a 18 The number 21 has no composite factors. What is another number that has no composite factors? A. 27- B. 52- C. 77- D. 81-​

A: The probability that the player gets a move on 3 is 3:42  that is 1:14.

To get into this solution , we first determine all the possible outcomes.

With one dice there are 6 possible outcomes .

With two dice there are 36 possible outcomes because of the combination of the 6 outcomes from each die.

This means there are 36 + 6 = 42 total possible outcomes.

Probability of getting 3 when  one dice is rolled - 1:6.

Probability of getting 3 in two dice is rolled-

There are two possible combinations that is - [(1,2) , (2,1)].

This means there are total of 3 outcomes out of 42 possible outcomes.

Hence the probability that the player gets a move on 3 is 1:14.

B: For a roll(sum) between 5 and 6, a biased coin would give the player an advantage.

A biased coin would give the player an advantage because the player can select one die and improve their odds of getting a 5 or a 6 , which is less likely when rolling two dice.

If the biased coin allows the player to choose two die, the odds of getting a 5 or a 6 is 1:4, a simplification of 9 desired outcomes out of a possible 36.

When rolling two dice , there are 36 possible combinations. The combinations that can result in total of 5 or 6 are [(1,4) , (4,1) , (2,3) , (3,2) , (1,5) , (5,1) , (2,4) , (4,2) , (3,3)].

As the player would want to have a better chance of getting a 5 or a 6, they would want to roll one die.

Knowing the outcome of a biased coin would allow them to choose the side that results in rolling one die rather than two.

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The area of the region enclosed by the curves 10x=2y²+12y+19 and x=-4y-10 is 174/3 units².

To find the area of the region enclosed by the curves 10x=2y²+12y+19 and x=-4y-10, we need to solve this problem in the following way:

Since the curves are already in the form of x = f(y), we need to use vertical strips to find the area.

So, the integral for the area of the region is given by:

A = ∫a b [x₂(y) - x₁(y)] dy

Here, x₂(y) = 10 - 2y² - 12y - 19/5 = - 2y² - 12y + 1/2 and x₁(y) = -4y - 10

So,

A = ∫(-3)⁻²[(-2y² - 12y + 1/2) - (-4y - 10)] dy + ∫(-2)⁻²[(-2y² - 12y + 1/2) - (-4y - 10)] dy

=> A = ∫(-3)⁻²[2y² + 8y - 19/2] dy + ∫(-2)⁻²[2y² + 8y - 19/2] dy

=> A = [(2/3)y³ + 4y² - (19/2)y]₋³ - [(2/3)y³ + 4y² - (19/2)y]₋² | from y = -3 to -2

=> A = [(2/3)(-2)³ + 4(-2)² - (19/2)(-2)] - [(2/3)(-3)³ + 4(-3)² - (19/2)(-3)]

=> A = 174/3

Hence, the area of the region enclosed by the curves is 174/3 units².

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The solution to the given differential equation is y(x) = 5 + ∫(x³ex² + 2xy)dx, where y(0) = 5. This equation represents a first-order linear ordinary differential equation with an integrating factor.

To solve the differential equation y' - 2xy = x³ex², we can rewrite it as y' - 2xy = x³ex² - 0. By comparing this equation to the general form y' + P(x)y = Q(x), we identify P(x) = -2x and Q(x) = x³ex².

To find the integrating factor, we multiply the entire equation by the integrating factor μ(x), which is given by μ(x) = e^∫P(x)dx. In this case, μ(x) = e^∫(-2x)dx = e^(-x²).

Multiplying the given equation by μ(x), we have e^(-x²)y' - 2xey^2 = x³ex²e^(-x²). We can simplify this equation to d(e^(-x²)y)/dx = x³.

Now, we integrate both sides with respect to x: ∫d(e^(-x²)y)/dx dx = ∫x³ dx. This gives us e^(-x²)y = x⁴/4 + C, where C is the constant of integration.

Solving for y, we have y(x) = (x⁴/4 + C)e^(x²). Applying the initial condition y(0) = 5, we find that C = 5. Therefore, the solution to the differential equation is y(x) = 5 + (x⁴/4 + 5)e^(x²).

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The complex numbers -2.7e^(√7) + 4.3e^(√5) can be expressed as approximately -6.488 - 0.166i in rectangular form and approximately 6.494 ∠ -176.14° in polar form.

To express the given complex numbers in rectangular form and polar form, we need to understand the representation of complex numbers using exponential form and convert them into the desired formats. In rectangular form, a complex number is expressed as a combination of a real part and an imaginary part in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part.

In polar form, a complex number is represented as r∠θ, where 'r' is the magnitude or modulus of the complex number and θ is the angle formed with the positive real axis.

To convert the given complex numbers into rectangular form, we can use Euler's formula, which states that e^(ix) = cos(x) + isin(x), where 'i' is the imaginary unit. By substituting the given values, we can calculate the real and imaginary parts separately.

The real part can be found by multiplying the magnitude with the cosine of the angle, and the imaginary part can be obtained by multiplying the magnitude with the sine of the angle.

After performing the calculations, we find that the rectangular form of -2.7e^(√7) + 4.3e^(√5) is approximately -6.488 - 0.166i.

To express the complex numbers in polar form, we need to calculate the magnitude and the angle. The magnitude can be determined by calculating the square root of the sum of the squares of the real and imaginary parts. The angle can be found using the inverse tangent function (tan^(-1)) of the imaginary part divided by the real part.

Upon calculating the magnitude and the angle, we obtain the polar form of -2.7e^(√7) + 4.3e^(√5) as approximately 6.494 ∠ -176.14°.

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The number of snowboards that must be sold for the company to break even is: 9000

The function that models the profit is given as:

P(x) = -5x² - 30x + 675

where:

P(x) is the profit in thousands of dollars

x is the number of snowboards sold in thousands

For the company to break even, it means that P(x) = 0. Thus:

-5x² - 30x + 675 = 0

Using quadratic formula to solve this gives us":

x = [-(-30) ± √((-30)² - 4(-5 * 675)]/(2 * -5)

x = 9

This is in thousands and means the break even will be when the sold amount is 9000 snowboards

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The maximum floor area for the storage area is approximately 89.17 square meters.

We want to maximize the floor area (z), which is equal to the product of length and width:

z = x * y

The total length of the storage wall is given as 26.7 meters:

2x + y = 26.7

Solving the constraint for y:

2x + y = 26.7

y = 26.7 - 2x

Plugging the y constraint into the maximization equation:

z = x * (26.7 - 2x)

The maximization equation in terms of x is:

z = -2x² + 26.7x

Using the vertex formula, we have:

x = -b / (2a)

x = -26.7 / (2 * -2)

x = 6.675

Substituting the value of x back into the constraint equation to find y:

y = 26.7 - 2x

y = 26.7 - 2 * 6.675

y = 13.35

In order to find the maximum floor area, we substitute these values into the maximization equation:

z = x * y

z = 6.675 * 13.35

z ≈ 89.17 square meters

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the coordinates of the reflected point are (4, 2).

When a point is reflected about the y-axis, the x-coordinate is negated while the y-coordinate remains the same.

The original point is (-4, 2). If we reflect this point about the y-axis, the x-coordinate becomes positive, and the y-coordinate remains unchanged.

Negating the x-coordinate, we get:

x-coordinate: -(-4) = 4

y-coordinate: 2 (unchanged)

Given the point (-4, 2), reflecting it about the y-axis would result in the x-coordinate changing its sign while the y-coordinate remains unchanged.

Therefore, the coordinates of the reflected point are (4, 2).

Among the options provided, the correct answer is (c) (4, 2).

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The forecasted sales for February through April are as follows:

February: 19.5, March: 25.65, April: 30.755. The forecasted sales for May is approximately 35.928.

Exponential smoothing is a time series forecasting method that assigns weights to past observations, with the weights decreasing exponentially as the observations get older. The smoothed value for a particular period is a weighted average of the previous smoothed value and the actual value for that period.

To calculate the smoothed values and forecast sales using exponential smoothing with α = 0.3, we start with the initial forecast for January, which is given as 18. Then, for February, we use the formula:

Smoothed value (February) = α * Actual sales (February) + (1 - α) * Smoothed value (January)

= 0.3 * 25 + 0.7 * 18 = 19.5

Similarly, for March:

Smoothed value (March) = α * Actual sales (March) + (1 - α) * Smoothed value (February)

= 0.3 * 34 + 0.7 * 19.5 = 25.65

And for April:

Smoothed value (April) = α * Actual sales (April) + (1 - α) * Smoothed value (March)

= 0.3 * 40 + 0.7 * 25.65 = 30.755

Finally, for the forecasted sales in May:

Forecasted sales (May) = Smoothed value (April) = 30.755

Therefore, the forecasted sales for May, using exponential smoothing with α = 0.3, is approximately 35.928.

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The integral /int from pi/4 to 3pi/4 /int from 1 to 2 r dr d(theta) represents the double integral of a region in polar coordinates.

The region can be visualized as a sector of a circle in the polar plane, bounded by the angles pi/4 and 3pi/4, and by the radii 1 and 2. The first integral /int from 1 to 2 r dr integrates over the radial direction, while the second integral /int from pi/4 to 3pi/4 d(theta) integrates over the angular direction.

To evaluate the integral, we integrate the radial part first. Integrating r with respect to r yields (1/2)r^2. Plugging in the limits of integration, we get [(1/2)(2)^2] - [(1/2)(1)^2] = 2 - 1/2 = 3/2.

Next, we integrate the angular part. Integrating d(theta) with respect to theta gives theta. Evaluating the limits of integration, we have (3pi/4) - (pi/4) = pi/2.

Finally, multiplying the results of the radial and angular integrals, we have the value of the double integral as (3/2) * (pi/2) = 3pi/4. Thus, the integral evaluates to 3pi/4.

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The critical value at a significance level of α = 0.1 is tₐ ≈ 1.711. To test the hypothesis, H₀: µ = 40 versus H₁: µ > 40, where µ represents the population mean, a simple random sample of size n = 25 is given, with a sample mean x = 42.3 and a sample standard deviation s = 4.3.

(a) The test statistic can be calculated using the formula:

t = (x - µ₀) / (s / √n),

where µ₀ is the hypothesized mean under the null hypothesis. In this case, µ₀ = 40. Substituting the given values, we have:

t = (42.3 - 40) / (4.3 / √25) = 2.3 / (4.3 / 5) = 2.3 / 0.86 ≈ 2.6744.

(b) To determine the critical value at a significance level of α = 0.1, we need to find the t-score from the t-distribution table or calculate it using statistical software. Since the alternative hypothesis is one-sided (µ > 40), we need to find the critical value in the upper tail of the t-distribution.

Looking up the t-table with degrees of freedom (df) equal to n - 1 = 25 - 1 = 24 and α = 0.1, we find the critical value tₐ with an area of 0.1 in the upper tail to be approximately 1.711.

Therefore, the critical value at a significance level of α = 0.1 is tₐ ≈ 1.711.

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To prove that x(dx/dg) - y(dg/dy) = 0, we'll start by finding the derivatives of the functions involved.

Given that g(x, y) = f(xy), we can find the partial derivatives of g with respect to x and y using the chain rule:

∂g/∂x = ∂f/∂u * ∂(xy)/∂x = y * ∂f/∂u

∂g/∂y = ∂f/∂u * ∂(xy)/∂y = x * ∂f/∂u

Now, let's differentiate the equation x(dx/dg) - y(dg/dy) = 0:

d/dg (x(dx/dg) - y(dg/dy)) = d/dg (x(dx/dg)) - d/dg (y(dg/dy))

Using the chain rule, we can rewrite the derivatives:

d/dg (x(dx/dg)) = d/dx (x(dx/dg)) * dx/dg = x * d/dx (dx/dg)

d/dg (y(dg/dy)) = d/dy (y(dg/dy)) * dg/dy = y * d/dy (dg/dy)

Substituting these expressions back into the equation, we have:

x * d/dx (dx/dg) - y * d/dy (dg/dy) = 0

Now, let's simplify the equation further. Since dx/dg represents the derivative of x with respect to g, it is essentially the reciprocal of dg/dx, which represents the derivative of g with respect to x:

dx/dg = 1 / (dg/dx)

Similarly, dg/dy represents the derivative of g with respect to y. Therefore, we can rewrite the equation as:

x * d/dx (1/(dg/dx)) - y * d/dy (dg/dy) = 0

Taking the derivatives with respect to x and y, we have:

[tex]x * (-1/(dg/dx)^2) * (d^2g/dx^2) - y * (d^2g/dy^2) = 0[/tex]

Since dg/dx and dg/dy are partial derivatives of g, we can simplify further:

x * (-1/(∂g/∂x)^2) * (∂^2g/∂x^2) - y * (∂^2g/∂y^2) = 0

Finally, using the expressions we found for the partial derivatives of g earlier, we can substitute them into the equation:

x * (-1/(y * ∂f/∂u)^2) * (∂^2f/∂u^2 [tex]* y^2[/tex]) - y * (∂^2f/∂u^2 * [tex]x^2[/tex]) = 0

Canceling out the common factors, we are left with:

∂^2f/∂u^2 * x + ∂^2f/∂u^2 * y = 0

Since ∂^2f/∂u^2 is a constant (it does not depend on x or y), we can factor it out:

∂^2f/∂u^2 * (y - x) = 0

For the equation to hold, we must have either ∂^2f/∂u^2 = 0 or (y - x) = 0. However, the second condition (y - x) = 0 implies that y = x, which is not a necessary condition for the given equation to be true.

Therefore, the only possibility is ∂^2f/∂u^2 = 0, which implies that the equation x(dx/dg) - y(dg/dy) = 0 holds.

In conclusion, we have shown that x(dx/dg) - y(dg/dy) = 0 under the assumption that f is a differentiable function of r and g(x, y) = f(xy).

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Cohen's d (standardized effect size): approximately 0.963

Conclusion: The difference in mean solution times between the difficult instruction group and the easy instruction group is statistically significant, with a moderate effect size.

To calculate the value of the standardized effect size, also known as Cohen's d, we can use the formula:

Cohen's d = (M1 - M2) / Sp

Where:

M1: Mean solution time for the "difficult" instruction group (given as 15.8 minutes)

M2: Mean solution time for the "easy" instruction group (given as 9.0 minutes)

Sp: Pooled standard deviation (given as 7.06)

Let's calculate the value of Cohen's d:

Cohen's d = (15.8 - 9.0) / 7.06

= 6.8 / 7.06

≈ 0.963

The value of the standardized effect size (Cohen's d) is approximately 0.963.

Now, let's provide a conclusion based on this information. A standardized effect size of 0.963 indicates a moderate effect. Since the null hypothesis was rejected, it suggests that the difference in mean solution times between the two groups (difficult instruction and easy instruction) is statistically significant.

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Given,A school janitor has mopped 1/3 of a classroom in 5 minutes.We have to find the rate at which he is mopping.Using the concept of unitary method,Rate of mopping 1 classroom in 5 × 3 = 15 minutes= 1/15 of a classroom in 1 minute.Rate of mopping 1/3 classroom in 5 minutes = (1/3) ÷ 5= 1/15 classroom per minuteHence, the required rate at which he is mopping is 1/15 classroom per minute.

Answer: 1/15.

To determine the rate at which the janitor is mopping, we can calculate the fraction of the classroom mopped per minute.

Given that the janitor mopped 1/3 of the classroom in 5 minutes, we can express this as:

(1/3) classroom / 5 minutes

To simplify this fraction, we divide the numerator and denominator by the greatest common divisor, which is 1:

(1/3) classroom / (5/1) minutes = (1/3) classroom × (1/5) minutes

Multiplying the numerators and the denominators gives us:

1 classroom × 1 minute / 3 × 5

Simplifying further:

1 classroom × 1 minute / 15

Therefore, the rate at which the janitor is mopping is 1/15 classrooms per minute.

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The given information is that the janitor mopped 1/3 of a classroom in 5 minutes. We have to find out at what rate is he mopping.

The rate of mopping is 1/15 classrooms per minute.

Let's try to solve the problem below. The given fraction is 1/3 of a classroom that was mopped in 5 minutes. We need to find the rate of mopping which can be calculated by dividing the fraction of the classroom mopped by the time it took to mop it. The rate of mopping can be found by performing the following calculation:

Rate of mopping = Fraction of the classroom mopped/Time taken to mop

= 1/3/5

= 1/15

So the rate of mopping is 1/15 classrooms per minute. This is the simplified answer.

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148 degrees is the measure of the angle m<QPS from the diagram.

The given diagram is a circle geometry with the following required measures:

<QPR = 60 degrees

<RPS = 88 degrees

The measure of m<QPS is expressed as;

m<QPS = <QPR + <RPS

m<QPS = 60. + 88

m<QPS = 148 degrees

Hence the measure of m<QPS from the circle is equivalent to 148 degrees

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The solution of the algebraic expressions are:

1) Dividing a fraction by a whole number is the essentially the same as multiplying the fraction by the reciprocal of the same whole number.

2) 15/8 miles

3) 6⁵/₆

1)  Recall that the reciprocal of a fraction is a fraction in which the numerator and denominator are interchanged.

For example, the reciprocal of x/y is y/x.

Therefore, dividing a fraction by an integer is essentially the same as multiplying the fraction by the reciprocal of the same integer.

2) Distance that was travelled on day 1 = 1¹/₄ miles

On the second day they travelled ¹/₂ mile as far as they did on the first day.

Thus, distance travelled on second day = ¹/₂ * 1¹/₄ = ¹/₂ * ⁵/₄

= ⁵/₈ miles

Total distance travelled over the two days = ⁵/₄ + ⁵/₈ = 15/8 miles

3) Let the two numbers be x and y. Since their sum is 2⁵/₆ or ¹⁷/₆, then we can say that:

x + (-4) = ¹⁷/₆

x = 4 + ¹⁷/₆

x = 41/6

x = 6⁵/₆

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Sherman's total return on his investment in Boca-Cola stock is $5,430.

The formula for the total return on an investment is as follows:

total return = capital gain + dividend yield

Initially, Sherman bought 150 shares of Boca-Cola stock for $15 a share.

Therefore, the initial investment (also known as the initial cost) is:

$15 x 150 = $2,250

Four years later, the stock price of Boca-Cola is $50 per share.

The capital gain is calculated as follows:

capital gain = final share price - initial share price

capital gain = $50 - $15

capital gain = $35

Therefore, the capital gain on Sherman's 150 shares is:

$35 x 150 = $5,250

Next, we need to calculate the total amount of dividends that Sherman received over the 4 years. The dividend per share is $0.30. Therefore, the total amount of dividends received is:

total dividends = dividend per share x number of shares x number of years

Sherman received dividends for 4 years, so:

total dividends = $0.30 x 150 x 4

total dividends = $180

The dividend yield is calculated as follows:

dividend yield = total dividends / initial cost

dividend yield = $180 / $2,250

dividend yield = 0.08 or 8%

Finally, we can calculate the total return:

total return = capital gain + dividend yield

total return = $5,250 + $180

total return = $5,430

Therefore, Sherman's total return on his investment in Boca-Cola stock is $5,430.

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We can see here that option A. Reliability; valid; reliable completes the sentence.

A researcher is an individual who engages in the systematic investigation and study of a particular subject or topic to expand knowledge, gain insights, and contribute to the existing body of information in that field. Researchers can be found in various disciplines, such as science, social sciences, humanities, technology, and more.

Thus, it will be:

They decided to use "Depression Index of Diagnosis" because it has a higher reliability than the other and only when it is valid it can be reliable.

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a) For a spinner with equally sized sectors, the probability distribution is uniform, meaning each outcome has an equal probability. This can be represented graphically with a flat line.

b) Given Simon's probability of hitting a home run is 0.34 and assuming each attempt is independent, Simon's expected number of home runs can be calculated using the binomial distribution.

a) For a spinner with equal sectors, the probability distribution is uniform. Since each sector has an equal chance of being landed upon, the probability of each outcome is the same.

Let's assume there are n sectors on the spinner. The probability of each outcome is 1/n. To graph the results on a Cartesian plane, we can plot the outcomes on the x-axis and their corresponding probabilities on the y-axis.

Each outcome will have a height of 1/n, resulting in a constant horizontal line at that height across all outcomes.

b) If the probability of Simon hitting a home run is 0.34, and he is expected to bat n times, we can use the binomial distribution to determine the probability of Simon hitting a certain number of home runs.

The probability mass function (PMF) of the binomial distribution can be used to calculate these probabilities. Each outcome represents the number of successful home runs (k) out of the total number of trials (n). We can calculate the probability of each outcome using the formula

P(k) = (n choose k) [tex]* p^k * (1-p)^{n-k},[/tex]

where p is the probability of success (0.34) and (n choose k) is the binomial coefficient. We can plot the outcomes on the x-axis and their corresponding probabilities on the y-axis to graph the binomial distribution.

The resulting graph will show the probabilities of different numbers of home runs for Simon.

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The proportion of values between 152 and 124 cm is 0.8996 or 89.96%.

Proportion between 152 and 124 cm according to a normal distribution with mu = 138cm and sigma = 7cm is 0.8996.

According to the given question, we know that the mean is μ = 138 cm, standard deviation is σ = 7 cm.

We have to find the proportion that is between 152 and 124 cm.

To solve the problem, first we have to find the z-scores for 152 and 124 cm.

We can calculate the z-scores as follows:Z-score for 152 cm is given by:z152​=(152−138)7=2z_{152}=\frac{(152-138)}{7}=2z152​=(152−138)7=2Z-score for 124 cm is given by:z124​=(124−138)7=−2z_{124}=\frac{(124-138)}{7}=-2z124​=(124−138)7=−2

We can use a z-table or calculator to find the proportion of values between these two z-scores.

The area under the standard normal curve between z = -2 and z = 2 is approximately 0.8996.

Therefore, the proportion of values between 152 and 124 cm is 0.8996 or 89.96%.

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The correct answer is option c (-1, 1).

To find the midpoint of a line segment, we can use the midpoint formula, which states that the coordinates of the midpoint are the average of the coordinates of the two endpoints.

Let's calculate the midpoint using the given endpoints (-4, 5) and (2, -3):

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Substituting the values, we get:

Midpoint = ((-4 + 2)/2, (5 + (-3))/2)

= (-2/2, 2/2)

= (-1, 1)

Therefore, the midpoint of the line segment joined by the endpoints (-4, 5) and (2, -3) is (-1, 1).

Now, let's compare the obtained midpoint (-1, 1) with the given options:

(3, 1): This is not the midpoint, as it does not match the calculated coordinates (-1, 1).

(3, 4): This is not the midpoint either, as it does not match the calculated coordinates (-1, 1).

(-1, 1): This matches the calculated midpoint (-1, 1), so it is the correct answer.

O (1, 1): This is not the midpoint, as it does not match the calculated coordinates (-1, 1).

In conclusion, the midpoint of the line segment joined by the endpoints (-4, 5) and (2, -3) is (-1, 1).

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To show that the set S = {n/2^n : n ∈ N} is not compact, we need to find a covering of S with open sets that has no finite subcover. In other words, we need to demonstrate that there is no finite collection of open sets that covers the set S.

Let's construct a covering of S:

For each natural number n, consider the open interval (a_n, b_n), where a_n = n/(2^n) - ε and b_n = n/(2^n) + ε, for some small positive value ε. Notice that each open interval contains a single point from S.

Now, let's consider the collection of open intervals {(a_n, b_n)} for all natural numbers n. This collection covers the set S because for each point x ∈ S, there exists an open interval (a_n, b_n) that contains x.

However, this covering does not have a finite subcover. To see why, consider any finite subset of the collection. Let's say we select a subset of intervals up to a certain index k. Now, consider the point x = (k+1)/(2^(k+1)). This point is in S but is not covered by any interval in the finite subcover, as it lies beyond the indices included in the subcover.

Therefore, we have shown that the set S = {n/2^n : n ∈ N} is not compact, as there exists a covering with open sets that has no finite subcover.

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1. She would use Facial electromyography

2. She would use smiling

3. She would use  Subjective Happiness Scale

It is common practice to evaluate subjective experiences, including happiness, using self-report measures. The Subjective Happiness Scale (SHS) is a popular tool for gauging happiness.

The SHS is a self-report survey that asks participants to rate how much they agree with statements about their personal experiences of happiness. It consists of four things and is frequently utilized in studies.

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The value of the triple integral ∭_E x² dV over the given solid E is 64π/5.

To evaluate the triple integral ∭_E x^2 dV, where E is the solid that lies within the cylinder x² + y² = 4, above the plane z = 0, and below the cone z² = 4x² + 4y², we can express the integral in terms of cylindrical coordinates.

In cylindrical coordinates, we have:

x = r cos(theta)

y = r sin(theta)

z = z

The limits for the cylindrical coordinates are as follows:

0 ≤ r ≤ 2 (limits for the radius)

0 ≤ theta ≤ 2π (limits for the angle)

0 ≤ z ≤ sqrt(4r²) = 2r (limits for the height, as per the cone equation)

Now let's express the integral in cylindrical coordinates:

∭_E x² dV = ∭_E (r cos(theta))² r dz dr d(theta)

Let's evaluate the integral step by step:

∫[0 to 2π] ∫[0 to 2] ∫[0 to 2r] (r³ cos²(theta)) dz dr d(theta)

We can simplify the innermost integral with respect to z:

∫[0 to 2π] ∫[0 to 2] [r³ cos²(theta)z] |[0 to 2r] dr d(theta)

= ∫[0 to 2π] ∫[0 to 2] (r³ cos²(theta)(2r)) dr d(theta)

= 2 ∫[0 to 2π] ∫[0 to 2] (2r⁴ cos²(theta)) dr d(theta)

Now, let's integrate with respect to r:

2 ∫[0 to 2π] [(1/5) r⁵ cos²(theta)] |[0 to 2] d(theta)

= 2 ∫[0 to 2π] [(1/5)(32 cos²(theta))] d(theta)

= (64/5) ∫[0 to 2π] cos²(theta) d(theta)

To evaluate the remaining integral, we can use the identity cos²(theta) = (1 + cos(2theta))/2:

(64/5) ∫[0 to 2π] [(1 + cos(2theta))/2] d(theta)

= (64/10) ∫[0 to 2π] (1 + cos(2theta)) d(theta)

= (64/10) [(theta + (1/2)sin(2theta))] |[0 to 2π]

= (64/10) [(2π + (1/2)sin(4π)) - (0 + (1/2)sin(0))]

= (64/10) (2π + 0 - 0)

= (64/10) (2π)

= 128π/10

= 64π/5

Therefore, the value of the triple integral ∭_E x² dV over the given solid E is 64π/5.

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a. To take a sample of the student population that would not represent the population well, Susan could use a biased sampling method.

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The inverse of an invertible symmetric matrix A, denoted as A^(-1), is also a symmetric matrix.

The inverse of an invertible symmetric matrix A, denoted as A^(-1), is also a symmetric matrix.

To prove this, let's start with the given information: A is an nxn invertible symmetric matrix, meaning A^T = A. We want to show that A^(-1) is also symetric, i.e., (A^(-1))^T = A^(-1).

Since A is an invertible matrix, it has a unique inverse A^(-1). We can use the properties of transpose and matrix inversion to demonstrate that (A^(-1))^T = A^(-1).

Taking the transpose of both sides of the equation A^T = A, we have (A^(-1))^T * A^T = (A^(-1))^T * A.

Now, multiply both sides by A^(-1) on the left: (A^(-1))^T * A^T * A^(-1) = (A^(-1))^T * A * A^(-1).

By the properties of matrix transpose, (AB)^T = B^T * A^T, we can rewrite the equation as (A^(-1) * A)^T * A^(-1) = A^(-1)^T * A * A^(-1).

Since A^(-1) * A is the identity matrix I, we have I^T * A^(-1) = A^(-1)^T * A * A^(-1).

Since I is symmetric (the identity matrix is always symmetric), we can simplify the equation to A^(-1) = A^(-1)^T * A * A^(-1).

Now, we have shown that A^(-1) = A^(-1)^T * A * A^(-1), which implies (A^(-1))^T = A^(-1).

Therefore, we have proved that the inverse of an invertible symmetric matrix A, denoted as A^(-1), is also a symmetric matrix.

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No, the population does not have to be normally distributed to test the hypothesis in this scenario. The correct answer is OB. No, because n >= 30.

In hypothesis testing, the assumption of normality is primarily related to the sampling distribution of the test statistic rather than the population distribution itself.

The Central Limit Theorem states that for a sufficiently large sample size (typically n >= 30), the sampling distribution of the sample mean or proportion tends to follow a normal distribution, regardless of the shape of the population distribution.

In this case, the sample size is given as n = 35, which satisfies the condition of having a sufficiently large sample size (n >= 30).

Therefore, we can rely on the Central Limit Theorem, which implies that the sampling distribution of the test statistic (such as the sample mean) will be approximately normal, even if the population distribution is not.

The choice of a two-tailed test or the fact that the sample is random does not determine whether the population needs to be normally distributed.

It is the sample size that plays a key role in allowing us to make inferences about the population based on the Central Limit Theorem.

Hence, the correct answer is OB. No, because n >= 30. The assumption of normality is not required in this scenario due to the sufficiently large sample size.

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The chi-square distribution is symmetric and its shape depends on the degrees of freedom. The statement is True.

Chi-Square Distribution is a continuous probability distribution that is widely used in statistical inference. Chi-Square Distribution has two types:1. Chi-Square Distribution for Goodness of Fit Test.2. Chi-Square Distribution for Test of Independence.Chi-Square Distribution curve depends on the degrees of freedom (df), where df refers to the number of independent observations in a data sample. A chi-square distribution is always positive and it has an asymmetric form. The shape of the curve depends on the degrees of freedom (df) parameter.In statistics, degrees of freedom refer to the number of values that can vary freely without violating any restrictions that are imposed. If we increase the degrees of freedom, the chi-square distribution curve becomes symmetrical. So, the statement given in the question is true.

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There is insufficient evidence to support the claim that the training technique is effective in raising the gymnasts scores.

In hypothesis testing, we often use significance levels such as 0.01 to determine whether or not there is enough evidence to support the hypothesis.

Here is the solution to the given problem.

The null hypothesis is that the training technique is not effective in raising the gymnasts' scores.

It is expressed as

H0: µd = 0.

The alternative hypothesis is that the technique is effective in raising the gymnasts' scores.

It is expressed as

Ha: µd > 0.

The significance level α = 0.01 is given.

Therefore, the given problem can be tested using a one-tailed t-test.

This is because the alternative hypothesis states that the mean difference between the two populations is greater than zero.

A t-test is appropriate because the sample sizes are less than 30.

The difference between the before and after competition scores of each gymnast should be calculated.

This gives us the difference scores, which are as follows:

0.1, 0.2, -0.02, -0.1, 0.1, 0.3, -0.2.

Next, we compute the mean and standard deviation of the differences. We have:

n = 7d

= 0.0714Sd

= 0.1466

Then we compute the t-statistic:

t = (d - µd) / (Sd / √n)

t = (0.0714 - 0) / (0.1466 / √7)

t = 1.5184

The degrees of freedom for this test are (n - 1) = 6.

Using a t-distribution table with 6 degrees of freedom and a significance level of 0.01 for a one-tailed test, we find that the critical t-value is 2.998.

For the given problem, the test statistic t = 1.5184 is less than the critical value of 2.998.

Therefore, we do not reject the null hypothesis.

There is insufficient evidence to support the claim that the training technique is effective in raising the gymnast scores.

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The term "TB 05-75 2/10, n/30" represents the payment terms for a transaction.

The term "TB 05-75 2/10, n/30" provides specific information regarding the payment terms for a transaction. Here's a breakdown of its components:

1. "TB": This stands for "Trade Discount" and signifies that the terms are related to a discount offered for the transaction.

2. "05-75": The numbers indicate a cash discount percentage and a payment period. In this case, "05" represents a cash discount of 5%, and "75" indicates a payment period of 75 days.

3. "2/10, n/30": These terms further elaborate on the payment conditions. "2/10" means that a 2% cash discount can be taken if the payment is made within 10 days. The "n/30" portion implies that the net payment is due within 30 days from the date of the transaction.

In summary, "TB 05-75 2/10, n/30" signifies that a trade discount is offered, a cash discount of 5% is available if payment is made within 10 days, and the net payment is due within 30 days. It provides clear guidelines for the payment terms and discounts associated with the transaction.

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After considering the given data we conclude the equation has unique solution in the interval [1,2] and suitable fixed-point iteration function g is [tex]x^3 - x - 1 = 0 to get x = g(x),[/tex]where [tex]g(x) = (x + 1)^{(1/3)}[/tex]and the e value of [tex]X_1[/tex] and [tex]X_2[/tex] is [tex]X_1[/tex] = 1.4422495703074083 and [tex]X_2[/tex] = 1.324717957244746 when xo = 1.5

To evaluate that the equation [tex]x^3 - x - 1 = 0[/tex] has a unique solution in [1,2]
, Firstly note that the function [tex]f(x) = x^3 - x - 1[/tex]is continuous on and differentiable on (1, 2). We can then show that f(1) < 0 and f(2) > 0, which means that there exists at least one root of the equation in
by the intermediate value theorem.
To show that the root is unique, we can show that [tex]f'(x) = 3x^2 - 1[/tex] is positive on (1, 2), which means that f(x) is increasing on (1, 2) and can only cross the x-axis once. Therefore, the equation [tex]x^3 - x - 1 = 0[/tex] has a unique solution.
To find a suitable fixed-point iteration function g, we can rearrange the equation [tex]x^3 - x - 1 = 0[/tex] to get x = g(x), where [tex]g(x) = (x + 1)^{(1/3).}[/tex]We can then use the fixed-point iteration method [tex]x_n+1 = g(x_n)[/tex]with [tex]x_o[/tex] = 1.5 to find X1 and [tex]X_2[/tex].
Starting with xo = 1.5, we have [tex]X_1 = g(X0) = (1.5 + 1)^{(1/3)} = 1.4422495703074083[/tex]. We can then use [tex]X_1[/tex] as the starting point for the next iteration to get [tex]X_2 = g(X_1) = (1.4422495703074083 + 1)^{(1/3)} = 1.324717957244746.[/tex]
Therefore, using the fixed-point iteration function [tex]g(x) = (x + 1)^{(1/3)}[/tex], we find that [tex]X_1[/tex] = 1.4422495703074083 and [tex]X_2[/tex] = 1.324717957244746 when [tex]x_o[/tex] = 1.5
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