solve the following initial value problems: dy /dt = − 2t a. y(0) = − 1 : y = b. y(0) = 5 : y =

These statistical measures help the researcher summarize, describe, and understand the data collected from the study participants.

As the researcher is describing the characteristics of the subjects, the appropriate statistical analysis would be descriptive statistics.

Descriptive statistics is a method that summarizes and describes the data collected in a study. It includes measures such as mean, median, mode, standard deviation, range, and frequency distribution.

These measures can provide a clear picture of the demographic data of the three groups of subjects and help in drawing meaningful conclusions.

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The personnel director in this scenario is performing a convenience sampling.

Convenience sampling is a type of non-probability sampling method where the researcher selects participants based on their accessibility and proximity to the researcher. In this case, the personnel director chose employees from the department nearest her office for the study.

This approach may not provide a fully representative sample of the entire company's employee population, as it doesn't account for possible differences in eating habits across various departments. Therefore, the findings from this study might not accurately represent the proportion of employees who buy lunch in the cafeteria, bring their own lunches, or go out to lunch for the entire company.

To obtain more accurate results, the director might consider employing a probability sampling method, such as simple random sampling or stratified sampling, to ensure a more representative sample of the company's employees.

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To find the cost of operating a 200-W lamp continuously for 1 week, we need to first calculate the amount of energy consumed by the lamp.

Energy consumed (in kWh) = Power (in W) x Time (in hours) / 1000

In this case, the power of the lamp is 200 W and the time it is used for is 1 week, which is 7 days x 24 hours/day = 168 hours.

Energy consumed = 200 x 168 / 1000 = 33.6 kWh

Now, to find the cost of this energy, we multiply the energy consumed by the power utility rate of 16 cents/kWh.

Cost of operating the lamp for 1 week = Energy consumed x Power utility rate
= 33.6 x 0.16
= $5.38

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The correlation is statistically significant, it is not a perfect relationship and there may be other factors at play that affect academic performance. Overall, the results suggest that studying for more hours per week may lead to better academic performance

Based on the given information, we can conclude that there is a positive correlation between the number of hours studied per week and academic performance of 25 UK University students. The correlation coefficient of 0.71 suggests a moderate to strong positive relationship between the two variables. This means that as the number of hours studied per week increases, the academic performance of the students also tends to increase. However, it is important to note that the "critical r" value of 0.87 with a significance level of p≤0.05 indicates that there is a chance of 5% that the observed correlation between the variables could be due to random chance. This means that while the correlation is statistically significant, it is not a perfect relationship and there may be other factors at play that affect academic performance. Overall, the results suggest that studying for more hours per week may lead to better academic performance, but it is not the only factor that contributes to success. Other variables such as natural ability, motivation, and study habits may also play a role in academic achievement.

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The total number of beads in the necklace is 1083. We can see that there are 3 beads in the first row, 10 beads in the second row, 17 beads in the third row, and so on until there are 122 beads in the last row.

To write the given series in summation notation, we can observe that each term in the series can be obtained by adding 7 to the previous term, starting with the first term 3. So, the nth term of the series can be given by:

[tex]a_n = 3 + (n-1)7[/tex]

= 7n - 4

The lower limit of the sum is the first term, which is a_1 = 3. The upper limit of the sum is the 18th term, which is [tex]a_{18} = 7(18) - 4 = 122[/tex]. Therefore, the series can be written in summation notation as:

[tex]$\sum_{n=1}^{18} (7n - 4)$[/tex]

The explicit formula for the sum of this series can be obtained using the formula for the sum of an arithmetic series:

[tex]$S_n = \frac{n}{2}(2a_1 + (n-1)d)$[/tex]

where S_n is the sum of the first n terms of the series, a_1 is the first term, and d is the common difference.

Substituting the values, we get:

[tex]$S_{18} = \frac{18}{2}(2(3) + (18-1)7)$[/tex]

= 9(6 + 119)

= 1083

Adding up all these beads gives us the total number of beads in the necklace.

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The probability that both Product A and Product B will be successful is 0.48 or 48%.

To calculate the probability that both Product A and Product B will be successful, we need to multiply the individual probabilities of success for each product, since their success is independent of each other.

Step 1: Identify the probability of success for Product A and Product B.
- Product A: 60% (0.6)
- Product B: 80% (0.8)

Step 2: Multiply the probabilities of success for both products.
- Probability of both being successful = (Probability of Product A success) × (Probability of Product B success)
- Probability of both being successful = (0.6) × (0.8)
P(A and B) = P(A) x P(B)

P(A) = 0.6 (given in the information)

P(B) = 0.8 (given in the information)

P(A and B) = 0.6 x 0.8 = 0.48

Step 3: Calculate the result.
- Probability of both being successful = 0.48

Therefore, the probability that both Product A and Product B will be successful is 48%.

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The action that involves reducing the impact of a risk event by reducing the probability of its occurrence is called risk mitigation. This involves identifying potential risks that could impact a project, product or organization and taking proactive steps to reduce the likelihood of those risks occurring. Risk mitigation can involve a range of activities, such as implementing safety procedures, conducting regular inspections and maintenance, investing in new technology or tools, improving communication and collaboration, and providing ongoing training and education to employees.

By reducing the probability of a risk event occurring, organizations can minimize the potential impact of those events and avoid the costly consequences of downtime, lost productivity, reputational damage, legal liability, and other negative outcomes. Effective risk mitigation requires a comprehensive and proactive approach that involves ongoing monitoring and evaluation of potential risks, as well as continuous improvement efforts to address emerging threats and challenges. Overall, risk mitigation is a critical component of any successful risk management strategy and helps to ensure the long-term success and sustainability of an organization.

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Meta-analysis is a research technique that involves systematically reviewing and statistically synthesizing data from multiple studies on a particular research question or topic.

The goal of meta-analysis is to provide a comprehensive summary of the existing evidence by combining the results of individual studies, which may have produced conflicting or inconclusive findings.

Meta-analysis typically involves a systematic search for relevant studies, followed by an evaluation of their quality and an extraction of relevant data. The data from the studies are then combined using statistical methods to produce an overall effect size estimate, such as a mean difference or odds ratio.

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The general solution for Py"(t) + 9ry'(t) + 17y(t) + 3ty'(t) + 5y(t) = 0 is y(t) = c1 e^(r1t/P) + c2 e^(r2t/P), where c1 and c2 are constants and r1 and r2 are the roots found above.


15. To find the general solution for y'(t) - y(t) + 5y(t) = 0, we first need to find the characteristic equation. The characteristic equation is r - 1 + 5 = 0, which simplifies to r + 4 = 0. Therefore, the characteristic roots are r1 = -4.

The general solution for y'(t) - y(t) + 5y(t) = 0 is y(t) = c1 e^(-4t), where c1 is a constant.

16. To find the general solution for y"(t) + 3ty'(t) + 6y(t) = 0, we first need to find the characteristic equation. The characteristic equation is r^2 + 3tr + 6 = 0, which has the roots r1 = (-3t + i sqrt(3))/2 and r2 = (-3t - i sqrt(3))/2.

The general solution for y"(t) + 3ty'(t) + 6y(t) = 0 is y(t) = c1 e^((-3t + i sqrt(3))/2) + c2 e^((-3t - i sqrt(3))/2), where c1 and c2 are constants.

17. To find the general solution for Py"(t) + 9ry'(t) + 17y(t) + 3ty'(t) + 5y(t) = 0, we first need to rearrange the equation as Py"(t) + (3t + 9r)y'(t) + (17 + 5P)y(t) = 0.

Then, we find the characteristic equation, which is Pr^2 + (3t + 9r)r + (17 + 5P) = 0. The roots of this equation are given by the quadratic formula:

r1 = (-3t - 9r + sqrt((3t + 9r)^2 - 4P(17 + 5P)))/(2P)
r2 = (-3t - 9r - sqrt((3t + 9r)^2 - 4P(17 + 5P)))/(2P)

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The point that symbolizes the shared point of various sets is the intersection of sets.

The intersection of sets A and B is represented by region II.

We deduce from the query that sets A and B share a set element at II.

This means that Region II is the intersection of both sets.

Hence, the region that represents the intersection of sets A and B is II.

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D. The t-distribution has a larger variance than the standard normal distribution. The t-distribution and the standard normal (z) distribution are both probability distributions used in statistical analyses. The main difference between them lies in their variance.

The t-distribution has a larger variance compared to the standard normal distribution, particularly when the sample size is small or the degrees of freedom are low. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution, and their variances become more similar.

In contrast to the other options:
A. Both t-distribution and standard normal distribution can be calculated with or without a known standard deviation, depending on the context.
B. The confidence intervals for the t-distribution are generally wider than those for the standard normal distribution, especially when sample sizes are small.
C. The standard normal distribution is not dependent on parameters like degrees of freedom, while the t-distribution is.

Therefore, option D is the correct answer as it highlights the key difference between the t-distribution and the standard normal distribution in terms of variance.

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There are 90 ways that a coach can select a starting team of one center, two forwards, and two guards if the basketball team consists of three centers, five forwards, and three guards.

To determine the number of ways to select a starting team, we first need to choose one center out of three centers, which can be done in 3 ways.

Then we need to choose two forwards out of five forwards, which can be done in 5C2 ways, where 5C2 represents the number of combinations of 2 items that can be chosen from a set of 5 items.

Finally, we need to choose two guards out of three guards, which can be done in 3C2 ways.

Using the multiplication principle, the total number of ways to select a starting team is:

3 x 5C2 x 3C2 = 3 x 10 x 3 = 90

Therefore, there are 90 ways.

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There are 604800 number of ways to paint seven rooms with different color

The number of ways to paint seven rooms such that no two rooms have the same color, given 10 different color paints, can be found using the permutation formula:

nPr = n! / (n-r)!

where n is the total number of options (in this case, the 10 different colors available) and r is the number of options chosen (in this case, the 7 rooms being painted).

Therefore, the number of ways to paint seven rooms with different colors can be calculated as:

10P7 = 10! / (10-7)! = 604800

So, there are 604800 ways to paint seven rooms with different colors if 10 different color paints are available.

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Similar to inference techniques used in earlier chapters regarding sample sizes, the conditions that allow us to use the Chi-square test protect us from having expected counts that are too small.

When conducting a Chi-square test, we are comparing observed counts to expected counts, and if the expected counts are too small, we may not have enough data to accurately determine if the observed counts are significantly different from what we would expect by chance alone. Therefore, one of the assumptions of the Chi-square test is that the expected counts in each cell of the contingency table should be at least 5, although some researchers suggest that expected counts should be at least 10 to increase the accuracy of the test.

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The probability of drawing a 3 or a heart from a standard deck of cards is 1/4 or 25%.

There are four suits in a standard deck of cards: Hearts, Clubs, Diamonds, and Spades. Each suit contains 13 cards with face values 2 through 10, plus a Jack, Queen, King, and Ace.

Since there are 13 hearts in a standard deck of cards, the probability of drawing a heart is 13/52 or 1/4.

There are four 3s in a standard deck of cards, one in each suit. Since we have already counted the 3 of hearts as a heart, there are three remaining 3s that are not hearts. So the probability of drawing a 3 is 3/52.

To find the probability of drawing a 3 or a heart, we add the probabilities of drawing a heart and drawing a 3, but then we need to subtract the probability of drawing the 3 of hearts twice (because it is both a heart and a 3) to avoid double-counting. So the probability of drawing a 3 or a heart is:

P(3 or Heart) = P(Heart) + P(3) - P(3 of Hearts)

= 1/4 + 3/52 - 1/52

= 13/52

= 1/4

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A watch that costs $12 US dollars would cost $14.40 Canadian dollars, option d is correct.

To convert US dollars to Canadian dollars using the exchange rate of 1.2 Canadian dollars per U.S. dollar, we multiply the amount of US dollars by the exchange rate:

$12 US dollars x 1.2 Canadian dollars per U.S. dollar = $14.40 Canadian dollars

An exchange rate is the rate at which one currency can be exchanged for another currency.

In this case, the exchange rate of 1.2 Canadian dollars per U.S. dollar means that for every U.S. dollar, you can exchange it for 1.2 Canadian dollars.

Therefore, a watch that costs $12 US dollars would cost $14.40 Canadian dollars at an exchange rate of 1.2 Canadian dollars per U.S. dollar.

The answer is option D, $14.40 Canadian dollars.

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Since the variable of interest is the proportion of employees who participate in volunteer activities, the data is categorical (either "participates" or "does not participate").  Option A is Correct.

To find the appropriate inferential statistic for this scenario, we need to consider two factors: the type of data and the sample size.

Additionally, since the sample size is relatively large (n = 60), we can assume that the sampling distribution of the proportion is approximately normal, even if the population distribution is not.

Based on these factors, the appropriate inferential statistic for this scenario would be a confidence interval for the population proportion. This would allow us to estimate the range of plausible values for the true proportion of employees who participate in volunteer activities, based on the sample data.

We could also calculate a hypothesis test to determine if the sample proportion is significantly different from a hypothesized population proportion, but this would require us to specify a null hypothesis and alternative hypothesis, and to choose a significance level. Without more information about the research question and goals of the survey, it is difficult to determine if a hypothesis test would be appropriate or necessary. Option A is Correct.

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Correct Question:

A bank surveyed all of its 60 employees to determine the proportion who participate in volunteer activities. Which of the following statements is true?

A. The bank should not use the data from this survey because this is an observational study.

B. The bank can use the result of this survey to prove that working for the bank causes employees to participate in volunteer activities.

C. The bank did not select a random sample of employees, so the survey will not provide the bank with useful information.

D. The bank would have to use the survey data to construct a confidence interval in order to estimate the proportion of employees who participate in volunteer activities.

E. The bank does not need to use an inference procedure to determine the proportion of employees who participate in volunteer activities because the survey was a census of all employees.

If David can unload a delivery truck in 20 minutes. Allie can unload the same truck in 35 minutes. If they work together, the time it will take to unload the truck  is: 15.56 minutes .

Since David can unload the truck in 20 minutes or 1/20 of the job, in just one minute. Similar to Allie she can finish 1/35 of the task in under one minute.

Both of them will collectively finish a portion of the work in one minute that is equal to the sum of their individual rates:

1/20 + 1/35

= 9/140

Together they will unload the vehicle in the following locations:

1 / (9/140)

= 15.56 minutes

Therefore the time is  15.56 minutes .

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In order to calculate the standard error of the sample mean, you'll need to use the following formula:

Standard Error (SE) = Standard Deviation (SD) / √(Sample Size)

In this case, you have the following information:
- The average refund in the 2011 tax year, according to the IRS, was $3,109 (this is not directly needed for the calculation).
- The standard deviation for these refunds is $874.
- The random sample size is 50 tax returns.

Now, plug these values into the formula:

SE = 874 / √(50)

SE ≈ 874 / 7.071

SE ≈ 123.64

The standard error of the sample mean for a random sample of 50 tax returns is approximately $123.64.

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Both equilibria for the given range of values of h are stable.

We are given that;

Nt+1 = 2:5Nt/1+Nt - hNt

Now,

To decide if these equilibria are stable or unstable, we need to find the slope of the model function at these equilibria and compare it to 1 in absolute value. The slope of the model function is given by:

dNt+1/dNt = [2.5 / (1 + Nt)^2] - h

At Nt = 0, we have:

dNt+1/dNt = [2.5 / (1 + 0)^2] - h dNt+1/dNt = 2.5 - h

This slope is less than 1 in absolute value for any value of h between 0 and 3.5, so this equilibrium is stable for those values of h.

At Nt ≈ 3.13, we have:

dNt+1/dNt = [2.5 / (1 + 3.13)^2] - 0.79 dNt+1/dNt ≈ -0.19

This slope is also less than 1 in absolute value, so this equilibrium is also stable.

Therefore, by the given slope the answer will be stable.

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The statistical method used in this example is likely to be survey sampling, where the piano teacher collects data on the most popular lesson.

Books among their students and uses this information to make an informed purchasing decision. The teacher may also use descriptive statistics to analyze and summarize the data collected from the survey.
In this example, the piano teacher purchases lesson books that are most popular with students. The statistical method used in this scenario is "mode." Mode refers to the most frequently occurring value in a dataset. In this case, the teacher would determine which lesson books are most popular among students by finding the books with the highest frequency of usage or preference.

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We can reject the null hypothesis at a 5% significance level (α=0.05), since the p-value is less than α.

To find the p-value for the given t-test, we need to look up the t-distribution table with 17 degrees of freedom (df). Since this is a two-tail test, we need to find the area in both tails of the t-distribution that corresponds to a t-value of 2.5 (and the negative of -2.5).

Looking up the t-distribution table with 17 df, we find that the critical t-value at a 5% significance level (α/2) is approximately ±2.110. Since our test statistic t=2.5 falls outside this critical value, the p-value will be less than 0.05.

To find the exact p-value, we can use a t-distribution calculator or statistical software. For a two-tailed t-test with 17 degrees of freedom and a test statistic of t=2.5, the p-value is approximately 0.021. Therefore, we can reject the null hypothesis at a 5% significance level (α=0.05), since the p-value is less than α.

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Jay's net weight change in 6 month is 30 pounds.

a) Given that, in February, the record low temperature for ST.Paul Minnesota was -3° F

In January temperature = -3×6

= -18 F

c) Jay went on a diet and last 5 pounds each month

Jay's net weight change in 6 months = 5×6

= 30 pounds

Therefore, Jay's net weight change in 6 month is 30 pounds.

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Yes, the 5x5 matrix is diagonalizable because it has two eigenvalues with eigenspaces of dimensions 3 and 2.

A matrix is diagonalizable if and only if the sum of the dimensions of its eigenspaces equals its size (the number of rows or columns).

a matrix is a rectangular array of numbers or symbols arranged in rows and columns. Each element in the matrix is

identified by its row and column index. Matrices are commonly used in linear algebra, where they are used to

represent linear transformations and systems of linear equations.

In this case, the three-dimensional eigenspace and the two-dimensional eigenspace have a combined dimension of 5, which is the size of the matrix.

Therefore, the 5x5 matrix is diagonalizable.

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Question

A is a 5 x 5 matrix with two eigenvalues. One eigenspace is three-dimensional, and the other eigenspace is two-dimensional. Is A diagonalizable?

The length of each shorter side is 10 centimeters in the given case.

Let's call the length of the shorter side "x".

The formula for the perimeter (distance around) of a rectangle is:

Perimeter = 2Length + 2Width

We are given that the perimeter is 44 centimeters, and that the length of each longer side is 12 centimeters.

The perimeter of a shape is the distance around its boundary.

The formula for the perimeter of a rectangle is:

Perimeter = 2 * (Length + Width)

where "Length" and "Width" are the dimensions of the rectangle.

The formula for the perimeter of a square is:

Perimeter = 4 * Length

where "Length" is the length of a side of the square.

The formula for the perimeter of a triangle is:

Perimeter = Side1 + Side2 + Side3 So we can plug in these values and solve for the length of the shorter side:

44 = 2(12) + 2x

44 = 24 + 2x

20 = 2x

x = 10

Therefore, the length of each shorter side is 10 centimeters.

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To solve this problem, we need to use the formula:

Distance = Rate x Time

Let's assume that t represents the number of hours that have passed since the trains left the station. We can then set up two equations:

Distance traveled by the southbound train = 62t
Distance traveled by the northbound train = 68t

To find out when the trains will be 455 kilometers apart, we can set up another equation:

Distance between the trains = 455

We can now use substitution to solve for t:

62t + 68t = 455
130t = 455
t = 3.5 hours

Therefore, the trains will be 455 kilometers apart 3.5 hours after they leave the station.

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Answer:

[tex]f(x)=16x^4+96x^{3}+216x^{2}+216x+81[/tex]

Step-by-step explanation:

The function f(x) = (2x + 3)⁴ is a fourth-degree polynomial function.

It can be expanded using the binomial theorem.

[tex]\boxed{\begin{minipage}{5cm} \underline{Binomial Theorem}\\\\$\displaystyle (a+b)^n=\sum^{n}_{k=0}\binom{n}{k} a^{n-k}b^{k}$\\\\\\where \displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!}\\\end{minipage}}[/tex]

Comparing the given function with (a + b)ⁿ:

Substitute these values into the binomial theorem formula:

[tex]\displaystyle (2x+3)^4=\binom{4}{0}(2x)^{4-0}3^{0}+\binom{4}{1}(2x)^{4-1}3^{1}+\binom{4}{2}(2x)^{4-2}3^{2}+\binom{4}{3}(2x)^{4-3}3^{3}+\\\\\\\phantom{wwww}\binom{4}{4}(2x)^{4-4}3^{4}[/tex]

Solve:

[tex]\begin{aligned}\displaystyle (2x+3)^4&=\binom{4}{0}(2x)^4\cdot3^0+\binom{4}{1}(2x)^{3}\cdot3^1+\binom{4}{2}(2x)^2\cdot3^2+\binom{4}{3}(2x)^{1}\cdot3^3+\binom{4}{4}(2x)^0\cdot3^4\\\\&=\binom{4}{0}16x^4\cdot1+\binom{4}{1}8x^3\cdot3+\binom{4}{2}4x^2\cdot9+\binom{4}{3}2x\cdot27+\binom{4}{4}\cdot81\\\\&=\binom{4}{0}16x^4+\binom{4}{1}24x^3+\binom{4}{2}36x^2+\binom{4}{3}54x+\binom{4}{4}81\\\\&=1\cdot16x^4+4\cdot24x^3+6\cdot36x^2+4\cdot54x+1\cdot81\\\\&=16x^4+96x^3+216x^2+216x+81\end{aligned}[/tex]

Therefore, the expanded function is:

[tex]f(x)=16x^4+96x^{3}+216x^{2}+216x+81[/tex]

[tex] \Large{\boxed{\sf F(x) = (2x + 3)^4 = 16x^4 + 96x^3 + 216x^2 + 216x + 81 }} [/tex]

[tex] \\ [/tex]

To expand the given function, we will apply the binomial theorem, which is the following:

[tex]\sf(a+b)^n =\sf\sum\limits_{k=0}^{n} \binom{n}{k}a^{n-k}b^{k} \\ \\ \sf \:Where\text{:} \\ \star \: \sf n \: is \: a \: positive \: integer. \: ( n \in \mathbb{N}) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf \star \: k \: is \: a \: positive \: integer \: less \: than \: or \: equal \: to \: n. \: (k \leqslant n, \: k \: \in \: \mathbb{ N}) \\ \\ \\ \sf \star \: \displaystyle\binom{ \sf \: n}{ \sf \: k} \: \sf is \: a \: \underline{binomial \: coefficient} \: and \: is \: calculated \: as \: follows\text{:} \\ \\ \\ \sf \displaystyle\binom{ \sf \: n}{ \sf \: k} = \sf \dfrac{n! }{(n - k)! k ! }[/tex][tex] \\ \\[/tex]

[tex] \\ [/tex]

[tex] \\ [/tex]

[tex] \sf F(x) = (\underbrace{\sf 2x}_{\sf a} + \underbrace{3}_{\sf b})^{\overbrace{\sf 4}^{n}} \\ \\ \implies \sf a = 2x \: \: ,b = 3 \: \: ,n = 4 [/tex]

[tex] \\ [/tex]

[tex] \\ [/tex]

[tex] \sf (2x + 3)^4 = \displaystyle\sum\limits_{ \sf k=0}^{ \sf 4} \binom{ \sf 4}{ \sf k}( \sf 2x)^{4-k}(3)^{k} \\ \\ \\ \sf = \binom{ \sf 4}{ \sf 0}( \sf 2x)^{4-0}(3)^{0} + \binom{ \sf 4}{ \sf 1}( \sf 2x)^{4-1}(3)^{1} + \binom{ \sf 4}{ \sf 2}( \sf 2x)^{4-2}(3)^{2} + \binom{ \sf 4}{ \sf 3}( \sf 2x)^{4-3}(3)^{3} + \binom{ \sf 4}{ \sf 4}( \sf 2x)^{4-4}(3)^{4} \\ \\ \\ \sf = \binom{ \sf 4}{ \sf 0}( \sf 2x)^{4}(3)^{0} + \binom{ \sf 4}{ \sf 1}( \sf 2x)^{3}(3)^{1} + \binom{ \sf 4}{ \sf 2}( \sf 2x)^{2}(3)^{2} + \binom{ \sf 4}{ \sf 3}( \sf 2x)^{1}(3)^{3} + \binom{ \sf 4}{ \sf 4}( \sf 2x)^{0}(3)^{4} \\ \\ \\ \sf = \binom{ \sf 4}{ \sf 0}( \sf 16 {x}^{4} ) + \binom{ \sf 4}{ \sf 1}( \sf 24 {x}^{3}) + \binom{ \sf 4}{ \sf 2}( \sf 36x^{2}) + \binom{ \sf 4}{ \sf 3}( \sf 54x) + \binom{ \sf 4}{ \sf 4}(81) [/tex]

[tex] \\ [/tex]

[tex] \\ [/tex]

[tex] \\ \star \: \displaystyle\binom{ \sf 4}{\sf \: 0} = \sf \dfrac{4! }{(4-0)!0 ! } = \dfrac{4!}{4!0!} = \dfrac{4!}{4!} = \boxed{\sf 1} \\ \\ \star\:\displaystyle\binom{ \sf 4 }{ \sf \: 1} =\sf \dfrac{4! }{(4 - 1)!1 ! } = \dfrac{4!}{3!1!}= \dfrac{2\times 3 \times 4}{2 \times 3} = \boxed{\sf 4} \\ \\ \star \: \displaystyle\binom{ \sf 4 }{ \sf \: 2} =\sf \dfrac{4! }{(4-2)!2!} = \dfrac{4!}{2!2!}=\dfrac{2 \times 3 \times 4}{2 \times 2} = \boxed{\sf 6} \\ \\ \star \:\displaystyle\binom{ \sf 4 }{ \sf \: 3}= \sf \dfrac{4! }{(4 - 3)!3!} =\dfrac{4!}{1!3!} = \dfrac{2 \times 3 \times 4}{2 \times 3 } = \boxed{\sf 4}\\ \\ \star \:\displaystyle\binom{ \sf 4 }{ \sf \: 4}= \sf \dfrac{4! }{(4 - 4)!4 !} =\dfrac{4!}{0!4!} = \dfrac{2 \times 3 \times 4}{2 \times 3 \times 4 } = \boxed{\sf 1} [/tex]

[tex] \\ [/tex]

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[tex] \sf (2x + 3)^4 = \binom{ \sf 4}{ \sf 0}( \sf 16 {x}^{4} ) + \binom{ \sf 4}{ \sf 1}( \sf 24 {x}^{3}) + \binom{ \sf 4}{ \sf 2}( \sf 36x^{2}) + \binom{ \sf 4}{ \sf 3}( \sf 54x) + \binom{ \sf 4}{ \sf 4}(81) \\ \\ \\ \sf = (1)(16x^4) + (4)(24x^3) + (6)(36x^2) + (4)(54x) + (1)(81) \\ \\ \\ \boxed{\boxed{\sf = 16x^4 + 96x^3 + 216x^2 + 216x + 81}} [/tex]

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Thus, the probability of rolling one 3 and two 2s when rolling a six-sided die three times and obtaining a sum of 7 is 1/72.

To solve this problem, we need to use the concept of probability. The probability of rolling a certain number on a six-sided die is 1/6. We can use this information to determine the probability of rolling a specific combination of numbers on three rolls of the die.

First, let's consider the total number of possible outcomes when rolling a six-sided die three times. Each roll has six possible outcomes, so there are a total of 6 x 6 x 6 = 216 possible outcomes when rolling the die three times.

Next, we need to determine how many of these outcomes result in a sum of 7. To do this, we can use a table to list all of the possible combinations of three rolls that add up to 7:

1-2-4
1-3-3
1-4-2
1-5-1
2-1-4
2-2-3
2-3-2
2-4-1
3-1-3
3-2-2
3-3-1
4-1-2
4-2-1
5-1-1

There are 14 possible combinations that add up to 7.

Now, we need to determine how many of these combinations consist of one 3 and two 2s. There are three possible ways that this can occur:

2-2-3
2-3-2
3-2-2

Therefore, there are three outcomes that result in one 3 and two 2s.

Finally, we can determine the probability of rolling one 3 and two 2s by dividing the number of outcomes that meet the desired criteria (three) by the total number of possible outcomes (216):

P(one 3 and two 2s) = 3/216 = 1/72

Therefore, the probability of rolling one 3 and two 2s when rolling a six-sided die three times and obtaining a sum of 7 is 1/72.

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we need to use the formula for the rate of change of distance between two moving objects: rate of change of distance = √[(rate of object 1)^2 + (rate of object 2)^2].

At noon, ship A is 10 nautical miles due west of ship B. Let's assume that ship A is at position (0,0) and ship B is at position (10,0) on a coordinate plane. Ship A is sailing west at 19 knots, which means its position at 7 PM is (-7*19,0) = (-133,0) miles from its starting point.

Ship B is sailing north at 15 knots, which means its position at 7 PM is (10,7*15) = (10,105) miles from its starting point.

Using the distance formula, we can find the distance between the two ships at noon: distance = √[(10-0)^2 + (0-0)^2] = √100 = 10 miles.

Using the formula for the rate of change of distance, we can find the rate at which the distance between the two ships is changing at 7 PM: rate of change of distance = √[(19)^2 + (15)^2] = √(361 + 225) = √586 = 24.18 knots, Therefore, the distance between the ships is changing at a rate of 24.18 knots at 7 PM.

At noon, the distance between Ship A and Ship B is 10 nautical miles. Ship A is sailing west at 19 knots, and Ship B is sailing north at 15 knots.

From noon to 7 PM, there are 7 hours of sailing. During this time, Ship A travels 7 hours * 19 knots/hour = 133 nautical miles west.

Ship B travels 7 hours * 15 knots/hour = 105 nautical miles north. Now, we can use the Pythagorean theorem to find the new distance between the ships: Distance^2 = (10 + 133)^2 + (105)^2, Distance^2 = 143^2 + 105^2, Distance = √(20449 + 11025) = √31474.

We know the rates at which the ships are moving west and north, so we can find d(Distance^2)/dt: d(Distance^2)/dt = 2*(10 + 133)*(-19) + 2*(105)*(15) = -5746 + 3150 = -2596,

Now, we can solve for the rate at which the distance is changing: d(Distance)/dt = -2596 / (2 * √31474) ≈ -0.73 knots, The distance between the ships is decreasing at a rate of approximately 0.73 knots at 7 PM.

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