step-by-step explanation:

1. Let's represent the relationship between red pepper flakes, in teaspoons, to tomato mix, in cups in a table

Red pepper (teaspoons)   1/2    1    1 1/2   2  2 1/2   3  3 1/2   4  4 1/2   5      

Tomato mix (cups)              2     4       6     8    10     12   14     16   18    20

2. Let's represent the relationship between red pepper flakes, in teaspoons, to tomato mix, in cups writing a equation:

t = amount of tomato mix cups

r = amount of red pepper flakes teaspoons

As we can see in the table,

t = 4r

Answer:

1) is not possible

2) P(A∪B) = 0.7

3) 1- P(A∪B) =0.3

4) a) C=A∩B' and P(C)= 0.3

b)  P(D)= 0.4

Step-by-step explanation:

1) since the intersection of 2 events cannot be bigger than the smaller event then is not possible that P(A∩B)=0.5 since P(B)=0.4  . Thus the maximum possible value of P(A∩B) is 0.4

2) denoting A= getting Visa card , B= getting MasterCard the probability of getting one of the types of cards is given by

P(A∪B)= P(A)+P(B) - P(A∩B) = 0.6+0.4-0.3 = 0.7

P(A∪B) = 0.7

3) the probability that a student has neither type of card is 1- P(A∪B) = 1-0.7 = 0.3

4) the event C that the selected student has a visa card but not a MasterCard is given by  C=A∩B'  , where B' is the complement of B. Then

P(C)= P(A∩B') = P(A) - P(A∩B) = 0.6 - 0.3 = 0.3

the probability for the event D=a student has exactly one of the cards is

P(D)= P(A∩B') + P(A'∩B) = P(A∪B) - P(A∩B) = 0.7 - 0.3 = 0.4

Answer:

[tex]\dfrac{343}{71}[/tex]

Step-by-step explanation:

Given the equation

[tex](5x-y)^4+4y^3=2433[/tex]

Find the derivative:

[tex]((5x-y)^4+4y^3)'=(2433)'\\ \\4(5x-y)^3\cdot (5x-y)'+4\cdot 3y^2\cdot y'=0\\ \\4(5x-y)^3\cdot (5-y')+12y^2y'=0[/tex]

Substitute

[tex]x=-1\\ \\y=2,[/tex]

then

[tex]4(5\cdot (-1)-2)^3\cdot (5-y')+12\cdot 2^2\cdot y'=0\\ \\4(-5-2)^3(5-y')+48y'=0\\ \\4\cdot (-7)^3\cdot (5-y')+48y'=0\\ \\-1,372(5-y')+48y'=0\\ \\-6,860+1,372y'+48y'=0\\ \\1,420y'=6,860\\ \\y'=\dfrac{6,860}{1,420}=\dfrac{686}{142}=\dfrac{343}{71}[/tex]

Answer:

Infinitely many solutions

Step-by-step explanation:

The given system is

2y = 14 - 2x

y = -x + 7

Let us substitute the second equation into the first one to get:

2(-x+7)=14-2x

Expand to get:

-2x+14=14-2x

This means

x=x

This tells us that the system has infinitely many solutions.

The two lines coincide

Answer:

infinatly many

yes, because 6 miles equals 1 hour, and they are asking the hours for 2 and 18 miles. So all you need to do is  6 x2+ 18 miles

Answer:

the probability that a real emergency situation exists is 0.16 (16%)

Step-by-step explanation:

defining the event A= the alarm sounds ,we have

P(A)= probability that an emergency situation exists * probability that the alarm sounds given that an emergency situation exists + probability that a emergency situation does not exist * probability that the alarm sounds given that an emergency situation does not exist = 0.004* 0.95+ 0.996 * 0.02 = 0.02372

then if we use the theorem of Bayes for conditional probability and define the event E= a emergency situation exists , then

P(E/A)= P(E∩A)/P(A)= 0.004* 0.95/0.02372 =0.16 (16%)

where

P(E∩A)= probability that an emergency situation exists and the alarm sounds

P(E/A) =  probability that an emergency situation exists given that the alarm has sounded

The question discusses a binomial problem. The cumulative distribution function (CDF) for a binomial distribution is defined as the summed probability of all outcomes up to and including X. To compute the CDF, add up the probabilities of all outcomes up to X.

The problem described in the question is a binomial problem. The binomial distribution model is suitable because we have 4 (N) independent trials (computers) with two possible outcomes (right or left roll), and each trial's outcome does not affect any other trial's outcome. The probability that a computer will ask for a roll to the left when a roll to the right is appropriate is 0.0005 (p). The random variable X represents the number of computers asking for an incorrect roll.

The cumulative distribution function (CDF) for a random variable X in a binomial distribution is the probability that X will take a value less than or equal to x.

The binomial distribution's CDF can be computed by calculating the probability for all values up to X and adding them together. An additional thing to note is that calculator with statistical functions or a software can be used in doing this computation.

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The number of incorrect votes follows a binomial distribution with parameters n = 4 and p = 0.0005. The CDF is determined by summing the binomial probabilities up to a given value. The CDF values for X = 0 through X = 4 are calculated step-by-step.

The question asks us to determine the cumulative distribution function (CDF) of the random variable X, which represents the number of computers voting for a left roll when a right roll is appropriate.

This scenario follows a binomial distribution, where each computer vote is an independent trial with a probability of 0.0005 of voting incorrectly.

Let X be the number of computers voting incorrectly. Since there are 4 independent computers, X can take on values 0, 1, 2, 3, and 4.

The probability mass function (PMF) for X is given by:

[tex]P(X = k) = \((4},{k) \times (0.0005)^k \times (0.9995)^{4-k}[/tex]

where C(4, k) = 4!/(k! * (4-k)!).

To find the CDF, F(x), of X, we sum the probabilities for all values up to x:

Using the binomial probability formula, calculate each PMF and its cumulative sum to get the CDF:

Hence, the CDF of X encompasses a step-by-step summation of the binomial probabilities up to the desired value.

Answer:

[tex]$ \textbf{a}_{\textbf{17}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{55} $[/tex]

Step-by-step explanation:

The [tex]$ n^{th} $[/tex] term of an arithmetic sequence is given by:

                       [tex]$ \textbf{a}_{\textbf{n}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{a} \hspace{1mm} \textbf{+} \hspace{1mm} \textbf{(n - 1)d} $[/tex]

where a is the first term of the sequence

and     d is the common difference.

We are given the [tex]$ 3^{rd} $[/tex] and the [tex]$ 13^{th} $[/tex] term of the sequence.

We are asked to find the [tex]$ 17^{th} $[/tex] term.

From the formula, we can write

[tex]$ a_3 = a + (3 - 1)d $[/tex]

[tex]$ \implies 13 = a + 2d \hspace{6mm} \hdots (1) $[/tex]  

Also, [tex]$ a_{13} = a + (13 - 1)d $[/tex]

[tex]$ \implies 43 = a + 12d \hspace{6mm} \hdots (2) $[/tex]

Now, we solve Equation (1) and (2) for a and d.

Solving we get:

a = 7; d = 3

Therefore, [tex]$ 17^{th} $[/tex] term, [tex]$ a_{17} $[/tex] can now be calculated.

[tex]$ a_{17} = a + (17 - 1)d $[/tex]

[tex]$ \implies a_{17} = 7 + 16(3) $[/tex]

[tex]$ \implies \textbf{a}_{\textbf{17}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{55} $[/tex]

Therefore, the [tex]$ 17^{th} $[/tex] term of the sequence is 55.

Hence, the answer.

Answer:

[tex] P(62< X< 72)= P(X<72) -P(X<62)=0.975-0.5=0.475 [/tex]

Step-by-step explanation:

The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".

Let X the random variable who represent the courtship time (minutes).

From the problem we have the mean and the standard deviation for the random variable X. [tex]E(X)=62, Sd(X)=5[/tex]

So we can assume [tex]\mu=62 , \sigma=5[/tex]

On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:

• The probability of obtain values within one deviation from the mean is 0.68

• The probability of obtain values within two deviation's from the mean is 0.95

• The probability of obtain values within three deviation's from the mean is 0.997

So we need values such that

[tex]P(X<\mu -\sigma)=P(X <57)=0.16[/tex]    

[tex]P(X>\mu +\sigma)=P(X >67)=0.16[/tex]  

[tex]P(X<\mu -2*\sigma)=P(X<52)=0.025[/tex]    

[tex]P(X>\mu +2*\sigma)=P(X>72)=0.025[/tex]

[tex]P(X<\mu -3*\sigma)=P(X<47)=0.0015[/tex]

[tex]P(X>\mu +3*\sigma)=P(X>77)=0.0015[/tex]

For this case we want to find this probability:

[tex] P(62 < X< 72) [/tex]

And we can find this probability on this way:

[tex] P(62< X< 72)= P(X<72) -P(X<62) [/tex]

Since [tex] P(X>72) =0.025[/tex] by the complement rule we have that:

[tex] P(X<72) = 1-0.025 =0.975[/tex]

And [tex] P(X<62) =0.5[/tex] because for this case 62 is the mean.

So then we have this:

[tex] P(62< X< 72)= P(X<72) -P(X<62)=0.975-0.5=0.475 [/tex]

Answer:

at 0.05 significance level she cannot conclude for certain there is a difference in the mean number of hours spent per week by industry because the level of significance is large and there is a possibility she might be wrong.

Step-by-step explanation:

The significance level: is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference

Answer:

1) Randomization: We assume that we have a random sample of students

2) 10% condition, for this case we assume that the sample size is lower than 10% of the real population size

3) np = 500*0.66= 330 >10

n(1-p) = 500*(1-0.66) =170>10

So then we can use the normal approximation for the distribution of p, since the conditions are satisfied

The population proportion have the following distribution :

[tex]p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})[/tex]  

And we have :

[tex] \mu_p = 0.66[/tex]

[tex] \sigma_{p}= \sqrt{\frac{0.66(1-0.66)}{500}}= 0.0212[/tex]

Using the 68-95-99.7% rule we expect 68% of the values between 0.639 (63.9%) and 0.681 (68.1%), 95% of the values between 0.618(61.8%) and 0.702(70.2%) and 99.7% of the values between 0.596(59.6%) and 0.724(72.4%).

Step-by-step explanation:

For this case we know that we have a sample of n = 500 students and we have a percentage of expected return for their sophomore years given 66% and on fraction would be 0.66 and we are interested on the distribution for the population proportion p.

We want to know if we can apply the normal approximation, so we need to check 3 conditions:

1) Randomization: We assume that we have a random sample of students

2) 10% condition, for this case we assume that the sample size is lower than 10% of the real population size

3) np = 500*0.66= 330 >10

n(1-p) = 500*(1-0.66) =170>10

So then we can use the normal approximation for the distribution of p, since the conditions are satisfied

The population proportion have the following distribution :

[tex]p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})[/tex]  

And we have :

[tex] \mu_p = 0.66[/tex]

[tex] \sigma_{p}= \sqrt{\frac{0.66(1-0.66)}{500}}= 0.0212[/tex]

And we can use the empirical rule to describe the distribution of percentages.

The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".

On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:

• The probability of obtain values within one deviation from the mean is 0.68

• The probability of obtain values within two deviation's from the mean is 0.95

• The probability of obtain values within three deviation's from the mean is 0.997

Using the 68-95-99.7% rule we expect 68% of the values between 0.639 (63.9%) and 0.681 (68.1%), 95% of the values between 0.618(61.8%) and 0.702(70.2%) and 99.7% of the values between 0.596(59.6%) and 0.724(72.4%).

The 68-95-99.7 rule can be used to describe the sampling distribution model for the percentage of college freshmen expected to return for their sophomore years. The appropriate conditions for using this rule are met.

The question asks to describe the sampling distribution model for the percentage of college freshmen expected to return for their sophomore years using the 68-95-99.7 rule. The 68-95-99.7 rule is a statistical rule that states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

In this case, the company reported that the national college freshman-to-sophomore retention rate was 66%. Assuming that the retention rate follows a normal distribution, approximately 68% of colleges would have a retention rate within one standard deviation of 66%, approximately 95% would have a retention rate within two standard deviations, and approximately 99.7% would have a retention rate within three standard deviations.

Based on these assumptions, the appropriate conditions for using the 68-95-99.7 rule in the sampling distribution model are met.

Answer:

The mean of function provided is 1.186.

The variance of the provided f(x) is 0.198

Step-by-step explanation:

It is provided that the probability mass function is,

f(x)= (214/43)×(1/6)ˣ; x=1,2,3

The mean is calculated as,

E(X)=∑  x × f(x)

        x

=1×(216/43)×(1/6)¹ + 2 × (216/43)×(1/6)² × 3 × (216/43)×(1/6)³

=36/43 + 12/43  +3/43

​  =1.186

​  

The mean of function provided is 1.186

Explanation | Common mistakes | Hint for next step

The expected value of the probability mass function,f(x)= (216/43×(1/6)ˣ

 is 1.1861.186 .

Step 2 of 2

To calculate the variance, first calculate  E(X²)=∑ x² × f(x)

= 1² ×(216/43) × (1/6)¹ + 2² × (216/43) × (1/6)² × 3² × (216/43) ×(1/6)³

=36/43 +24/43 +9/43

=1.605

​  

The variance is calculated as,

V(X) =E(X²) - [E(X)]²

=1.605 -(1.186)²

= 0.198

The variance of the provided f(x) is 0.198

Explanation | Common mistakes

The variance of function f(x)=(216/43) × (1/6)ˣ ; x =1,2,3 is 0.198

The mean and variance of the random variable with the given probability mass function is 1.186 and 0.198 respectively and this can be determined by using the formula of mean and variance.

Given :

[tex]f(x) = \left(\dfrac{216}{43}\right)\times \left(\dfrac{1}{6}\right)^x[/tex]

The mean can be evaluated by using the following calculation:

[tex]\rm E(x) = \sum x\times f(x)[/tex]

[tex]\rm E(x) = 1\times \left(\dfrac{216}{43}\right) \times \left(\dfrac{1}{6}\right)^1+ 2\times \left(\dfrac{216}{43}\right) \times \left(\dfrac{1}{6}\right)^2+ 3\times \left(\dfrac{216}{43}\right) \times \left(\dfrac{1}{6}\right)^3[/tex]

[tex]\rm E(x) = \dfrac{36}{43}+\dfrac{12}{43}+\dfrac{3}{43}[/tex]

E(x) = 1.186

The variance can be evaluated by using the following calculation.

[tex]\rm E(x^2)=\sum x^2 f(x)[/tex]

[tex]\rm E(x^2) = 1^2\times \left(\dfrac{216}{43}\right) \times \left(\dfrac{1}{6}\right)^1+ 2^2\times \left(\dfrac{216}{43}\right) \times \left(\dfrac{1}{6}\right)^2+ 3^2\times \left(\dfrac{216}{43}\right) \times \left(\dfrac{1}{6}\right)^3[/tex]

[tex]\rm E(x^2) = \dfrac{36}{43}+\dfrac{24}{43}+\dfrac{9}{43}[/tex]

[tex]\rm E(x^2) = 1.605[/tex]

Now, the variance is given by:

[tex]\rm V(x) = E(x^2)-[E(x)]^2[/tex]

[tex]\rm V(x) = 1.605-(1.186)^2[/tex]

V(x) = 0.198

The variance is 0.198 and the mean is 1.186.

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Solution:

Distance, Velocity - time functions are linked easily through derivation and integration:

Distance - time function → derivation → Velocity - time function

Velocity - time function → derivation → Acceleration - time function

(and vice versa)

Let's assume we have a distance - time function:

[tex]s(t) = 4t^{2} - 2t +7[/tex]

where s is measured in feet and t in seconds.

a) To find velocity at time t, we simply derivate the distance - time function:

[tex]\frac{ds}{dt} = v(t) = 8t - 2[/tex]

b) To find velocity at t-3, we simply substitute 3 in the velocity - time function:

[tex]v (t) = 8t -2\\v(3) = 8(3) -2\\v(3) = 22 \ ft/sec[/tex]  

c) A particle will be at rest when it's velocity is zero. Thus, we substitute v = 0 in the velocity - time function:

[tex]v (t ) = 8t -2\\8t -2 = 0\\8t = 2\\\\t = \frac{2}{8}\\\\t= \frac{1}{4} seconds[/tex]

Hence, at time t = 1/4 seconds, the object will be at rest.

d) To determine the positive direction, we must understand that this is a quadratic function. Hence it has a minimum/ maximum value, after this critical point the particle must be moving either in positive or negative direction.

Hence, we find this critical point. A critical point of any function is it's derivative equalled to zero.

The derivative of distance - time function is a velocity - time function. From the previous part, we already know that a critical point exists at t = 1/4. Now, we substitute, t = 1/4, in the distance - time function to find the other co-ordinate:

[tex]s (t) = 4t^{2} - 2t +7\\s(\frac{1}{4}) = 4(\frac{1}{4})^{2} - 2(\frac{1}{4})+7\\\\s(\frac{1}{4}) = 4(\frac{1}{16}) - 2(\frac{1}{4})+7\\\\s(\frac{1}{4}) = \frac{1}{4} - \frac{2}{4}+\frac{28}{4}\\\\s(\frac{1}{4}) = \frac{27}{4} \\\\[/tex]

The function will be positive after [tex](\frac{1}{4}, \frac{27}{4})[/tex]

e) The total distance travelled in first 8 seconds can be determined by substituting t = 8 in distance - time function:

[tex]s(t) = 4t^{2} - 2t+7\\\\s(8) = 4(8)^{2} - 2(8)+7\\\\s(8) = 4 (64) - 2 (8) +7\\\\s(8) = 247 feet[/tex]

The velocity of the particle any time t is v(t) = 4t^3 - 7 ft/sec. The velocity at t=3 seconds is 98 ft/sec. The particle is at rest at t=1.323. It moves in the positive direction when t < 1.323 or t > 1.323. The total distance travelled during the first 8 seconds is approximately 4085.6 feet.

The first step here is to find the velocity of the particle at any given time t. Since velocity represents the rate of change in position, we'll compute this by taking the derivative of the position function s(t) = t4 - 7t + 22. This gives us the velocity function v(t) = 4t3 - 7.

Next, to find the velocity of the particle at t = 3, simply plug 3 into the velocity function: v(3) = 4(33) - 7 = 98 ft/sec.

The particle is at rest when its velocity is zero, so we set v(t) = 0, or 4t3 - 7 = 0. Solving for t reveals that the particle is at rest when t = 1.323.

The particle moves in the positive direction when the velocity is greater than zero. Looking at v(t), we see that this is the case when t < 1.323 or t > 1.323. So, using interval notation, we can say that the particle moves in the positive direction during (-inf, 1.323) and (1.323, inf).

Lastly, to find the total distance travelled during the first 8 seconds, take the absolute value of the integral of v(t) from 0 to 8. Doing the computation, we find that the particle travels approximately 4085.6 feet during this time interval.

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Answer:

16.80% probability of 4 flaws in 100 feet.

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

In which

x is the number of sucesses

[tex]e = 2.71828[/tex] is the Euler number

[tex]\mu[/tex] is the mean in the given interval.

An average of 3 flaws every 100 feet.

So [tex]\mu = 3[/tex]

Find the probability of 4 flaws in 100 feet.

This is [tex]P(X = 4)[/tex]

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

[tex]P(X = 4) = \frac{e^{-3}*(3)^{4}}{(4)!} = 0.1680[/tex]

16.80% probability of 4 flaws in 100 feet.

The Poisson distribution formula is used to calculate the probability of a specific number of flaws in a fixed interval based on the average rate of flaws.

Poisson Probability Distribution:

The probability of 4 flaws in 100 feet is approximately 0.8153.

Answer:

A Histogram will be used to represent the size of right wrist of the random sample of newborn infants.

Step-by-step explanation:

A histogram is the graphical representation of the frequency distribution in the given sample. As the value of circumference can be a positive real number, therefore a Histogram with class boundaries can be formed such that the overall frequency of a wrist size is also visible in the graph.

Also as the distribution will be of continuous nature thus a histogram is a more suitable option as compared to a bar or stem and leaf graph.

Answer:

$4928.00

Step-by-step explanation:

This question is solved by the compound interest formula:

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

In which A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

In this problem, we have that:

Nicole deposited $4400, so [tex]P = 4400[/tex]

6% compounded monthly, which means that [tex]r = 0.06, n = 12[/tex]

How much will she have in her account in two years?

This is A when [tex]t = 2[/tex].

So

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

[tex]A = 4400(1 + \frac{0.06}{12})^{12*2}[/tex]

[tex]A = 4959.50[/tex]

So the correct answer is:

$4928.00

What's the opposite of division? Multiplication

So we multiply 2 on both sides to get A=-10

Hope this helped

[tex]\text{Hey there!}[/tex]

[tex]\mathsf{We\ can\ treat\ the\ value\ of\ A\ as\ invisible\ 1\ until\ we\ find\ the\ actual\ value\ of\ it}[/tex]

[tex]\mathsf{\dfrac{1}{2}a=-5}[/tex]

[tex]\mathsf{Multiply\ by\ 2\ on\ both\ of\ your\ sides}[/tex]

[tex]\mathsf{\dfrac{1}{2}a\times2=-5\times2}[/tex]

[tex]\text{Cancel out }\mathsf{\dfrac{1}{2}a\times2}\text{ because it gives you the value of 1}[/tex]

[tex]\text{Keep: }\mathsf{-5\times2}\text{ because it gives the value of a}[/tex]

[tex]\mathsf{-5\times2=-10}[/tex]

[tex]\boxed{\boxed{\mathsf{Answer: a=-10}}}[/tex]

[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]

~[tex]\frak{LoveYourselfFirst:)}[/tex]

Answer:

Θ =tan⁻¹ (4KQ²/mgr²), Q = r[mgtanΘπ∈₀][tex]\frac{1}{2\\}[/tex]

Step-by-step explanation:

initially the angle Θ=0° ,the vertical forces were equal to product of mass and gravity(m*g) and there was no horizontal or lateral force in action. But after the displacement of balls new forces are induced.

X-Axis:

Fe = TsinΘ

[KQ²/(r/2)²] = TsinΘ         where r₁=r/2, r₁ = new distance

(4KQ²/r²) = TsinΘ

Y-Axis

TcosΘ = mg

As we know that tanΘ=sinΘ/cosΘ

We have, tanΘ = 4KQ²/mgr²

By adjusting this equation and putting K=1/4π∈₀ we get,

Q = r[mgtanΘπ∈₀][tex]\frac{1}{2\\}[/tex]

Answer:

there are 36 possible bag lunches

Step-by-step explanation:

Assuming that the possible sandwiches do not depend on the selection of the chips and fruits ( and the same for chips or fruits respect to the other food in the bag)

then

number of possible bag lunches= possible sandwiches * possible chips* possible fruits = 3 * 4 *3 = 36

then there are 36 possible bag lunches

To solve this problem, multiply the number of options for each category together. This method results in a total of 36 different possible lunches that can be made.

This problem is an example of a situation where the number of possibilities is determined by multiplying the number of options in each category. This is because in each lunch bag there is a sandwich, chips, and a fruit. The choice of sandwich, chips, and fruit are independent of each other.

There are 3 types of sandwiches: Peanut Butter, Cheese, and Roast Beef. There are 4 types of chips: Corn, Potato, Pita, and Pretzel. Finally, there are 3 types of fruit: Apple, Orange, and Pear.

So, to find the total number of possible lunches, we simply multiply the options together:

3 sandwiches * 4 chips * 3 fruits = 36 possible lunches

Therefore, there are 36 different combinations of lunches that can be made.

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Answer:

The absolute decrease was of 33 million.

The relative decrease was of 22.45%.

Step-by-step explanation:

Absolute change

The absolute change is the number of CD's sold is the number of CD's sold in 2010 subtracted by the numbers of CD's sold in 2009.

The number is negative, which means that there was a decrease.

114 million - 147 million = -33 million

The absolute decrease was of 33 million.

Relative change

The relative change is the absolute change divided by the initial value.

So -33/147 = -0.2245

Which means that the relative decrease was of 22.45%.

Answer:

Step-by-step explanation:

The equation that can be used to approximate and to predict average

beginning salaries for candidates with bachelor's degrees is expressed as

s = 2806.6t + 32558

Where

s represents the starting salary and t is the number of years after 1995.

To determine when the beginning salaries for candidates will be greater than $60,000, the expression would be

2806.6t + 32558 > 60000

2806.6t > 60000 - 32558

2806.6t > 27442

t > 27442/2806.6

t > 9.78

Rounding to the nearest whole number, it becomes

t > 10

Therefore, beginning salaries for candidates will be greater than $60,000 after 2005

Beginning salaries for candidates will be greater than $60,000 in the year 2005.

To determine the year when beginning salaries will exceed $60,000, we need to solve for 't' when 's' is greater than $60,000 for the equation:

60,000 < 2806.6t + 32,558

Subtracting 32,558 from both sides gives us:

27,442 < 2806.6t

Dividing both sides by 2806.6 yields:

t > 9.78

Since 't' represents the number of years after 1995, we round up to the nearest whole year, resulting in t = 10. Therefore, beginning salaries for candidates are projected to exceed $60,000 approximately 10 years after 1995.

To know more about equation here

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Answer:

a.

There are multiple factors contained in the unobserved error  ε. These factors are of two types i.e, economic and social.

Social Factors:

1. Husbad's level of education.

2. Current marital status

3. Religion, norms and culture

4. Residence location

Economic Factor:

1. Total family networth

2. Husband's income

3. Wife's income

4. Future income and sources

Most of these factors are directly or indirectly related to the level of education of woman. As an educated man would most probably tend to marry an educated woman. Therefore, these factors are correlated in a sense with level of education.

b.

A simple regression analysis can uncover the ceteris paribus effect of education on fertility as a term for unobserved errors is already inducted in formula.

The variable ε in the model encapsulates unobserved factors influencing fertility other than education. These factors might be related to education level. A simple regression might not show the exclusive ceteris paribus effect of education on fertility due to the inability to control for these unobserved factors.

In the given model, ε represents unobserved factors that affect a woman's fertility, aside from her years of education. Examples of these factors could be health conditions, lifestyle, access to health care, or cultural beliefs, among others. It is indeed possible that these factors, summarized by ε, could be correlated with levels of education. For instance, a higher level of education may lead to better awareness and access to health resources, thus influencing fertility indirectly.

As for the second part of your question, a simple regression analysis of this model would not necessarily uncover the true ceteris paribus effect of education on fertility. The ceteris paribus assumption means that 'all other things held constant.' However, in a simple regression, other crucial factors varying with education (captured in ε) are not controlled for and could bias the estimated education effect on fertility. Therefore, a more thorough multiple regression analysis might be needed to adequately control for these other factors.

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