On one of its routes across Asia, Alpha Airlines flies an aircraft with checked-in luggage capacity of 8500 lbs. There are 121 seats on the flight.
The average (per passenger) weight of checked-in luggage is 68 lbs with a standard deviation of 11 lbs.
What is the probability that on a randomly selected full flight on this route the checked-in luggage capacity will be exceeded?

The large puppy weighs 4 1/2 x 42/3 pounds.

4 1/2 x 42/3 = 9/2 x 42/3 = 63 pounds

answer: 63 pounds for the large puppy

Answer:the large puppy weighs 63 pounds.

Step-by-step explanation:

A vet weighs 2 puppies. The small puppy weighs 4 1/2 pounds. Converting 4 1/2 pounds to improper fraction, it becomes 9/2 pounds.

The large puppy weighs 42/3 times as much as the small puppy. This means that the number of pounds that the large puppy weighs would be

42/3 × 9/2 = 63

Answer:

Step-by-step explanation:

The average of 2.605, 24.04, 13.3, and 201.64 is gotten by adding the values and dividing by 4 since we are dealing with 4 digits.

[tex]Average =\frac{2.605+24.04+13.3+201.64}{4}[/tex]

average = 241.585/4

average=60.39625

Answer:a) μ = 5 and σ = 16

b) z-score are -0.3125, 0.0625, -0.0625, -0.125, 0.4375

c) New population of N=5 scores are 93.75, 101.25, 98.75, 97.5, 108.75

Step-by-step explanation:

The detailed explanation can be found in the attached pictures

The new population of N = 5 scores with a mean of µ = 100 and a standard deviation of σ = 20 are 125, 105, 95, 90 and 135

(a) Compute µ and σ for the population.

The dataset is given as:

0, 6, 4, 3, and 12.

The mean is calculated as:

[tex]\mu = \frac{\sum x}n[/tex]

So, we have:

[tex]\mu = \frac{0 + 6 + 4 + 3 + 12}5[/tex]

[tex]\mu = 5[/tex]

The standard deviation is calculated as:

[tex]\sigma = \sqrt{\frac{\sum(x - \bar x)^2}n}[/tex]

This gives

[tex]\sigma = \sqrt{\frac{(0 - 5)^2 + (6- 5)^2 + (4- 5)^2 + (3- 5)^2 + (12- 5)^2}5[/tex]

[tex]\sigma = \sqrt{\frac{80}5[/tex]

[tex]\sigma = \sqrt{16[/tex]

[tex]\sigma = 4[/tex]

Hence, the values of μ and σ are μ = 5 and σ = 4

(b) The z-scores

This is calculated as:

[tex]z = \frac{x - \mu}{\sigma}[/tex]

When x = 0, 6, 4, 3, and 12.

We have:

[tex]z = \frac{0 - 5}{4} = 1.25[/tex]

[tex]z = \frac{6 - 5}{4} = 0.25[/tex]

[tex]z = \frac{4 - 5}{4} = -0.25[/tex]

[tex]z = \frac{3 - 5}{4} = -0.5[/tex]

[tex]z = \frac{12 - 5}{4} = 1.75[/tex]

Hence, the z-scores are 1.25, 0.25, -0.25, -0.5 and 1.75

(c) Transform the new population

We have:

N = 5, µ = 100 and σ = 20.

In (b), we have:

[tex]z = \frac{x - \mu}{\sigma}[/tex]

Make x the subject

[tex]x = \mu + z\sigma[/tex]

This gives

[tex]x_i = \mu + z_i\sigma[/tex]

So, we have:

[tex]x_1 = 100 + 1.25* 20 = 125[/tex]

[tex]x_2 = 100 + 0.25* 20 = 105[/tex]

[tex]x_3 = 100 - 0.25* 20 = 95[/tex]

[tex]x_4 = 100 - 0.5* 20 = 90[/tex]

[tex]x_5 = 100 + 1.75* 20 = 135[/tex]

Hence, the new population of N = 5 scores with a mean of µ = 100 and a standard deviation of σ = 20 are 125, 105, 95, 90 and 135

Read more about z-scores at:

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Answer:

a) 1820 ways

b) 43680 ways

Step-by-step explanation:

When the order of the choices is relevant we use the permutation formula:

[tex]P_{n,x}[/tex] is the number of different permutations of x objects from a set of n elements, given by the following formula.

[tex]P_{n,x} = \frac{n!}{(n-x)!}[/tex]

When the order of choices is not relevant we use the combination formula:

[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

In this problem, we have that:

[tex]x = 4, n = 16[/tex]

(a) How many ways can this be done, if the order of the choices is not relevant?

[tex]C_{16,4} = \frac{16!}{4!(12)!} = 1820[/tex]

(b) How many ways can this be done, if the order of the choices is relevant?

[tex]P_{16,4} = \frac{16!}{(12)!} = 43680[/tex]

We can choose 4 objects from 16 in 1820 ways if order doesn't matter (combination), and in 43680 ways if order does matter (permutation).

The subject of this question is combinatorial mathematics. You're being asked to calculate combinations and permutations.

(a) If the order of the choices is not relevant, we are dealing with a combination. The formula for a combination is C(n, r) = n! / [r!(n-r)!], where n is the number of objects and r is the number of objects chosen. In this case, n = 16 and r = 4, so C(16, 4) = 16! / [4!(16-4)!] = 1820 combinations.

(b) If the order of the choices is relevant, we are dealing with a permutation. The formula for a permutation is P(n, r) = n! / (n-r)!. Again, n = 16 and r = 4, so P(16, 4) = 16! / (16-4)! = 43680 permutations.

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Final answer:

To calculate the probability of winning on the sixth ticket, multiply the probability of not winning on the first five tickets by the probability of winning on the sixth ticket.

Explanation:

To calculate the probability of winning on the sixth ticket, we multiply the probability of not winning on the first five tickets (0.65)^5 by the probability of winning on the sixth ticket (0.35). Here's the step-by-step calculation:

Probability of not winning on the first five tickets: (0.65)^5

Probability of winning on the sixth ticket: 0.35

Therefore, the probability of first winning on the sixth ticket is equal to (0.65)^5 * 0.35.

Step-by-step explanation:

If x is the volume of 39% solution, and y is the volume of 70% solution, then:

x + y = 40

0.39x + 0.70y = 0.55(40)

Solve the system of equations.

0.39x + 0.70(40 − x) = 0.55(40)

0.39x + 28 − 0.70x = 22

6 = 0.31x

x = 19.4

y = 20.6

The chemist needs 19.4 liters of 39% solution and 20.6 liters of 70% solution.

To find the orthonormal basis using the Gram-Schmidt process, we calculate the first vector by dividing the first given vector by its magnitude and normalize it. Then, we subtract the projection of each subsequent vector onto the previously found orthonormal vectors and normalize the resulting vector.

To find an orthonormal basis for the subspace of R4 spanned by the given vectors using the Gram-Schmidt process, we will start by finding the first vector of the orthonormal basis. Let's call the given vectors v1, v2, and v3, respectively. The first vector of the orthonormal basis, u1, is equal to v1 divided by its magnitude, which is ||v1||. So, u1 = v1 / ||v1||. We can calculate ||v1|| as √(1^2 + 0^2 + 1^2 + 1^2) =  √3.

Therefore, u1 = (1/√3, 0/√3, 1/√3, 1/√3).

Now, we need to find u2, the second vector of the orthonormal basis. To do this, we subtract the projection of v2 onto u1 from v2, then divide the result by its magnitude. We calculate the projection of v2 onto u1 as proj_u1(v2) = u1 * dot(u1, v2), where dot(u1, v2) represents the dot product of u1 and v2.

Finally, we subtract proj_u1(v2) from v2 to get v2' = v2 - proj_u1(v2), and then normalize v2' to get u2 = v2' / ||v2'||.

We can repeat this process to find u3, the third vector of the orthonormal basis. Subtract proj_u1(v3) and proj_u2(v3) from v3, then normalize the result to get u3 = v3' / ||v3'||.

Therefore, the orthonormal basis for the subspace spanned by the given vectors is (1/√3, 0/√3, 1/√3, 1/√3), (0, 0, 0, 1), and (-1/√3, 0/√3, 1/√3, 1/√3).

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Answer:

The answer to your question is 5028 votes

Step-by-step explanation:

Data

Winner 8392 votes

Second-place = 7480 votes

Total votes = 20900

Third candidate = ?

Process

1.- Write an equation

       Total votes = Winner + Second-place + Third-place

Solve for Third-place

       Third-place = Total votes - Winner - Second-place

2.- Substitution

        Third-place = 20900 - 8392 - 7480

3.- Simplification

        Third-place = 20900 - 15872

4.- Result

         Third-place = 5028 votes

Answer:

5028

Step-by-step explanation:

If everyone in the town voted for 3 candidates and the total vote is 20900, then

a + b + c = 20900

If stands for the first contestant, b for the second contestant and c for the last contestant.

If a = 8392

b = 7480

c = 20900-(a+b)

c = 20900-15872

c = 5028

Answer:

0.6547 or 65.47%

Step-by-step explanation:

One minute equals 1/60 of an hour, the mean number of occurrences in that interval is:

[tex]\lambda =\frac{400}{60}=6.6667[/tex]

The poisson distribution is described by the following equation:

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

The probability that more than 5 vehicles will arrive is:

[tex]P(x>5)= 1-(P(0)+P(1)+P(2)+P(3)+P(4)+P(5))\\P(x>5) = 1-(\frac{6.667^{0}*e^{-6.667}}{1}+\frac{6.667^{1}*e^{-6.667}}{1}+\frac{6.667^{2}*e^{-6.667}}{2}+\frac{6.667^{3}*e^{-6.667}}{3*2}+\frac{6.667^{4}*e^{-6.667}}{4*3*2}+\frac{6.667^{5}*e^{-6.667}}{5*4*3*2})\\P(x>5)=1-(0.00127+0.00848+0.02827+ 0.06283+0.10473+0.13965)\\P(x>5)=0.6547[/tex]

The probability that more than five vehicles will arrive in a one-minute interval is 0.6547 or 65.47%.

The probability that more than five vehicles will arrive in a one-minute interval is approximately 0.6582.

Step 1

Given that vehicles arrive at an intersection at a rate of 400 vehicles per hour, and this follows a Poisson distribution, we want to find the probability that more than five vehicles will arrive in a one-minute interval.

First, convert the arrival rate to a one-minute interval. Since there are 60 minutes in an hour, the arrival rate per minute is:

[tex]\[ \lambda = \frac{400 \, \text{veh/h}}{60} = \frac{400}{60} \approx 6.67 \, \text{veh/min} \][/tex]

The Poisson distribution formula for the probability of observing k events in an interval is:

[tex]\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \][/tex]

We need the probability that more than five vehicles arrive in one minute:

[tex]\[ P(X > 5) = 1 - P(X \leq 5) \][/tex]

Step 2

First, calculate [tex]\( P(X \leq 5) \)[/tex] by summing the probabilities for k = 0 to k = 5 :

[tex]\[ P(X \leq 5) = \sum_{k=0}^{5} \frac{e^{-6.67} 6.67^k}{k!} \][/tex]

Let's calculate these probabilities:

- For k = 0 :

 [tex]\[ P(X = 0) = \frac{e^{-6.67} 6.67^0}{0!} = e^{-6.67} \][/tex]

 - For k = 1 :

 [tex]\[ P(X = 1) = \frac{e^{-6.67} 6.67^1}{1!} = e^{-6.67} \times 6.67 \][/tex]

 - For k = 2 :

 [tex]\[ P(X = 2) = \frac{e^{-6.67} 6.67^2}{2!} = e^{-6.67} \times \frac{6.67^2}{2} \][/tex]

 - For k = 3:

 [tex]\[ P(X = 3) = \frac{e^{-6.67} 6.67^3}{3!} = e^{-6.67} \times \frac{6.67^3}{6} \][/tex]

 - For k = 4 :

 [tex]\[ P(X = 4) = \frac{e^{-6.67} 6.67^4}{4!} = e^{-6.67} \times \frac{6.67^4}{24} \][/tex]

 - For k = 5 :

 [tex]\[ P(X = 5) = \frac{e^{-6.67} 6.67^5}{5!} = e^{-6.67} \times \frac{6.67^5}{120} \][/tex]

Sum these probabilities to find [tex]\( P(X \leq 5) \)[/tex].

Step 3

Next, we calculate [tex]\( e^{-6.67} \)[/tex] and the terms:

[tex]\[e^{-6.67} \approx 0.00126\][/tex]

[tex]\[P(X = 0) \approx 0.00126\][/tex]

[tex]\[P(X = 1) \approx 0.00126 \times 6.67 = 0.0084\][/tex]

[tex]\[P(X = 2) \approx 0.00126 \times \frac{6.67^2}{2} = 0.0280\][/tex]

[tex]\[P(X = 3) \approx 0.00126 \times \frac{6.67^3}{6} = 0.0622\][/tex]

[tex]\[P(X = 4) \approx 0.00126 \times \frac{6.67^4}{24} = 0.1037\][/tex]

[tex]\[P(X = 5) \approx 0.00126 \times \frac{6.67^5}{120} = 0.1382\][/tex]

Sum these probabilities:

[tex]\[P(X \leq 5) \approx 0.00126 + 0.0084 + 0.0280 + 0.0622 + 0.1037 + 0.1382 = 0.34176\][/tex]

Finally, the probability that more than five vehicles will arrive in a one-minute interval is:

[tex]\[P(X > 5) = 1 - P(X \leq 5) = 1 - 0.34176 = 0.65824\][/tex]

The probability that more than five vehicles will arrive in a one-minute interval is approximately 0.6582.

Answer:

The common factor is -1/4

Step-by-step explanation:

Let's find out the common ratio for the geometric sequence:

24, - 6, 3/2, -3/8

Relationship between first and second term:

24 * -1/4 = -24/4 = -6

Relationship between second and third term:

-6 * -1/4 = 6/4 = 3/2

Relationship between third and fourth term:

3/2 * -1/4 = -3/8

The common factor is -1/4

Answer:

Wednesday

Step-by-step explanation:

A 18 sessions and we know that a session lasts 7 days. We also know that the sessions are grouped and that there is only a break after every 2 sessions. The sessions can only be held on weekdays which is 5 days. The first session starts on Friday. We need to determine the last day of the 18 sessions.

WE can assume that a break is a one day.

The first two sessions will be a total of 14 days and then a break. Friday adding 14 days will result in the first two sessions ending on Wednesday and a break day. The next two sessions will start Friday again.

Therefore the sessions are even number of 18 and therefore will always end on a Wednesday

Answer:2.5 breaks

Step-by-step explanation:

Answer:

53.36

Step-by-step explanation:

The mean of binomial distribution is calculated by multiplying number of trials to probability of success. It can be denoted as

E(x)=mean=np

Where n is the fixed number of trails and p is the probability of success.

Here, n=58 and p=0.92

E(x)=np

E(x)=58*0.92

E(x)=53.36

So, the mean of the given binomial distribution is 53.36.

The mean of this binomial distribution is 53.36.

The mean of a binomial distribution can be calculated using the formula µ = np, where µ represents the mean, n is the number of trials, and p is the probability of success in each trial.

In this case, the problem mentions 58 identical engine tests with a probability of success, p, being 0.92. Therefore, the mean of this binomial distribution would be µ = 58 * 0.92 = 53.36.

Answer:

Step-by-step explanation:

A. Price is 1.75

Total revenue, if number of galsses equals X is 1.75X

Total cost function of X glasses of lemonade is 25+0.1X

Profit function is Revenue - Cost, P=1.75X-25-0.1X=1.65X-25

C. Marginal profit= d/dx(1.65x-25) = 1.65-0=1.65

X=300 glasses, P'(300)=1.65

X=700 glasses, P'(700)=1.65

The length of the angle bisector of angle ∠A is approximately 4.62 feet.

To find the length of the angle bisector of angle ∠A in triangle ΔABC, we can use the Angle Bisector Theorem, which states that in a triangle, the angle bisector of a vertex divides the opposite side into segments proportional to the adjacent sides.

In triangle ΔABC, let AD be the angle bisector of ∠A, where D lies on BC. According to the Angle Bisector Theorem:

AC/CD = AB/BD

Given AC = 5 ft and AB = 13 ft, we can plug in these values:

5/CD = 13/BD

To find BD, we use the Pythagorean theorem:

BD = √(AB² - AD²) = √(13² - 5²) = √(169 - 25) = √144 = 12 ft

Now, using the Angle Bisector Theorem:

5/CD = 13/12

Cross-multiply:

5 × 12 = 13 × CD

CD = (5 × 12) / 13 = 60 / 13 ≈ 4.62 ft

Answer:

The solution of the two give equations is (-1,4)

Step-by-step explanation:

i) First equation is        y = -x + 3

ii) Second equation is y = x + 5

iii) If we add the two equations we get 2y = 8    ∴ y = 4

iv) Substituting the value of y obtained in iii) in equation i) we get

    4 = -x + 3   ∴ -x = 4 - 3     ∴ -x = 1      ∴ x = -1

v) substituting x = -1 and y = 4 in the second equation we see that the equation is satisfied.

Answer:

[tex] \frac{dP}{dt}=k_1 P -k_2 P= P(k_1 -k_2)[/tex]

Step-by-step explanation:

For this case we know that the birth rate is given by [tex]b[/tex] and the death rate is given by [tex] d[/tex].

We also know that these rates are proportional to the population size, so then we have this:

[tex] b \propto P(t)  [/tex]

[tex] d \propto P(t)[/tex]

And in order to have expression with the sign= we have the proportional constants given [tex]k_1[/tex] for b and [tex]k_2[/tex] for d, so then we convert the system of equations on this:

[tex] b = k_1 P(t) [/tex]

[tex] d = k_2 P(t) [/tex]

And then the change in the population respect to the time would be calculated on this way:

[tex] \frac{dP}{dt} = b-d[/tex]

And if we replace what we found we got:

[tex] \frac{dP}{dt}=k_1 P -k_2 P= P(k_1 -k_2)[/tex]

And we can solve the differential equation reordering the terms like this:

[tex] \frac{dP}{P}= (k_1 -k_2) dt[/tex]

And if we integrate both sides we got:

[tex] ln |P| = (k_1 -k_2) t +C[/tex]

Using exponentials we got:

[tex] P(t) = e^{(k_1 -k_2)t} *e^c[/tex]

And we can rewrite this expression like this:

[tex] P(t) = P_o e^{(k_1 -k_2)t}[/tex] where [tex] e^c = P_o[/tex]

Final answer:

The differential equation for the population change accounting for both birth and death rates when both are proportional to the population at time t is given by dP/dt = k1P - k2P, where k1 is the birth rate constant, k2 is the death rate constant, and P is the population at time t.

Explanation:

The question concerns the modeling of a population where the rate of change in population (dP/dt) is considered to be the difference between the birth rate and the death rate, both of which are proportional to the current population. This is a scenario often explored in the field of ecology and employs principles of differential calculus. If we let k1 be the proportionality constant for the birth rate and k2 be the proportionality constant for the death rate, the differential equation representing the rate of change in population can be expressed as follows:

dP/dt = k1P - k2P

In this model, P represents the population at time t, where t > 0. Here, k1P represents the total birth rate and k2P represents the total death rate in the community. As such, the net change in population (dP/dt) is determined by the intrinsic rate of increase (r), which is k1 - k2.

Answer:

a) For this case we see that the cumulative % for 600 is 77.1% so then we will have 0.771 of the values below 600

b) For this case we know that the cumulative percent for 1200 is 98.2% so then the percentage above would be 100-98.2 = 1.8%, so then the proportion above 1200 would be 0.018.

c) For this case we can add the percentages obtained from the previous parts and we got : 77.1% +1.8% = 78.9% and then the proportion that are less than 600 and at least 100 would be 0.789

Step-by-step explanation:

Assuming the cumulative percentages in the figure attached.

What proportion of fire loads are less than 600?

For this case we see that the cumulative % for 600 is 77.1% so then we will have 0.771 of the values below 600

What proportion of fire loads are At least 1200?

For this case we know that the cumulative percent for 1200 is 98.2% so then the percentage above would be 100-98.2 = 1.8%, so then the proportion above 1200 would be 0.018.

What proportion of fire loads are less than 600 at least 120?

For this case we can add the percentages obtained from the previous parts and we got : 77.1% +1.8% = 78.9% and then the proportion that are less than 600 and at least 100 would be 0.789

Answer:

answer is -3 just subtract 4 from each side

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

B ⊂ A

Hope it helps!

Answer:

This is useful to choose which calculation to perform.

Step-by-step explanation:

1) Firstly, let's consider that the Binomial Distribution tends to the Poisson Distribution given certain conditions:

[tex]n\rightarrow \infty, p\rightarrow 0, \lambda =np[/tex]

Roughly, they tend to the same value.

2) The Binomial Probability is calculated through this formula:

[tex]Binomial: P(X=x)=\binom{n}{x}p^{x}(1-p)^{n-x}[/tex]

Poisson Distribution this way:

[tex]Poisson:P(X=x)=\frac{\lambda^{x} e^{-\lambda }}{x!}[/tex]

3) If we plug

[tex]p=\frac{\lambda }{n}[/tex]

In the Binomial formula, given an "n" a very large quantity we'll have a closer outcome to Poisson.

[tex]P(X=x)=\binom{n}{x}\left ( \frac{\lambda }{n} \right )^{x}(1-\frac{\lambda }{n})^{n-x} \approx \frac{\lambda^{x} e^{-\lambda }}{x!}[/tex]

4) This is useful especially due to the convenience of calculating.

Because operating with exponentials and factorials, is hard and sometimes 'n' and 'p' may also be unknown, and sometimes the known parameter is the Mean.

Final answer:

Identifying if dataset variables follow a Binomial or Poisson distribution aids in selecting appropriate statistical models and sampling methods for ecological count data, leading to more accurate analyses and conclusions.

Explanation:

Identifying whether variables in datasets are Binomial or Poisson in nature can be incredibly helpful in statistical analyses, particularly in the field of ecology where data often consists of counts of organisms. These statistical models help determine appropriate sampling protocols and confirm the distribution of the observed data, which is essential for choosing the correct statistical tests and making accurate probability statements.

The Binomial distribution is used for data representing the number of successes in a fixed number of independent trials with a constant success probability, such as sex ratios or ratios of juveniles to adults. Conversely, the Poisson distribution is suitable for data representing counts over an interval of time or space, and is typically applied to model random occurrences in a fixed interval, like the count of organisms in a particular habitat.

Logistic regression is another analytical tool used for binary (yes/no) categorical data. It is based on a different premise, allowing researchers to predict occurrence probabilities by modeling the relationship between species detection and various explanatory variables. When using any statistical model, it is crucial to validate that the data align with the assumptions inherent to the model chosen. This is because using inappropriate statistical models could lead to incorrect conclusions, affecting research validity and the understanding of the ecological phenomena being studied.

Final answer:

To compute the covariance between Z1 and Z2, we need to calculate the covariance between X and X, X and Y, Y and X, and Y and Y individually. Using the given variances and calculations, we can find Cov(Z1, Z2) = 308 + 0.1(144 + 2Cov(X, Y) + 64).

Explanation:

To compute the covariance between Z1 and Z2, we need to calculate the covariance between X and X, X and Y, Y and X, and Y and Y first.

Now, we can calculate Cov(Z1, Z2) using the following formula:

Cov(Z1, Z2) = Cov(X + Y, X + 0.1X + 5) = Cov(Z1, Z1 + 0.1X + 5) = Cov(Z1, Z1) + Cov(Z1, 0.1X) + Cov(Z1, 5) = Var(Z1) + 0.1Cov(Z1, X) + 0 = Var(X + Y) + 0.1Cov(Z1, X) + 0 = 308 + 0.1(Cov(X, X) + Cov(X, Y) + Cov(Y, X) + Cov(Y, Y)) + 0 = 308 + 0.1(144 + Cov(X, Y) + Cov(Y, X) + 64) + 0 = 308 + 0.1(144 + 2Cov(X, Y) + 64)

The value of L is approximately 2.609.

Given curve [tex]y = \frac{x^3}{12} + \frac{1}{x}[/tex], from x = 1 to x = 2.

The arc length of a curve defined by a function y = f(x) from x = a to x = b can be calculated using the arc length formula:

[tex]L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} dx[/tex]

To calculated the value of L, to find the derivative f'(x) and substitute it into the arc length formula.

First, let's find the derivative of y with respect to x:

[tex]f'(x) = \dfrac{d}{dx} (\dfrac{x^3}{12} + \dfrac{1}{x})\\f'(x) = \dfrac{1}{4}x^2 - \dfrac{1}{x^2}[/tex]

Now, we can substitute this derivative into the arc length formula:

[tex]L = \int_{1}^{2} \sqrt{1 + (\frac{1}{4}x^2 - \frac{1}{x^2})^2} dx[/tex]

Now, represent this arc length integral:

[tex]L = \int_{1}^{2} \sqrt{1 + \left(\frac{1}{4}x^2 - \frac{1}{x^2}\right)^2} dx[/tex]

Expanding the square inside the square root:

[tex]L = \int_{1}^{2} \sqrt{1 + \left(\frac{1}{16}x^4 - \frac{1}{2} + \frac{1}{x^4}\right)} dx[/tex]

Combining the terms inside the square root:

[tex]L = \int_{1}^{2} \sqrt{\frac{1}{16}x^4 + \frac{1}{2} - \frac{1}{x^4} + 1} dxL = \int_{1}^{2} \sqrt{\frac{1}{16}x^4 + \frac{1}{x^4} + \frac{3}{2}} dx[/tex]

Now, let's integrate this expression:

[tex]L = \int_{1}^{2} \sqrt{\frac{1}{16}x^4 + \frac{1}{x^4} + \frac{3}{2}} dx[/tex]

So, the value of L is approximately 2.609.

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Complete question:

Find the arc length of the curve  [tex]Y= ((x^3)/12)) + 1/x[/tex]  from x = 1 to x = 2.

Arc Length [tex]\[Arc\ Length = \int_1^2 \sqrt{1 + \frac{1}{x^4}} \, dx\][/tex]

To find the arc length of the curve [tex]\(y = \frac{3}{12} + \frac{1}{x}\)[/tex] on the interval [tex]\([1, 2]\)[/tex], you can use the arc length formula for a function [tex]\(y = f(x)\)[/tex] on the interval [tex]\([a, b]\)[/tex]:

[tex]\[Arc\ Length = \int_a^b \sqrt{1 + (f'(x))^2} \, dx\][/tex]

First, calculate the derivative of the function [tex]\(y = \frac{3}{12} + \frac{1}{x}\):\[y' = 0 - \frac{1}{x^2} = -\frac{1}{x^2}\][/tex]

Now, we can set up the integral:

[tex]\[Arc\ Length = \int_1^2 \sqrt{1 + \left(-\frac{1}{x^2}\right)^2} \, dx\][/tex]

Simplify the expression inside the square root:

[tex]\[Arc\ Length = \int_1^2 \sqrt{1 + \frac{1}{x^4}} \, dx\][/tex]

This integral does not have a simple closed-form solution, so you may need to use numerical methods or a calculator to approximate the value of the arc length.

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Answer:

sin C = 7/27, cos C = 24/27 and tan C = 7/24

Step-by-step explanation:

sin C = opposite/hypotenuse

         = 7/27

cos C = adjacent/hypotenuse

          = 24/27

tan C = opposite/ adjacent

         = 7/24

Answer:

sin c - 25/7

cos c - 7/24

tan c - 25/24

Step-by-step explanation:

Answer:

3.2 pi

Step-by-step explanation:

Given 3 curves are:

y = square root ( x / 2)

y = 0

x = 2

Use washer method for hollow volumes.

Step 1: Compute A (y)

A ( y ) = pi * ( f_1 (y) ^2 - f_2 (y) ^2)

where,

f_1 (y) is the function further away from y axis

f_2 (y) is the function closer to y axis

f_1 (y) = 2

f_2 (y) = 2*y^2

A ( y ) = pi * ( 2 ^2 - (2*y) ^2)

A (y) = pi * (4 - 4*y^2)

A (y) = 4*pi * (1 - y^2)

Step 2: Compute V (y)

[tex]V = \int\limits^1_0 {A (y)} \, dy \\V = 4*pi\int\limits^1_0 {1 - y^2} \, dy\\\\V = 4 * pi* (y - 0.2 y^5) \limits^1_0\\\\V = 4*pi*(1 - 0.2)\\\\V = 3.2 pi[/tex]

Answer: V = 3.2 pi

Answer:

a) The physicians since we want to measure about behaviors in the natural clinical settings from successful physicians. And the obervational unit would be the patients from these physicians.

b) They have in total 10 physicians who have been rated highly by patients but they use random sampling and they select just 5 physician-patient encounters from the 10 physicians

c) For this case based on the survey, we only have 10 doctors who have been highly qualified by the patients, so our goal is to analyze the information on these 10, and the sample size is 5. and that represent 50% of the original objective, so in this case we can say that this sample size would be sufficient to extrapolate the sample size information from 5 to the total number of doctors 10 of interest.

Step-by-step explanation:

Assuming this complete question: "Suppose you are interested in the behaviors of physicians that have high ratings of patient satisfaction. The research goal is to identify the behaviors in the natural clinical settings of these successful physicians so that these behaviors can be built into the curricula of medical preparation programs. The main data were collected by the video recording of five randomly selected physician-patient encounters from 10 physicians who have been rated highly by patients in a reliable satisfaction survey. In analyzing the recordings, what would you define as the unit for analysis? Why? How many data units (in rough estimates) are you likely to get based on this decision? Does the estimated number of data units seem adequate? Why or why not? "

In analyzing the recordings, what would you define as the unit for analysis? Why?

The physicians since we want to measure about behaviors in the natural clinical settings from successful physicians. And the obervational unit would be the patients from these physicians.

How many data units (in rough estimates) are you likely to get based on this decision?

They have in total 10 physicians who have been rated highly by patients but they use random sampling and they select just 5 physician-patient encounters from the 10 physicians

Does the estimated number of data units seem adequate? Why or why not?

For this case based on the survey, we only have 10 doctors who have been highly qualified by the patients, so our goal is to analyze the information on these 10, and the sample size is 5. and that represent 50% of the original objective, so in this case we can say that this sample size would be sufficient to extrapolate the sample size information from 5 to the total number of doctors 10 of interest.

Answer:

Step-by-step explanation:

Triangle RST is a right angle triangle.

From the given right angle triangle

RT represents the hypotenuse of the right angle triangle.

With 26 degrees as the reference angle,

ST represents the adjacent side of the right angle triangle.

RS represents the opposite side of the right angle triangle.

1) To determine RS, we would apply trigonometric ratio

Sin θ = opposite side/hypotenuse Therefore,

Sin 26 = RS/9.1

RS = 9.1Sin26 = 9.1 × 0.4384

RS = 4.0

2) To determine ST, we would apply trigonometric ratio

Cos θ = adjacent side/hypotenuse Therefore,

Cos 26 = ST/9.1

ST = 9.1Cos26 = 9.1 × 0.8988

ST = 8.1

3) The sum of the angles in a triangle is 180 degrees. Therefore,

∠R + 26 + 90 = 180

∠R = 180 - (26 + 90)

∠R = 64 degrees

Answer:

1) The equation of a line is  [tex]y=-\frac{1}{2}x+\frac{17}{2}[/tex].

2) The equation of the line that passes through the points (2,4) and (3,7) is [tex]y=3x-2[/tex]

3) The equation of the line that has slope m = −2 and y-intercept equal to -1 is [tex]y=-2x-1[/tex]

4) The slope-intercept form of [tex]y-7=0[/tex] is [tex]y=7[/tex], where the slope is zero and the y-intercept is (0, 7).

Step-by-step explanation:

1) The equation of a line with slope m, passing through the point [tex](x_1,y_1)[/tex], is

[tex]y-y_1=m(x-x_1)[/tex]

We know that [tex]m = -\frac{1}{2}[/tex] and the point is (1,8). Therefore, the equation of the line is

[tex]y-8=-\frac{1}{2} (x-1)\\\\y-8+8=-\frac{1}{2}\left(x-1\right)+8\\\\y=-\frac{1}{2}x+\frac{1}{2}+8\\\\y=-\frac{1}{2}x+\frac{17}{2}[/tex]

2) The equation of a line is typically written as

[tex]y=mx+b[/tex]

where m is the slope and b is the y-intercept.

The slope of a line is a measure of how fast the line "goes up" or "goes down" and is given by

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

To find the equation of the line that passes through the points (2, 4) and (3, 7), the first step is to find the slope.

Applying the definition of the slope, we get that

[tex]m=\frac{7-4}{3-2}=3[/tex]

Now, we find the y-intercept with the help of point (2, 4) and the general form of the equation of a line

[tex]4=3(2)+b\\b=-2[/tex]

The equation of the line that passes through the points (2,4) and (3,7) is [tex]y=3x-2[/tex]

3) The equation of the line that has slope m = −2 and y-intercept equal to -1 is

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

4) The slope-intercept form of [tex]y-7=0[/tex] is [tex]y=7[/tex], where the slope is zero and the y-intercept is (0, 7).

Final answer:

An equation of a line through the point (1, 8) with a slope of -1/2 is y - 8 = (-1/2)(x - 1). A line with a slope of -2 and a y-intercept of -14 is represented by y = -2x - 14. The equation y - 7 = 0 represents a horizontal line with a slope of 0 and a y-intercept of (0, 7).

Explanation:

Equations of Lines: Slope and Y-Intercept

To find an equation of a line that passes through the point (1, 8) with a slope m of -1/2, we use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the given point. Plugging in the values, we get:

y - 8 = (-1/2)(x - 1)

For a line with a slope m of −2 and a y-intercept b of −14, we can write the equation directly in slope-intercept form, which is y = mx + b. Therefore, the equation is:

y = -2x − 14

For the equation y − 7 = 0, this is a horizontal line where the slope is 0 because there's no change in y as x varies. The y-intercept of this line is (0, 7), as the line crosses the y-axis at y = 7.

Answer:

Step-by-step explanation:

Hello!

Three samples of components manufactured are taken per day. They are classified as:

D: "Conforming (suitable for its use)"

E: "Downgraded (unsuitable for the intended purpose but usable for another purpose)"

F: "Scrap (not usable)"

This classification includes the three events that may occur in your sample space S. The experiment consists in recording the categories of the three parts tested in a day.

a. List the 27 outcomes in the sample space.

The possible outcomes in the space sample are the combinations of the three events. To avoid using the same letters as in the following questions I've named the evets as D, E, and F

S={DDD, DED, DFD, DEF, DFE, DEE, DFF, DDE, DDF , EDE, EEE, EFE, EED, EEF, EDF, EFD, EDD, EFF , FDF, FEF, FFF, FFE, FFD, FDE, FED, FDD, FEE}

b. Let A be the event that all the parts fall into the same category. List the outcomes in A.

A: "All the parts fall into the same category"

You have three possible outcomes for this event, that the three compounds are conforming, "DDD", that the three are unconforming, "EEE", or that the three compounds are scrap, "FFF". There are only three possible outcomes for this event.

S={DDD, EEE, FFF}

c. Let B be the event that there is one part in each category. List the outcomes in B.

B: "There is a part in each category"

This means, for example, The first one is conforming "D", the second one is unconforming "E" and the third one is scrap "F", then the first one may be unconforming "E", the second one is conforming "D" and the thirds one is scrap "F", and so on, you have 6 possible outcomes for this event:

S={DEF, DFE, EDF, EFD, FDE, FED}

d. Let C be the event that at least two parts are conforming. List the outcomes in C.

C: "At least two parts are conforming"

For this event, you can have two of the compounds to be considered conforming or the three of them.

S={DDD, DED, DFD, DDE, DDF , EDD, FDD}

A total of 7 combinations fit this event.

I hope you have a SUPER day!

Answer:

The world population in 2030 will be of 9.0062 billion.

The would population in 2060 will be of 13.71 billion.

Step-by-step explanation:

The exponential model for population growth is as follows.

[tex]P(t) = P(0)e^{rt}[/tex]

In which P(t) is the population in t years from now, P(0) is the population in the current year and r(decimal) is the growth rate.e = 2.71 is the Euler number.

If the world population is 7.0 billion in 2012.

2012 is the initial year, so P(0) = 7.

P(t) will be measured in billions of people.

The growth rate is constant at 1.4%.

This means that [tex]r = 0.014[/tex]

Calculate the population in 2030.

2030 is 2030-2012 = 18 years after 2012, so this is P(18).

[tex]P(t) = 7e^{rt}[/tex]

[tex]P(18) = 7e^{0.014*18} = 9.0062[/tex]

So the world population in 2030 will be of 9.0062 billion.

What will be the population in 2060.

This is 2060-2012 = 48 years after 2012. So this is P(48).

[tex]P(t) = 7e^{rt}[/tex]

[tex]P(48) = 7e^{0.014*48} = 13.71[/tex]

The would population in 2060 will be of 13.71 billion.

Answer:

187 trees

Step-by-step explanation:

50 trees per acre in an orchard and

the orchard is 1.7 x 2.2 km,

we calculate the orchard dimension = 1.7 x 2.2km = 3.74

to calculate how many trees are in the entire orchard = 3.74 x 50 = 187 trees

The number of trees in the entire orchard will be 46,208.

The analysis of mathematical representations is algebra, and the handling of those symbols is logic.

PEMDAS rule means for the Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This rule is used to solve the equation in a proper and correct manner.

If there are 50 trees per acre in an orchard and the orchard is 1.7 x 2.2 km.

The area of the entire orchard will be

A = 1.7 x 2.2

A = 3.74 square km

We know that 1 square km = 247.105 acre

A = 3.74 x 247.105

A = 924.17

Then the number of trees in the entire orchard will be

⇒ 50 x 924.17

⇒ 46,208

The number of trees in the entire orchard will be 46,208.

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Answer:

a) FFFF, FVFF, FFVF, FFFV,

FVFV, FFVV,FVVF,VVFF,

VFVF, VFFF,VFFV,FFVV

VVVV, VFVV, VVFV, VVVF

b) FVFF, FFVF, FFFV,VFFF

c) VVVV or FFFF

d) FFFF, FVFF, FFVF, FFFV, VFFF

e) FFFF, FVFF, FFVF, FFFV ,VFFF, VVVV

Step-by-step explanation:

For this case we define some notation:

F= mortgage classified as fixed rate

V= mortgage classified as variable rate

We select a sample of 4 mortgages.

Part a

We have 2*2*2*2= 16 possible combinations defined below:

FFFF, FVFF, FFVF, FFFV,

FVFV, FFVV,FVVF,VVFF,

VFVF, VVFF,VFFV,FFVV

VVVV, VFVV, VVFV, VVVF

Part b

Which outcomes are in the event that exactly three of the selected mortgages are fixed rate

We need to see in the possible outcomes from part a) how many we have exactly three F's .If we analyze the possible options the possible combinations are:

FVFF, FFVF, FFFV, VFFF

Part c

Which outcomes are in the event that all four mortgages are of the same type?

For this case we have just two possible values: VVVV or FFFF

Part d

Which outcomes are in the event that at most one of the four is a variable ­rate mortgage?

We need to see in the possible outcomes from part a) how many have at least one V. After analyze we see that the possible values:

FFFF, FVFF, FFVF, FFFV, VFFF

Part e

The union represent all four mortgages are of the same type or outcomes are in the event that at most one of the four is a variable ­rate. So we are looking for the possible outcomes VVVV and FFFF and the outcomes with just one V ( FVFF, FFVF, FFFV ,VFFF) so then the union would be:

FFFF, FVFF, FFVF, FFFV ,VFFF, VVVV