What are some expressioms equivalent to -49y-14

The measure of angle ITE is 44 degrees. In a kite, the diagonals intersect at a right angle and bisect each other. Let's label the points in the kite as follows:

Point P is the intersection of the diagonals.

Point K is at one end of the kite.

Point E is at the other end of the kite, opposite to point K.

Point I is between points K and P.

Point T is between points E and P.

Since the diagonals bisect each other, we know that IP = KP and EP = TP. Let's use this fact to find the measure of angle ITP, which is equal to ITE:

We know that angle KIP = 46 degrees.

Since IP = KP, angle KIP is isosceles, so angle KPI is also 46 degrees.

Similarly, since KEP = 34 degrees, we know that angle KPE is also 34 degrees.

Since EP = TP, angle EPT is also 34 degrees.

We also know that angles KPI and EPT add up to 90 degrees, since they are complementary angles formed by the intersection of perpendicular lines.

Therefore, angle IPT is 90 - 46 = 44 degrees.

Since IP = TP, angles ITP and IPT are congruent, so angle ITP is also 44 degrees.

Therefore, the measure of angle ITE is 44 degrees.

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Answer:

[tex]x \approx 60.4[/tex]

Step-by-step explanation:

We can solve for x using the trigonometric ratio cosine:

[tex]\cos(\theta) = \dfrac{\text{adjacent}}{\text{hypotenuse}}[/tex]

↓ plugging in the given values

[tex]\cos(22\°) = \dfrac{56}{x}[/tex]

↓ taking the reciprocal of (flipping) both sides

[tex]\dfrac{1}{\cos(22\°)} = \dfrac{x}{56}[/tex]

↓ multiplying both sides by 56

[tex]\dfrac{56}{\cos(22\°)} = x[/tex]

↓ plugging into a calculator

[tex]\boxed{x \approx 60.4}[/tex]

The value of x is x= ± 2√5 (option a).

To solve this equation, we need to isolate x on one side of the equation. We can do this by taking the square root of both sides of the equation. However, we need to keep in mind that when we take the square root of a number, there are always two possible solutions, one positive and one negative.

So, taking the square root of both sides of x² = 20, we get:

x = ± √20

Simplifying √20, we get:

x = ± √(4 × 5)

Using the property of square roots that √(a × b) = √a × √b, we can simplify further to get:

x = ± 2√5

Therefore, the two solutions to the equation x² = 20 are x = 2√5 and x = -2√5.

However, we also need to check if any of these solutions make sense in the context of the problem. In this case, we are looking for the value of x, which is a measure of length, so we can discard the negative solution since lengths cannot be negative.

Therefore, the only valid solution is x = 2√5.

Hence the correct option is (a).

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We can say with 95% confidence that the true proportion of checking account customers who also have savings accounts in the population lies between 0.25 and 0.35.

a. To find the sample proportion of checking account customers who also have savings accounts, we need to divide the number of customers who have both types of accounts by the total number of checking account customers in the sample. Let's say the bank selected 500 checking account customers and found that 150 of them also had savings accounts. Then, the sample proportion would be:

150/500 = 0.3

So, 30% of the checking account customers in the sample also had savings accounts.

b. To find the standard error of the sample proportion, we use the formula:

SE = sqrt(p*(1-p)/n)

where p is the sample proportion (0.3 in this case), and n is the sample size (500). Plugging in the numbers, we get:

SE = sqrt(0.3*(1-0.3)/500) = 0.025

So, the standard error is 0.025.

c. To find a 95% confidence interval for the population proportion of checking account customers who also have savings accounts, we use the formula:

CI = p ± z*(SE)

where z is the z-score corresponding to a 95% confidence level (which is 1.96), and SE is the standard error we calculated in part b. Plugging in the numbers, we get:

CI = 0.3 ± 1.96*(0.025) = (0.25, 0.35)

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Answer: 142

Step-by-step explanation:

P(X > 20) has a rounded value of 0.0475.

By dividing the sum of the given numbers by the entire number of numbers, the mean—the average of the given numbers—is determined.

To find P(X > 20), where X is a normal random variable with mean μ = 15 seconds and standard deviation σ = 3 seconds, we need to standardize the variable and use the standard normal distribution.

Let Z be a standard normal random variable, then we can standardize X as follows:

Z = (X - μ) / σ = (20 - 15) / 3 = 1.67

Using a standard normal table or calculator, we can find the probability:

P(Z > 1.67) = 0.0475 (rounded to four decimal places)

Therefore, P(X > 20) has a rounded value of 0.0475.

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A company seeking to estimate the time required to produce 100 units of a product based on past production rates should utilize regression analysis. This statistical method is the most effective among the given choices because it focuses on identifying the relationship between variables, such as production rates and time, and uses this relationship to make predictions.

Regression analysis will enable the company to develop a model that quantifies the relationship between the production rates (independent variable) and the time taken to produce units (dependent variable). By analyzing historical data, the company can establish a mathematical equation to predict future production times based on the past performance.

Hypothesis testing and confidence intervals are less suited for this purpose, as they primarily focus on determining the significance of relationships or differences between groups rather than predicting future outcomes. Similarly, correlation analysis measures the strength of a relationship between variables but does not predict future values based on past data.

In summary, regression analysis is the most effective statistical method for a company to estimate the time required to produce 100 units of a product based on past production rates. This method enables the company to create a predictive model, which can help optimize production processes and enhance overall efficiency.

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computing a confidence interval for the mean of the differences in paired data.

Compute the differences between the two measurements for each pair of data points.

Calculate the mean and standard deviation of the differences.

Compute the standard error of the mean of the differences by dividing the standard deviation of the differences by the square root of the sample size.

Determine the appropriate confidence level and degrees of freedom based on the sample size and type of test being conducted.

Use a t-distribution to find the t-value associated with the desired confidence level and degrees of freedom.

Compute the confidence interval by adding and subtracting the product of the t-value and the standard error of the mean of the differences from the sample mean of the differences.

Regarding the conclusion, if the confidence interval does not include zero, it means that there is a statistically significant difference between the two devices, and the biologist's claim is supported at the chosen level of significance. If the confidence interval includes zero, it means that there is no statistically significant difference between the two devices, and the biologist's claim is rejected at the chosen level of significance.

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The error sum of squares is 0.32. So the answer is (B) 0.32.

The total sum of squares represents the variation in the dependent variable that can be explained by the independent variable(s) and the error term.

The regression sum of squares represents the variation in the dependent variable that can be explained by the independent variable(s) in the regression model.

The error sum of squares represents the variation in the dependent variable that cannot be explained by the independent variable(s) in the regression model and is often referred to as the residual variation.

The formula we can use is SSE = SST - SSR

Here we have

In a study where the least squares estimates were based on 34 sets of sample observations,

The total sum of squares SST = 4.53

The regression sum of squares SSR = 4.21

To find the error sum of squares, we use the formula:

=> SSE = SST - SSR

Plugging in the given values, we have:

=> SSE = SST - SSR = 4.53 - 4.21 = 0.32

Therefore,

The error sum of squares is 0.32. So the answer is (B) 0.32.

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Complete Question:

In a study where the least squares estimates were based on 34 sets of sample observations, the total sum of squares and regression sum of squares were found to be: SST = 4.53 and SSR = 4.21. What is the error sum of squares?

Multiple Choice

A. 1.07   B. 0.32     C. 0.92   D. 8.74

Tthe temperature first reaches 42 degrees 7.67 hours after midnight, or approximately at 7:40 AM.

The temperature variation over a day can be represented as a sinusoidal function in the form of y = A sin(Bx - C) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.

In this case, the midline of the temperature function is (33 + 57)/2 = 45 degrees. Therefore, D = 45.

The amplitude of the function is (57 - 33)/2 = 12 degrees. Therefore, A = 12.

Since the average temperature first occurs at 10 AM, which is 10 hours after midnight, the phase shift can be determined as C = (10/24) * 2π.

To find the frequency B of the function, we need to use the fact that the temperature function repeats every 24 hours. Therefore, B = 2π/24 = π/12.

Putting all the values in the equation y = 12 sin(π/12(x - 5/3)) + 45, we need to solve for x when y = 42.

42 = 12 sin(π/12(x - 5/3)) + 45

-3 = 12 sin(π/12(x - 5/3))

-1/4 = sin(π/12(x - 5/3))

π/2 = π/12(x - 5/3)

x - 5/3 = 6

x = 23/3

Therefore, the temperature first reaches 42 degrees 7.67 hours after midnight, or approximately at 7:40 AM.

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Bayes' Theorem is a mathematical theorem that enables the revision of prior probabilities based on new information or evidence.

This theorem is widely used in statistics, machine learning, and other fields that deal with uncertainty and probabilistic reasoning. Contrary to the first option mentioned in the question,

Bayes' Theorem can be used when conditional probabilities are known, and it enables the calculation of posterior probabilities, which is the probability of a hypothesis or event given the available evidence.

Therefore, the third option is correct; Bayes' Theorem enables the use of sample information to revise prior probabilities. This theorem is highly valuable because it allows the integration of new data or knowledge into the decision-making process,

which can lead to more accurate predictions and better-informed decisions. In summary, Bayes' Theorem is a powerful tool that requires knowledge of probabilities and enables the calculation of posterior probabilities based on new evidence or information.

By combining prior probabilities with likelihoods (based on new data), we can calculate posterior probabilities, which represent our updated knowledge.

This process is crucial in making informed decisions in various fields, such as data science, finance, and medical diagnosis.

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The approximate probability that at least 94 rolls are needed to get a sum greater than 354 is 0.852.

X be the number of rolls needed to obtain a sum greater than 354. We are interested in finding P(X ≥ 94).

We can use the fact that the sum of two fair six-sided dice is uniformly distributed between 2 and 12. Thus, the sum of n rolls of a fair six-sided die is uniformly distributed between n and 6n.

Let Yn be the sum of the first n rolls of the die. Then Yn is uniformly distributed between n and 6n, and we have:

P(Yn > 354) = P(Yn - n > 354 - n) = P((Yn - n)/5 > (354 - n)/5)

Now, (Yn - n)/5 is uniformly distributed between 1 and 6, and (354 - n)/5 is between 1 and 70. So we have:

P(Yn > 354) = P((Yn - n)/5 > (354 - n)/5) = P(U > (354 - n)/5)

where U is a uniform random variable on [1,6].

We want to find P(X ≥ 94) = P(Y94 ≤ 354) = 1 - P(Y94 > 354) = 1 - P(U > (354 - 94)/470) = 1 - (70/471) = 0.852.

Therefore, the approximate probability that at least 94 rolls are needed to get a sum greater than 354 is 0.852.

Probability is the likelihood or chance of an event. Occurring for example, the probability of flipping a coin and it being heads is ½, because there is 1 way of getting a head and the total number of possible outcomes is 2 (a head or tail). We write P(heads) = ½ .

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The value of the z-score of the normally distributed scores is z = -0.2

Given data ,

To find the z-score of a person who scored 390 on the exam, we can use the formula for z-score:

z = (X - μ) / σ

where:

X = the score of the person = 390

μ = the mean of the distribution = 400

σ = the standard deviation of the distribution = 50

On simplifying , we get

z = (390 - 400) / 50

z = -10 / 50

z = -0.2

Hence , the z-score of a person who scored 390 on the exam is -0.2

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The expected number of people who will drop out due to either relapse or moving away is 10 + 6 = 16. However, taking into account the standard deviations, it is 16 +/- 5.

To find the expected number of people who will drop out due to either relapse or moving away, we need to add the expected number of people who will drop out due to relapse (10) and the expected number of people who will drop out due to moving away (6).

Expected number of people who will drop out due to either relapse or moving away = 10 + 6 = 16.

However, we also need to take into account the standard deviations for each of these groups. To do this, we can use the square root of the sum of the variances (SD squared) for each group, squared.

Variances:
- Relapse: 4 squared = 16
- Moving away: 3 squared = 9

Square root of the sum of the variances:
- sqrt(16 + 9) = 5

Therefore, the expected number of people who will drop out due to either relapse or moving away, taking into account the standard deviations, is 16 +/- 5.

This means that we can expect anywhere between 11 and 21 people to drop out due to either relapse or moving away during the two-year cohort study.

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Since the culture is observed to double every 6 hours, we know that the growth rate is constant at r = ln(2)/6 per hour.

To calculate growth rates, divide the difference between the starting and ending values for the period under study by the starting value. The most frequent time intervals for growth rates are annually, quarterly, monthly, and weekly.

We can use the formula for exponential growth to model the number of bacteria in the culture after t hours:

n(t) = n0e^(rt)where n0 is the initial number of bacteria.

Substituting in the values given in the problem, we get:

n(t) = 25e^[(ln(2)/6)t]Simplifying this expression using the properties of logarithms, we can rewrite it in the form:

n(t) = 25(2)^(t/6)This is the exponential model for the number of bacteria in the culture after t hours.

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The exponential model for population of bacteria, [tex]n(t) = n_0{2}^{\frac{t}{a} }[/tex] can be written [tex]n(t) = 25 \times {2}^{\frac{t}{6} }[/tex] for the number of bacteria in the culture after t hours. The estimate number of bacteria after 13 hours is equals to the 112. In 92 hours, the bacteria count will reach to 1 million.

We have a certain culture of the bacterium Rhodobacter.

Initial population, n₀ = 25

The population become doubles in every 6 months. The exponential model

[tex]n(t) = n_0{2}^{\frac{t}{a} }[/tex] for the number of bacteria in the culture after t hours. Now, the population become double in 6 hours, so a = 6 , then exponential equation is [tex]n(t) = 25 \times {2}^{\frac{t}{6} }[/tex].

We have to estimate the number of bacteria after 13 hours. That is t = 13 hours, [tex]n( t) = 25( 2)^{\frac{t}{6}}[/tex]

Substitute t = 13 hours

[tex] = 25( 2)^{\frac{13}{6}}[/tex]

[tex]= 25( 2)^{2.16}[/tex]

= 111.728713807 ~ 112

So, n(13) = 112

We have to determine the value of t in hours for n(t) = 1 million = 1000000, using the above equation, [tex]1000000 = 25( 2)^{\frac{t}{6}}[/tex]

[tex]40000 = ( 2)^{\frac{t}{6}}[/tex]

Taking natural logarithm both sides

=>[tex] ln( 40000) = ln(( 2)^{\frac{t}{6}})[/tex]

=> [tex]ln(40000) = \frac{t}{6} ln(2)[/tex]

=> [tex]t = \frac{6 ln( 40000)}{ ln(2)}[/tex]

= 91.7262742773 ~ 92

Hence, required value is 92 hours..

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The value of x is 7[tex]\sqrt{3}[/tex] inches

The correct answer is an option (d)

Let us assume that in the attached diagram of right triangle the angle A measures 30 degrees.

Here, the hypotenuse measures 7 in.

We know that in right triangle, the tangent of angle θ is nothing but the ratio of opposite side of angle θ to the adjacent side of angle θ.

Consider the tan of angle A

tan(A) = opposite side of angle A / adjacent side of angle A

tan(30°) =  7 / x

We know that from the standard trigonometric table the value of tan(30°) is [tex]\frac{1}{\sqrt{3} }[/tex]

Substitute this value in above equation we get,

[tex]\frac{1}{\sqrt{3} }[/tex] =  7/x

We solve this equation to find the value of x.

x = 7 × [tex]\sqrt{3}[/tex]

x = 7[tex]\sqrt{3}[/tex] in.

Therefore, the correct answer is an option (d)

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The sampling method is not specified. The probability a customer spends less than $14 each visit is approximately 0.8413. The probability a customer spends more than $7 each visit is approximately 0.9082. The survey question is biased as it is leading and implies a comparison with a specific competitor.

The type of sampling method used is not specified in the given information.

Using the standard normal distribution, the probability that a customer spends less than $14 can be found as

P(X < 14) = P(Z < (14-11)/3) = P(Z < 1) = 0.8413

Therefore, the probability that a customer spends less than $14 each visit is 0.8413.

Similarly, the probability that a customer spends more than $7 can be found as

P(X > 7) = P(Z > (7-11)/3) = P(Z > -1.33) = 0.9082

Therefore, the probability that a customer spends more than $7 each visit is 0.9082.

The survey question "How does our food compare to PhillyBurger?" may be biased because it suggests a comparison between two specific brands, which may influence the customer's response. A more neutral question would be "How do you rate the quality of our food?"

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An even number of data values will always have two middle numbers, and an odd number of data values will always have one middle value. Therefore, the true statements are:

An even number of data values will always have two middle numbers.

An odd number of data values will always have one middle value.

What is even number?

An even number is an integer that is divisible by 2, i.e., when divided by 2, the remainder is 0. Examples of even numbers are 2, 4, 6, 8, 10, 12, etc.

The statement "an even number of data values will always have two middle numbers" is true. When there is an even number of data values, there is no single middle number because there are two values in the center.

For example, in the set {1, 2, 3, 4}, the middle numbers are 2 and 3. In general, if there are an even number of data values, the middle two values are found by taking the average of the two values in the center of the set. This is different from the case when there is an odd number of data values, where there is a single middle value.

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The weighted average rating for movie 3285 can be calculated by multiplying each rater's similarity with the given rating and then dividing the sum by the total sum of similarities.

In this case, we have two raters with similarity values of 15 and 30, respectively. For the first rater with an ID of 30, the similarity value is 15 and the given rating for movie 3285 is 7.0. Therefore, the contribution of this rater to the weighted average rating is (15 x 7.0) = 105.0.

For the second rater with an ID of 20, the similarity value is 30 and no rating is given for movie 3285. Therefore, the contribution of this rater to the weighted average rating is 0. The total sum of similarities is 15 + 30 = 45.0. Thus, the weighted average rating for movie 3285 is (105.0 + 0) / 45.0 = 2.33.

So, the weighted average rating for movie 3285 is 2.33. It's important to note that this calculation assumes that these are the only two raters who rated the movie, and that their similarity values accurately reflect their taste in movies.

we'll use the following formula: Weighted Average = (Rating1 * Similarity1 + Rating2 * Similarity2) / (Similarity1 + Similarity2), Plugging in the values: Weighted Average = (15 * 7.0 + 30 * 30.0) / (7.0 + 30.0)
Weighted Average = (105 + 900) / 37
Weighted Average = 1005 / 37
Weighted Average ≈ 27.16

So, the weighted average rating for the movie with ID 3285 is approximately 27.16.

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Answer:

c) range from -1.00 to 1.00

In a case whereby angela walked 12 mile. She took a break for water and then walked some more. In all, she walked 1110 miles then the distance that Angela walk after her break is 1098 miles

The distance that she walks initially = 12 mile

The total distance that she wlaked =1110 miles.

Then the distance she walked after the break =1110 miles -  12 mile

=1098 miles

Then we can come into conclusion that she was able to navigate 1098 miles after she rest during the break 1098 miles which implies that the distance she walked after the break was more.

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After the new resident with an income of $2 million per year builds a mansion in the community, the median household income remains unchanged.

This is because the median household income is the middle value in a list of incomes, and the new resident's income is much higher than any of the other household incomes, so it does not affect the middle value.

However, the mean household income will increase significantly. The mean is the sum of all the incomes divided by the total number of households, and the new resident's income is much larger than any of the other households.

Therefore, when the new resident's income is added to the total income, the mean will increase significantly.

Before the new resident moved in, the total income for all eight households was $240,000 (4 households x $30,000 + 4 households x $40,000). After the new resident moves in, the total income becomes $2,240,000 ($240,000 + $2,000,000).

Dividing this by the total number of households (now nine, with the addition of the new resident) gives a new mean household income of approximately $248,888.

In conclusion, the median household income remains unchanged, while the mean household income increases significantly after the new resident with a high income moves in.

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The derivative of F(x) is [tex]F'(x) = x^{(1/3)[/tex].

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output.

To find the derivative of the given function F(x), we will apply the fundamental theorem of calculus and differentiate the integral with respect to x.

Let's compute F'(x):

F(x) = ∫[0 to x] [tex]t^{(1/3)} dt[/tex]

To differentiate the integral with respect to x, we'll use the Leibniz integral rule:

F'(x) = d/dx ∫[0 to x] [tex]t^{(1/3)} dt[/tex]

According to the Leibniz integral rule, we have to apply the chain rule to the upper limit of the integral.

[tex]F'(x) = x^{(1/3)} d(x)/dx - 0^{(1/3)} d(0)/dx[/tex]   [applying the chain rule to the upper limit]

Since the upper limit of the integral is x, the derivative of x with respect to x is 1, and the derivative of 0 with respect to x is 0.

[tex]F'(x) = x^{(1/3)} (1) - 0^{(1/3)} (0)[/tex]

[tex]F'(x) = x^{(1/3)[/tex]

Therefore, the derivative of F(x) is [tex]F'(x) = x^{(1/3)[/tex].

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Answer:

A)  Rotate ΔABC 90° clockwise about the origin.

Step-by-step explanation:

From inspection of the given diagram, the coordinates of the vertices of triangle ABC are:

The coordinates of the vertices of triangle XYZ are:

The mapping rule for a rotation of 90° clockwise about the origin is:

[tex]\boxed{(x, y) \rightarrow (y, -x)}[/tex]

Therefore:

The mapping rule for a rotation of 90° clockwise about a point P is:

[tex]\boxed{\left([y - y_P + x_P], [x_P - x + y_P]\right)}[/tex]

So the mapping rule if the point of rotation is A (-1, 1) is:

[tex]\boxed{(y - 2 , -x)}[/tex]

Therefore:

The mapping rule for a reflection across the y -axis is:

[tex]\boxed{(x, y) \rightarrow (-x, y)}[/tex]

Therefore:

The mapping rule for a reflection across the line y = x is:

[tex]\boxed{ (x, y) \rightarrow (y, x)}[/tex]

Therefore:

Comparing the different transformations, we can see that the rigid motion that could be used to map triangle ABC onto triangle XYZ is:

The total cost for both the truck driver and the engineer this week is $4,320.

To solve this problem, we first need to determine how many hours the truck driver worked this week. We can do this by dividing the weekly earnings of the truck driver by their hourly rate of $40:

1440 / 40 = 36 hours

Therefore, the truck driver worked for 36 hours this week.

Now, we need to determine the total earnings of the engineer for the same number of hours. We know that the engineer earns $80 per hour, so for 36 hours of work, they will earn:

80 x 36 = $2,880

Therefore, the total cost for both the truck driver and the engineer this week will be the sum of their earnings:

1440 (truck driver) + 2880 (engineer) = $4,320

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Complete Question:

A truck driver earns $40 per hour, and an engineer earns $80 per hour. If the truck driver made 1,440 this week, then the total cost does he made is?

Poisson distribution of number of requests for a popular Web page,

a) Web server has a capacity of C requests per minute is equals to the 206.

b) The probability of overload in a one-second interval is approximately equal to 1.

Let x denotes the number of requests for a popular web page. Now, X = number of requests per minute ~ Poisson (180)

Now, by central limit theorem the distribution of x can be approximated by Normal diet with mean = 180 and variance 180 and we denote the approximated variable by Y, that is [tex]Y \: \tilde \: \: N(180, 180)[/tex].

If number of requests in a one minute interval is greater than C, then probability of overload is less than 0.055, that is P[ X > C] < 0.055

P[ X > C] ~ P[ Y > C + 0.5] ( by continuity )

so, P[ Y > C + 0.5] < 0.055

[tex]P[ \frac{ Y - 180}{ \sqrt{180}} > \frac{C + 0.5 - 180}{ \sqrt{180} }] < 0.055[/tex]

According to normal distribution, [tex] P[ \frac{ Y - 180}{180} ] = Z ≃N(1,0)[/tex]

Therefore, [tex]P[ Z > \frac{C + 0.5 - 180}{ \sqrt{180} }] < 0.055[/tex]

=> [tex][\frac{C + 0.5 - 180}{ \sqrt{180} }] < Z_{0.055}[/tex]

= 0.478069 ~ 0.4781.

=> [tex]C - 199.5 < 0.4781 × \sqrt{ 180} [/tex]

=> C = 199.5 + 0.4781 × 13.4164

=> C = 205.91 ~ 206.

b) Now, we have to determine the probability of overload in a one-second interval, using the value of C obtained in part(a), so, C = 206 so, [ C/60] = 3

Probability of overload, P = P( X> 3)

= 1 - P( X≤ 3)

[tex]= 1 - \sum_{x = 0}^{3} e^{-180} \frac{ 180^x}{x!} [/tex]

= 1

Hence, required probability is 1.

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To prove that for some n > 0, xi > 0 for all i > n, we can use the fact that (xi) is a sequence that converges to x.

This means that given any ε > 0, there exists an N such that |xi - x| < ε for all i > N.  we can use the fact that (xi) is a sequence that converges to x.

Let's choose ε = x/2. Since x > 0, ε > 0 as well. Then there exists an N such that |xi - x| < ε = x/2 for all i > N. Rearranging this inequality, we get:

x - xi < x/2

xi > x/2

Since xi > x/2 and x > 0, we have shown that for some N > 0, xi > 0 for all i > N.

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The exact length of the curve Y = [tex]x^{3/3}[/tex] + 4x, 1 ≤ x ≤ 2 is approximately 4.526 units. The length is found using the formula for arc length integration, which involves taking the square root of the sum of squares of the first derivative of the function.

To find the exact length of the curve, we use the arc length formula

L = ∫ √[1 + (dy/dx)²] dx, where y = [tex]x^{3/4}[/tex] and 1 ≤ x ≤ 2.

Taking the derivative of y with respect to x, we get

dy/dx = 3[tex]x^{2/4}[/tex]

Substituting into the formula, we get

L = ∫ √[1 + (3[tex]x^{2/4}[/tex])²] dx

L = ∫ √[1 + 9[tex]x^{4/16}[/tex]] dx

Making the substitution u = 9[tex]x^{4/16}[/tex] + 1, du/dx = (9/4)x³, we get

L = (4/9) ∫ √(u) du

L = (4/9) * (2/3) * [tex]u^{3/2}[/tex] + C

L = (8/27) * [tex](9x^4 + 16)^{3/2}[/tex] + C

Since the curve is between x = 1 and x = 2, the exact length of the curve is

L = (8/27) * [[tex](9(2^4) + 16)^{3/2} - (9(1^4) + 16)^{3/2}[/tex]]

L = (8/27) * [[tex](160)^{3/2} - (25)^{3/2}[/tex]]

L ≈ 4.526.

Therefore, the exact length of the curve is approximately 4.526.

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