Find functions f and g where fo g(x) = √3x² + 4x - 5.

The time needed to fix a furnace follows a uniform distribution between 1.5 and 4 hours. The mean time is 2.75 hours, and the standard deviation is approximately 0.5208 hours. The probability that a repairman takes more than 2.5 hours is 60%.

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The area of the region inside the circle r = 12 cosθ is 18π square units.To sketch the region and find its area, let's first analyze the equation of the circle:

r = 12 cosθ

We can rewrite this equation in Cartesian coordinates using the conversion formulas:

x = r cosθ

y = r sinθ

Substituting the value of r from the equation of the circle, we have:

x = 12 cosθ cosθ = 12 cos²θ

y = 12 cosθ sinθ = 6 sin(2θ)

Now we can plot the region on a graph. Let's focus on the interval θ ∈ [0, π/2] to cover the entire region inside the circle:

Note: Please refer to the attached graph for a visual representation of the region.

We observe that the region is a semi-circle with a diameter along the x-axis, centered at (6, 0) with a radius of 6 units. The curve starts at the point (12, 0) when θ = 0 and ends at the point (0, 0) when θ = π/2.

To find the area of the region, we integrate over the appropriate interval:

A = ∫[0, π/2] 1/2 * r² dθ

Substituting the value of r = 12 cosθ, we have:

A = ∫[0, π/2] 1/2 * (12 cosθ)² dθ

A = ∫[0, π/2] 1/2 * 144 cos²θ dθ

A = 72 ∫[0, π/2] cos²θ dθ

Using the trigonometric identity cos²θ = (1 + cos(2θ))/2, we can simplify the integral:

A = 72 ∫[0, π/2] (1 + cos(2θ))/2 dθ

A = 36 ∫[0, π/2] (1 + cos(2θ)) dθ

A = 36 [θ + (1/2) sin(2θ)] evaluated from θ = 0 to θ = π/2

A = 36 [(π/2) + (1/2) sin(π)] - 36 [0 + (1/2) sin(0)]

A = 36 (π/2)

Therefore, the area of the region inside the circle r = 12 cosθ is 18π square units.

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The closest point(s) on the curve to the origin is/are (2^(2/3), ∛2).To find the points on the curve x^2y = 8 nearest to the origin,

we need to minimize the distance between the origin (0, 0) and the points on the curve.

To do this, we can use the distance formula:

Distance = √((x - 0)^2 + (y - 0)^2) = √(x^2 + y^2)

Since we want to minimize the distance, we can minimize the square of the distance, which is equivalent to minimizing x^2 + y^2.

Now, let's find the points on the curve that minimize x^2 + y^2. We can do this by finding the critical points of the function x^2 + y^2 subject to the constraint x^2y = 8.

Using Lagrange multipliers, we set up the following equations:

2x = λ(2xy)

2y = λ(x^2)

We also have the constraint equation x^2y = 8.

From the first equation, we can rewrite it as 2x - 2λxy = 0, and solve for λ:

2x = 2λxy

1 = λy

Substituting this value of λ in the second equation, we get:

2y = x^2

Now, we can substitute this relationship between x and y into the constraint equation x^2y = 8:

(2y)^2y = 8

4y^3 = 8

y^3 = 2

y = ∛2

Substituting this value of y into 2y = x^2, we find:

2(∛2) = x^2

x = 2^(2/3)

So, the closest point(s) on the curve to the origin is/are (2^(2/3), ∛2).

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The probability distribution is not valid.  A probability distribution must satisfy certain conditions: The probability of each possible outcome must be greater than or equal to 0.

The sum of all probabilities must be equal to 1. In this case, the question does not provide the probability distribution for the number of new clients for marriage counseling. It only states that West Virginia has an annual divorce rate of approximately 5 divorces per 1000 people. This divorce rate does not directly provide the probability distribution for the number of new clients.

To determine the validity of the probability distribution, we would need the specific probabilities for different values of x, which are missing from the given information. Without this information, we cannot determine if the probability distribution is valid or not. Therefore, the answer to the question is "No," as the probability distribution is not provided.

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The expression that represents the money that the league raises is $7x - $55, where 'x' represents the number of shirts sold.

The expression that represents the money that the league raises can be calculated by subtracting the total cost from the total revenue.

To find the total revenue, we need to multiply the selling price of each shirt ($15) by the number of shirts sold. Let's denote the number of shirts sold as 'x'.

The revenue from selling shirts is given by the expression:

Revenue = Selling price per shirt [tex]\times[/tex] Number of shirts sold

Revenue = $15 [tex]\times[/tex] x

The total cost is the sum of the cost to make each shirt and the advertising cost.

Given that each shirt costs $8 to make and the advertising cost is $55, the total cost is:

Total Cost = Cost per shirt [tex]\times[/tex] Number of shirts + Advertising cost

Total Cost = $8 [tex]\times[/tex] x + $55

To find the money that the league raises, we subtract the total cost from the total revenue:

Money Raised = Revenue - Total Cost

Money Raised = $15x - ($8x + $55)

Money Raised = $15x - $8x - $55

Money Raised = $7x - $55

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The solution to the given Bernoulli equation t²y + 8ty - y³ = 0, with the substitution v = y^(1-n), is y = ((10t)/3)^(1/n).

To solve the given Bernoulli equation, t²y + 8ty - y³ = 0, we can make the substitution v = y^(1-n). Here, n ≠ 0, 1. Let's proceed with the solution using this substitution.

First, we differentiate both sides of the substitution v = y^(1-n) with respect to t:

dv/dt = (1-n)y^(-n) * dy/dt.

Next, we differentiate the original equation t²y + 8ty - y³ = 0 with respect to t:

d/dt(t²y) + d/dt(8ty) - d/dt(y³) = 0.

Differentiating each term separately:

2ty + 2t(dy/dt) + 8y + 8t(dy/dt) - 3y²(dy/dt) = 0.

Rearranging the equation:

2ty + 8y - 3y²(dy/dt) + 2t(dy/dt) + 8t(dy/dt) = 0.

Simplifying further:

(2ty + 8y) + (2t + 8t)(dy/dt) - 3y²(dy/dt) = 0.

Factoring out common terms:

2y(t + 4) + 10t(dy/dt) - 3y²(dy/dt) = 0.

Now, substitute v = y^(1-n) into the equation:

2(1-n)v(t + 4) + 10t(dy/dt) - 3(y^n)(dy/dt) = 0.

Rearranging terms and dividing through by (1-n):

2v(t + 4)/(1-n) + 10t(dy/dt) - 3(y^n)(dy/dt) = 0.

Simplifying further:

2v(t + 4)/(1-n) + (10t - 3(y^n))(dy/dt) = 0.

To eliminate the derivative term, we can set the expression in parentheses equal to zero:

10t - 3(y^n) = 0.

Solving for y^n:

3(y^n) = 10t.

y^n = (10t)/3.

Taking the n-th root of both sides:

y = ((10t)/3)^(1/n).

None of the given options (A, B, C, D) match this result.

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The correct answer is (b) categorical and nominal, categorical and ordinal, categorical and nominal.

Level of agreement is a variable that is categorical and ordinal. In a survey or study, responses are usually measured on a scale that has the following categories:

strongly disagree, disagree, neither agree nor disagree, agree, and strongly agree.Number of different tree species in a forest is a variable that is categorical and ordinal as well. This is because the different tree species can be counted and arranged in order of abundance or rarity.Basic Film Genres is a variable that is categorical and nominal. Film genres are broad categories that are used to categorize films based on similar narrative structures and themes. They are not arranged in a specific order; hence, they are nominal variables.

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A 95% confidence interval for the mean blood alcohol concentration (BAC) in fatal crashes where drivers had a positive BAC is (0.133, 0.167) g/dL

The lower bound of the interval, rounded to three decimal places, is 0.133 g/dL, and the upper bound, also rounded to three decimal places, is 0.167 g/dL.

To calculate the confidence interval, we use the formula:

Confidence Interval = sample mean ± (critical value * standard deviation / sqrt(sample size))

Given that the sample mean BAC is 0.15 g/dL, the standard deviation is 0.080 g/dL, and the sample size is 60, we can determine the critical value for a 95% confidence level, which is approximately 1.96 for a large sample size.

Plugging in the values, we have:

Confidence Interval = 0.15 ± (1.96 * 0.080 / sqrt(60))

Confidence Interval = 0.15 ± 0.01714

Therefore, the 95% confidence interval for the mean BAC in fatal crashes where drivers had a positive BAC is (0.133, 0.167) g/dL.

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To determine whether 1 girl in 10 births is a significantly low number of girls using the range rule of thumb, we need to find the range of the probability distribution provided in the table.

a. The maximum value in this range is 0.241.

b. The minimum value in this range is 0.002.

The range rule of thumb states that if the range of a probability distribution is relatively small, the data is considered typical or not significantly different. On the other hand, if the range is relatively large, the data is considered atypical or significantly different.

In this case, the range of the probability distribution is 0.241 - 0.002 = 0.239. Since the range is relatively small, we can conclude that the occurrence of 1 girl in 10 births is not significantly low based on the range rule of thumb.

However, it is important to note that the range rule of thumb is a rough guideline and does not provide a definitive statistical test. To make a more accurate determination, hypothesis testing or other statistical tests should be conducted.

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1. The probability that a labor-management dispute in this industry will be resolved without a strike is 0.475 or 47.5%.

2. The probability that if a labor-management dispute in this industry is resolved without a strike, it was over wages is approximately 0.5684 or 56.84%.

To solve these questions, we can use the information given and apply basic probability concepts.

Let's denote:

- W: Dispute over wages

- C: Dispute over working conditions

- F: Dispute over fringe issues

- S: Dispute resolved without a strike

1. To find the probability that a labor-management dispute will be resolved without a strike, we need to calculate P(S), the probability of resolving a dispute without a strike.

P(S) = P(S|W) × P(W) + P(S|C) × P(C) + P(S|F) × P(F)

Given information:

- P(S|W) = 45% (45 percent of disputes over wages are resolved without strikes)

- P(S|C) = 70% (70 percent of disputes over working conditions are resolved without strikes)

- P(S|F) = 40% (40 percent of disputes over fringe issues are resolved without strikes)

- P(W) = 60% (60 percent of all labor-management disputes are over wages)

- P(C) = 15% (15 percent of all labor-management disputes are over working conditions)

- P(F) = 25% (25 percent of all labor-management disputes are over fringe issues)

Plugging in the values:

P(S) = 0.45 × 0.60 + 0.70× 0.15 + 0.40 × 0.25

     = 0.27 + 0.105 + 0.1

     = 0.475

Therefore, the probability that a labor-management dispute in this industry will be resolved without a strike is 0.475 or 47.5%.

2. To find the probability that if a labor-management dispute is resolved without a strike, it was over wages, we need to calculate P(W|S), the probability of a dispute being over wages given that it was resolved without a strike. We can use Bayes' rule to calculate this.

P(W|S) = (P(S|W) × P(W)) / P(S)

Using the values already known:

P(W|S) = (0.45 × 0.60) / 0.475

      = 0.27 / 0.475

      ≈ 0.5684

Therefore, the probability that if a labor-management dispute in this industry is resolved without a strike, it was over wages is approximately 0.5684 or 56.84%.

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We have proven that the function is not uniformly continuous using the sequential definition of uniform continuity.

To prove that the function f: (0, [infinity]) → R defined by f(x) = x^(3/2) = (x)^(3) is not uniformly continuous, we will use the sequential definition of uniform continuity.

According to the sequential definition of uniform continuity, a function f is uniformly continuous on a given interval if and only if for any two sequences (xn) and (yn) in the interval such that the limit of (xn - yn) is zero, the limit of (f(xn) - f(yn)) is also zero.

Let's consider two sequences: (un) = ((n + n^(2)/n)^(2)) and (vn) = (n^(2)).

Now, we will show that the limit of (un - vn) is zero, but the limit of (f(un) - f(vn)) is not zero, indicating that the function is not uniformly continuous.

1. Limit of (un - vn):

  lim(n→∞) (un - vn)

  = lim(n→∞) ((n + n^(2)/n)^(2) - n^(2))

  = lim(n→∞) (n^(2) + 2n + n - n^(2))

  = lim(n→∞) (2n)

  = ∞

Since the limit of (un - vn) is not zero, we continue to evaluate the second limit.

2. Limit of (f(un) - f(vn)):

  lim(n→∞) (f(un) - f(vn))

  = lim(n→∞) ((un)^(3/2) - (vn)^(3/2))

  = lim(n→∞) (((n + n^(2)/n)^(2))^(3/2) - (n^(2))^(3/2))

  = lim(n→∞) ((n^(2) + 2n + n)^(3/2) - n^(3))

  = lim(n→∞) ((n^(2) + 3n)^(3/2) - n^(3))

  = lim(n→∞) ((n^(2))(1 + 3/n)^(3/2) - n^(3))

  = lim(n→∞) ((n^(2))(1 + 3/n)^(3/2) - n^(3))

  = lim(n→∞) ((n^(2))(1 + (3/n)(1 + o(1))) - n^(3))

  = lim(n→∞) (n^(2) + 3n^(1) + o(n) - n^(3))

  = lim(n→∞) (n^(2) + 3n^(1) - n^(3))

  By comparing the powers of n, we can see that the term with the highest power is -n^(3), which does not converge to zero.

Since the limit of (f(un) - f(vn)) is not zero, the function f(x) = x^(3/2) = (x)^(3) is not uniformly continuous on the interval (0, [infinity]).

Therefore, we have proven that the function is not uniformly continuous using the sequential definition of uniform continuity.

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Calculate the population standard deviation `σ`.The population standard deviation `σ` is not given in the question. Therefore, we have to calculate the sample standard deviation `s` and assume it as the population standard deviation.

The formula to calculate the confidence interval of the population mean is given below:

`Confidence interval = X ± Z × σ/√n` Where

X = sample mean

Z = Z-score at the confidence level

σ = population standard deviation

n = sample size The sample mean `X` is calculated by summing up all values and dividing by the number of values in the sample. `X` is the average of the sample mean. The sample size `n` is the total number of values in the sample. A 96% confidence level means that Z-score at a 96% confidence level is 1.750.1. Calculate the sample mean `X`.

Sum of the given sample values = $294,200

Sample size `n` = 20

X = Sum of the given sample values / Sample size

`X = $294,200 / 20 = $14,710` The sample mean `X` is $14,710.2. Calculate the population standard deviation `σ`.The population standard deviation `σ` is not given in the question. Therefore, we have to calculate the sample standard deviation `s` and assume it as the population standard deviation. `s` is the square root of the sample variance, which is calculated as follows: Step 1: Calculate the sample mean `X`. Already calculated above. Step 2: Calculate the difference between each value and the sample mean. Step 3: Square the above difference. Step 4: Sum the above squared differences. Step 5: Divide the above sum by the sample size minus one. Step 6: Find the square root of the above result. The result is the sample standard deviation `s`. The calculations are shown in the table below: Amount of Money Deviation (X - Mean)Squared.

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The regression equation is useful if the model has a high r-squared value and a low error rate, implying that the regression line is an excellent match for the data.

a. Compute the three sums of squares, SST, SSR, and SSE, using the defining formulas.

SST= 4783.44

SSR= 3196.09

SSE= 1587.35

b. Verify the regression identity,

SST =SSR+SSE.

Is this statement correct

Yes

c. Determine the value of r2, the coefficient of determination. r2= 0.6688

d. Determine the percentage of variation in the observed values of the response variable that is explained by the regression. 66.88%

e. State how useful the regression equation appears to be for making predictions.

The usefulness of the regression equation in predicting future values depends on its strength and accuracy.

The regression equation is useful if the model has a high r-squared value and a low error rate, implying that the regression line is an excellent match for the data.

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To estimate the standard error of b1 (the slope coefficient), we need to calculate the residuals first. Residuals are the differences between the actual sales and the predicted sales based on the regression model. Then, we calculate the sum of squared residuals (SSR) and the total sum of squares (SST).

SSR measures the variation explained by the regression model, while SST measures the total variation in the dependent variable. The coefficient of determination (R-squared) is the ratio of SSR to SST, indicating the proportion of total variation explained by the model. Finally, we can calculate the correlation coefficient by taking the square root of R-squared. However, the actual calculations require more data than what is provided in the question.

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We are asked to find the probability that the mean annual precipitation during 25 randomly picked years will be less than 111.8 inches.

Since the annual precipitation follows a normal distribution, we can use the properties of the normal distribution to calculate the probability.

By standardizing the random variable and using the standard normal distribution table or a calculator, we can find the corresponding probability.

The mean annual precipitation is μ = 109 inches, and the standard deviation is σ = 10 inches. Since the sample size is large (25), we can assume that the distribution of the sample mean will be approximately normal according to the Central Limit Theorem.

To calculate the probability that the mean annual precipitation is less than 111.8 inches, we need to standardize the random variable. We calculate the standard error of the mean (σ/√n) as 10/√25 = 2 inches.

Next, we standardize the random variable using the formula z = (x - μ) / (σ/√n), where x is the value we want to find the probability for. Substituting the given values, we have z = (111.8 - 109) / 2 = 1.4.

To find the probability, we can look up the corresponding value in the standard normal distribution table or use a calculator. The area to the left of z = 1.4 represents the probability that the mean annual precipitation is less than 111.8 inches.

By consulting the standard normal distribution table or using a calculator, we find that the probability is approximately 0.9192 or 91.92%.

The probability that the mean annual precipitation during 25 randomly picked years will be less than 111.8 inches is approximately 0.9192 or 91.92%.

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The probability of getting at least 14 correct guesses in 25 attempts is approximately 0.696.

To find the probability of at least 14 correct guesses in 25 attempts, we can use the binomial probability formula.

The probability of getting exactly k successes in n independent trials, where the probability of success is p, is given by:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

where C(n, k) represents the number of combinations of n items taken k at a time.

In this case, we want to find the probability of getting at least 14 correct guesses, which is equivalent to finding the probability of getting 14, 15, 16, ..., 25 correct guesses.

Let's calculate this probability step by step:

P(at least 14 correct) = P(X = 14) + P(X = 15) + ... + P(X = 25)

P(correct) = 0.5 (probability of success in each guess)

n = 25 (number of guesses)

Using the formula above, we can calculate each individual probability and sum them up:

P(at least 14 correct) = P(X = 14) + P(X = 15) + ... + P(X = 25)

P(at least 14 correct) = Σ[P(X = k)] from k = 14 to 25

P(at least 14 correct) = Σ[C(25, k) * 0.5^k * (1 - 0.5)^(25 - k)] from k = 14 to 25

Using a calculator or software, we can calculate this sum:

P(at least 14 correct) ≈ 0.696 (rounded to three decimal places)

Therefore, the probability of getting at least 14 correct guesses in 25 attempts is approximately 0.696.

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a) The level of v is 5.b) The parent's address is 3.4.5.2.c) v can have 0 siblings.d) At least 5 vertices in T.e) Possible sibling addresses: 3.4.5.2.4.1 to 3.4.5.2.4.9.



a) The level of vertex v can be determined by counting the number of digits in its address. In this case, vertex v has 5 digits in its address (3.4.5.2.4), so it is at level 5. b) To find the address of the parent of v, we need to remove the last digit from v's address. In this case, the parent's address would be 3.4.5.2. c) The least number of siblings v can have is 0, indicating that v is the only child of its parent. d) The smallest possible number of vertices in T can be determined by counting the total number of digits in the address of v. In this case, v's address has 5 digits, so there are at least 5 vertices in T.

e) To find the other addresses that must occur, we can consider the digits in v's address. The possible addresses would be 3.4.5.2.4.1, 3.4.5.2.4.2, 3.4.5.2.4.3, and so on, up to 3.4.5.2.4.9. These addresses represent the possible siblings of v.



In summary, a) The level of v is 5.b) The parent's address is 3.4.5.2.c) v can have 0 siblings.d) At least 5 vertices in T.e) Possible sibling addresses: 3.4.5.2.4.1 to 3.4.5.2.4.9.

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The correct answer is construct 90% and 95% confidence intervals for the population mean based on the given sample data. The widths of the confidence intervals differ, with the 95% interval being wider, indicating a higher level of confidence in capturing the true population mean within that range.

To construct confidence intervals for the population mean, we can use the formula:

Confidence Interval = Sample Mean ±[tex](Z * (Population Standard Deviation / \sqrt{Sample Size})[/tex]

For a 90% confidence interval, we need to find the Z-score corresponding to a confidence level of 90%. The Z-score can be obtained from the standard normal distribution table  For a 90% confidence level, the Z-score is approximately 1.645.

Using the given values:

Sample Mean ([tex]x^-[/tex]) = 86.80°F

Population Standard Deviation (σ) = 14.63°F

Sample Size (n) = 78

For the 90% confidence interval:

Confidence Interval = 86.80 ± [tex](1.645 * (14.63 / \sqrt{78}))[/tex]

Confidence Interval = 86.80 ± 2.5227

The 90% confidence interval for the population mean is (84.2773, 89.3227). This means that we are 90% confident that the true population mean falls within this interval.

Similarly, for a 95% confidence interval, we need to find the Z-score corresponding to a confidence level of 95%. The Z-score for a 95% confidence level is approximately 1.96.

For the 95% confidence interval:

Confidence Interval = 86.80 ± (1.96 * (14.63 / √78))

Confidence Interval = 86.80 ± 2.7538

The 95% confidence interval for the population mean is (84.0462, 89.5538). We can say with 95% confidence that the true population mean lies within this interval.

Comparing the widths of the confidence intervals, we can see that the 95% confidence interval is wider than the 90% confidence interval. This is because a higher confidence level requires a wider interval to capture a larger range of possible population means.

Therefore, construct 90% and 95% confidence intervals for the population mean based on the given sample data. The widths of the confidence intervals differ, with the 95% interval being wider, indicating a higher level of confidence in capturing the true population mean within that range.

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In a sample of 19 observations drawn from a normal population with a mean (μ) of 970 and a standard deviation (σ) of 220, we need to find the probabilities of three events: A. P(X > 1045) ≈ 0.3669:B. P(X < 874) ≈ 0.3336:C. P(X > 929) ≈ 0.5735

To find the probabilities, we need to standardize the values using the z-score formula: z = (X - μ) / σ.

A. P(X > 1045):

First, we calculate the z-score for 1045:

z = (1045 - 970) / 220 = 0.34

Using a standard normal distribution table or a calculator, we can find the probability associated with z = 0.34. In this case, the probability is approximately 0.6331. So, P(X > 1045) = 1 - 0.6331 = 0.3669.

B. P(X < 874):

Next, we calculate the z-score for 874:

z = (874 - 970) / 220 = -0.4364

Using the standard normal distribution table or a calculator, we find the probability associated with z = -0.4364, which is approximately 0.3336. Therefore, P(X < 874) = 0.3336.

C. P(X > 929):

The z-score for 929 is calculated as follows:

z = (929 - 970) / 220 = -0.1864

Using the standard normal distribution table or a calculator, we find the probability associated with z = -0.1864, which is approximately 0.4265. Hence, P(X > 929) = 1 - 0.4265 = 0.5735.

In conclusion, the probabilities are as follows:

A. P(X > 1045) ≈ 0.3669

B. P(X < 874) ≈ 0.3336

C. P(X > 929) ≈ 0.5735

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Using mathematical induction, we can prove that tₙ > 20 for all natural numbers n ≥ 2, starting from the base case t₂ = 11.

To prove that tₙ > 20 for all natural numbers n ≥ 2 using induction, we will follow the steps of the induction proof:

1. Base case: We need to show that t₂ > 20. Using the given recursive formula, we have t₂ = 2t₁ + 1 = 2(5) + 1 = 11. Since 11 > 20, the base case holds.

2. Inductive hypothesis: Assume that for some k ≥ 2, tₖ > 20.

3. Inductive step: We need to show that tₖ₊₁ > 20 using the assumption from the inductive hypothesis. Using the recursive formula, we have tₖ₊₁ = 2tₖ + 1. Since tₖ > 20 (from the inductive hypothesis), we can write tₖ₊₁ = 2tₖ + 1 > 2(20) + 1 = 41.

4. Conclusion: By completing the base case and the inductive step, we have shown that if tₖ > 20 for some k ≥ 2, then tₖ₊₁ > 20. This establishes that tₙ > 20 for all natural numbers n ≥ 2 by mathematical induction.

Therefore, we have proven that tₙ > 20 for all natural numbers n ≥ 2.

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