What is the y-value of the vertex?

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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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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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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The root-mean-square of this group of molecules can be calculated using the formula: RMS = √[(16² + 17² + 18² + 19² + 20² + 21² + 22² + 23² + 24² + 25² + 26²)/11].

RMS = √[6726/11] = √611.45 = 24.7 m/s (rounded to one decimal place)

Therefore, the root-mean-square of this group of molecules is 24.7 meters per second.
To calculate the root-mean-square (RMS) speed of the group of molecules, follow these steps:

1. Square each speed: 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676
2. Find the average of these squared speeds: (256 + 289 + 324 + 361 + 400 + 441 + 484 + 529 + 576 + 625 + 676) / 11 = 4951 / 11 = 450.091

3. Take the square root of the average: √450.091 ≈ 21.2 m/s, So, the root-mean-square speed of this group of molecules is approximately 21.2 m/s with one decimal place.

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Applying the Pythagorean, the length of the diagonal of the square is calculated as: x ≈ 4.2 [This is between 4 and 5].

To find the length of the diagonal of the square, apply the Pythagorean theorem which states that: c² = a² + b², where c is the diagonal and a and b is other legs.

Given the following:

a = 3

b = 3

x = ?

Plug in the values:

x² = 3² + 3²

x² = 18

x = √18

x ≈ 4.2

The value of x is between 4 and 5.

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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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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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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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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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The expected value E(X) for this distribution is 1.0, which corresponds to option B.

To find the expected value E(X) for the given discrete random variable X, we need to multiply each value of X by its corresponding probability and then sum up the products. Here's the calculation:

E(X) = (0 * 0.4) + (1 * 0.3) + (2 * 0.2) + (3 * 0.1)
E(X) = (0) + (0.3) + (0.4) + (0.3)
E(X) = 1.0

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There is insufficient evidence to reject the null hypothesis, and we cannot conclude that the actual overall mean diameter of the ball bearings is not 4.300 mm.

The standard deviation (s) of the ball bearings' diameters is given as 0.005 mm, indicating the variability in the measurements. The overall mean diameter (µ) is specified as 4.300 mm. A sample of 81 ball bearings (n) has a sample mean of 4.299 mm. To determine whether there's reason to believe that the actual overall mean diameter is not 4.300 mm, we need to conduct a hypothesis test.

We begin with stating the null hypothesis (H₀) as: µ = 4.300 mm, and the alternative hypothesis (H₁) as: µ ≠ 4.300 mm. To conduct the hypothesis test, we can use the Z-test since the sample size is large (n ≥ 30). The Z-test statistic is calculated as:

Z = (sample mean - µ) / (s / √n)

Plugging in the values:

Z = (4.299 - 4.300) / (0.005 / √81) ≈ -1.8

Now, we need to find the p-value associated with this Z-score. The p-value helps us to determine the likelihood of observing a sample mean as extreme as 4.299 mm, given that the null hypothesis is true. A low p-value (typically, p < 0.05) would indicate that there is evidence to reject the null hypothesis in favor of the alternative hypothesis.

In this case, the p-value associated with a Z-score of -1.8 is approximately 0.072, which is greater than 0.05. Therefore, there is insufficient evidence to reject the null hypothesis, and we cannot conclude that the actual overall mean diameter of the ball bearings is not 4.300 mm.

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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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To find the uncertainty in the slope of a linear trend line, your group-mates can use the uncertainties listed in each measurement by following these steps:

1. Plot the data points on a graph, including the uncertainties as error bars for each point. The error bars represent the range of possible values for each measurement due to uncertainty.

2. Fit a linear trend line to the data points, either by using a statistical software or by drawing a best-fit line manually.

3. For each data point, calculate the vertical deviation from the fitted trend line. This is the difference between the observed value (including the uncertainty) and the value predicted by the trend line.

4. Square each deviation and sum them to get the total sum of squares.

5. Calculate the uncertainty in the slope by dividing the total sum of squares by the number of data points minus two. This is known as the "degrees of freedom" (n-2).

6. Take the square root of the result to get the standard deviation of the slope, which represents the uncertainty in the slope.

By following these steps, your group-mates can accurately determine the uncertainty in the slope of a linear trend line using the uncertainties listed in each measurement.

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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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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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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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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 probability of having zero errors is approximately 0.711.

The probability distribution that can be used to calculate the probability that zero errors will be found in a 255-word bond transaction is the Poisson distribution.

The Poisson distribution is a discrete probability distribution that is used to model the number of events occurring within a fixed interval of time or space, given the average rate of occurrence of the events.

In this case, the average rate of occurrence of errors is 6 per hour, which can be converted to 0.1 errors per minute. Therefore, the expected number of errors in a 255-word bond transaction is (255/75)*0.1 = 0.34 errors.

Using the Poisson distribution, the probability of having zero errors in a 255-word bond transaction can be calculated as:

[tex]P(X = 0) = (e^{(-\lambda)} * \lambda^0) / 0! = e^{(-0.34)} * 0.34^0 / 1! \approx 0.711[/tex]

where λ is the expected number of errors in the 255-word bond transaction.

Therefore, the probability distribution used to calculate the probability of having zero errors in a 255-word bond transaction is the Poisson distribution, and the probability of having zero errors is approximately 0.711.

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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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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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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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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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The density function of the maximum of X1, ..., Xn i.i.d. uniform(0, 1) random variables can be found using the cumulative distribution function (CDF) and then taking its derivative. Let Y be the maximum of X1, ..., Xn, and let F(y) denote the CDF of Y.

Since the random variables are i.i.d., their joint CDF can be expressed as a product of individual CDFs:

F(y) = P(Y ≤ y) = P(X1 ≤ y) * ... * P(Xn ≤ y).

Since each Xi is a uniform(0, 1) random variable, its CDF is given by:

P(Xi ≤ y) = y for 0 ≤ y ≤ 1.

So the CDF of Y is:

F(y) = y^n for 0 ≤ y ≤ 1.

Now, to find the probability density function (PDF) of Y, take the derivative of F(y) with respect to y:

f(y) = dF(y)/dy = d(y^n)/dy = n*y^(n-1) for 0 ≤ y ≤ 1.

Therefore, the density function of the maximum of X1, ..., Xn i.i.d. uniform(0, 1) random variables is f(y) = n*y^(n-1) for 0 ≤ y ≤ 1.

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1) To find A ∩ B, we need to identify the elements that are common to both sets A and B. A = {a, d, j, o, z} and B = {a, d, f, g, o, u}. A ∩ B = {a, d, o}

2) Let A = {5, 3, 4, 1, 2, 7}, B = {6, 3, 1, 9}, and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. First, find A ∩ B, which is the set of elements common to both A and B. A ∩ B = {3, 1}. To find (A ∩ B)', we need to identify the elements in the universal set U that are not in the intersection A ∩ B. (A ∩ B)' = {2, 4, 5, 6, 7, 8, 9, 10}.

3) This question is identical to question 2, so the answer is the same. (A ∩ B)' = {2, 4, 5, 6, 7, 8, 9, 10}.

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For an electric car's home battery charger uses 10.7 kiloWatt for 6 hour, the total cost of electricity used by it is equals to the $29.532.

We have an electric car's home battery charger. The amount of power used by charger, P = 10.7 kilowatt

Time taken by charger to use power of 10.7 kilowatt, t = 6 hours

The rate of cost of electricity, r = $0.46 per kilowatt - hour

We have to determine the cost to charge the car's battery. Now, first we calculate the total energy used for charging, E= P × t

=> E = 10.7 kilowatt × 6 hours

= 64.2 kilowatt- hour

Also, Total cost of electricity = E × r

= 0.46 per kilowatt- hour × 64.2 kilowatt- hour

= $ 0.46 × 64.2

= $ 29.532

Hence, required value is $29.532.

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Answer:

1.65 is the right number of SDs to use when constructing a 90% confidence interval because it corresponds to the upper 5th percentile of a normal distribution, which gives a 5% chance of the true population parameter being outside the interval.

The concept of multiple regression and its application in predicting income among individuals with at least a high school education using age and years of schooling as predictors.

Multiple regression is a statistical technique used to study the relationship between one dependent variable (in this case, income) and multiple independent variables (here, age and years of schooling). By analyzing the data, we can determine if the independent variables have a significant effect on the dependent variable and how they influence it.

In this particular question, multiple regression was applied to examine if income among those with at least a high school education could be predicted from their age and number of years of schooling. The overall regression would involve collecting data on the individuals' income, age, and years of schooling. The data would then be entered into a statistical software program to perform the multiple regression analysis.

The steps in conducting the multiple regression analysis are as follows:

1. Define the dependent variable (income) and independent variables (age and years of schooling).
2. Collect data on each variable for a sample of individuals with at least a high school education.
3. Input the data into a statistical software program.
4. Perform the multiple regression analysis to determine the significance of the independent variables in predicting the dependent variable.
5. Interpret the results to assess if age and years of schooling are significant predictors of income.

The overall regression will provide valuable information about the relationships between income, age, and years of schooling. If the results show that age and years of schooling are significant predictors of income, we can conclude that income among individuals with at least a high school education can be predicted based on their age and number of years of schooling. This information can be used for various purposes, such as informing educational and economic policies, career guidance, and more.

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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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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]

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[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]

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[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]

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[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]

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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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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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