What is π and explain how it is used in finding the circumference of the circle.​

To find the limit as t approaches infinity of the solution y(t) to the initial value problem dy/dt = 2y(1 - y^5) with boundary condition y(0) = 1, follow these steps:

1. First, recognize that the given equation is a first-order, separable differential equation. Separate the variables y and t:
  dy/y(1 - y^5) = 2dt

2. Integrate both sides:
  ∫(1/y(1 - y^5))dy = ∫2dt

3. Evaluate the integrals:
  The left side requires partial fraction decomposition or a substitution (let u = 1 - y^5):
  ∫(1/y(u))dy = ∫2dt

  The right side is simpler:
  2t + C1

4. Solve for y(t) and apply the initial condition y(0) = 1:
  y(t) = some function of t and C1
  y(0) = 1 implies C1 = some value

5. Determine the limit as t approaches infinity:
  lim t→∞ y(t)

Following these steps will give you the solution to the problem and the limit as t approaches infinity.

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The probability of z being less than 0.25 as 0.5987 based on population proportion.

To use the normal approximation, we first need to check if the conditions are met. For this, we need to check if np and n(1-p) are both greater than or equal to 10.

np = 98 x 0.56 = 54.88

n(1-p) = 98 x 0.44 = 43.12

Since both np and n(1-p) are greater than 10, we can use the normal approximation.

Next, we need to find the mean and standard deviation of the sampling distribution of proportion.

Mean = np = 54.88

Standard deviation = sqrt(np(1-p)) = sqrt(98 x 0.56 x 0.44) = 4.43

Now we can standardize the variable X and find the probability:

z = (X - mean) / standard deviation = (56 - 54.88) / 4.43 = 0.25

Using a standard normal table or calculator, we can find the probability of z being less than 0.25 as 0.5987.

Therefore, P(X < 56) = P(Z < 0.25) = 0.5987.

Note that we rounded the mean and standard deviation to two decimal places, but you should keep the full values in your calculations to minimize rounding errors.

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By Rolle's theorem, there are at least two points on the interval [0,5] where the derivative of f(x) is equal to zero, namely x = 5/3 and x = 5/2.

Now, let's apply this theorem to the given function f(x) = x(x-5)^2 on the interval [0,5]. First, we need to check if the function satisfies the conditions of Rolle's theorem.

The function f(x) is a polynomial, and we know that polynomials are continuous and differentiable everywhere. Therefore, f(x) is continuous on the interval [0,5] and differentiable on the open interval (0,5).

Next, we need to check if f(0) = f(5). Evaluating the function at the endpoints of the interval, we get:

f(0) = 0(0-5)² = 0

f(5) = 5(5-5)² = 0

Since f(0) = f(5) = 0, we can conclude that Rolle's theorem applies to the function f(x) on the interval [0,5].

Finally, we need to find the point(s) that are guaranteed to exist by Rolle's theorem. According to the theorem, there must be at least one point c in (0,5) such that f'(c) = 0.

To find the derivative of f(x), we need to use the product rule and the chain rule:

f'(x) = (x-5)² + x(2(x-5)) = 3x² - 20x + 25

Now, we need to find the value(s) of x in (0,5) that make f'(x) = 0:

3x² - 20x + 25 = 0

Using the quadratic formula, we get:

x = (20 ± √(20² - 4(3)(25))) / (2(3)) = (20 ± 5) / 6

x = 5/3 or x = 5/2

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The total mass of the fishing rod is approximately 33.18 grams.

To find the total mass of the fishing rod, we need to integrate the density function δ(x) over the entire length of the rod.

The mass of an infinitesimal element of length dx located at a distance x from the handle is given by:

dm = δ(x) × dx

So the total mass of the fishing rod is given by

M = [tex]\int\limits^1_0[/tex] δ(x) dx

M = [tex]\int\limits^1_0[/tex] (100/(1+x²)) dx

Using the substitution u = 1 + x^2, du/dx = 2x, the integral becomes:

M = [tex]\int\limits^2_1[/tex] (100/u) du/2x

M = 50  [tex]\int\limits^2_1 u^{-1/2}[/tex]  du

M = 50 [2[tex]u^{1/2}[/tex]]

M = 50 [2([tex]2^{1/2}[/tex] - 1)]

M = 50 × ([tex]2^{1/2}[/tex] - 1)

M ≈ 33.18 grams

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The given question is incomplete, the complete question is:

The density function of a 1 meter long fishing rod is δ(x) = 100/ (1+x²)grams per meter, where x is the distance from the handle in meters, find the total mass of the fishing rod

Answer:

x² + 5x + 11

------------------------

3x - 4 | 3x³ + 17x² - 16x - 16

- (3x³ - 4x²)

---------------

21x² - 16x

- (21x² - 28x)

---------------

12x - 16

- (12x - 16)

---------

0

Therefore, the result when 3x³ + 17x² - 16x - 16 is divided by 3x - 4 is x² + 5x + 11.

To find the Maclaurin series for f(x) = 2 + ∫(0 to x) arctan(t) dt, we first need to find the power series representation of the integrand, arctan(t). The Maclaurin series for arctan(t) is given by:

arctan(t) = t - (t^3)/3 + (t^5)/5 - (t^7)/7 + ...

Now, we need to find the integral of arctan(t) with respect to t:

f(x) = 2 + ∫(0 to x) (t - (t^3)/3 + (t^5)/5 - (t^7)/7 + ...) dt

Integrating term by term, we get:

f(x) = 2 + (t^2)/2 - (t^4)/12 + (t^6)/30 - (t^8)/56 + ...

Now, replacing t with x, we obtain the Maclaurin series for f(x):

f(x) = 2 + (x^2)/2 - (x^4)/12 + (x^6)/30 - (x^8)/56 + ...

The first five non-zero terms, in order of increasing degree, are:

2, (x^2)/2, -(x^4)/12, (x^6)/30, -(x^8)/56.

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This indicates that there might be an error in the joint pdf or the limits of integration.

To determine the probability that the random variable Y lies between 0 and 1, we need to integrate the joint pdf over the region where 0 < Y < 1.

P(0 < Y < 1) = ∫∫fxy(x,y) dx dy, where the limits of integration are 0 < Y < 1 and 0 < X < ∞.

= ∫0^1 ∫0^∞ xy exp(-x) dx dy, since fxy(x,y) = xy exp(-x).

= ∫0^1 y [(-x) exp(-x)]|0^∞ dy, using integration by parts.

= ∫0^1 y (0 - 1) dy, since [(-x) exp(-x)]|0^∞ = 0.

= -1/2.

Therefore, the probability that the random variable Y lies between 0 and 1 is -1/2, which is not a valid probability. This indicates that there might be an error in the joint pdf or the limits of integration.

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The population density of Norway is 14.31 people per square kilometer.

What is Density?

Density is a  property of matter that describes how much mass is contained in a given volume. It is a measure of the amount of matter (mass) per unit of volume.

The formula for density is:

Density = Mass / Volume

Where:

Mass is the amount of matter in an object, usually measured in grams (g) or kilograms (kg).

Volume is the amount of space that an object occupies, usually measured in cubic meters (m³), cubic centimeters (cm³), or liters (L).

To find the population density of Norway, we need to divide the total population by the total area:

Population density = Population / Area

Population density = 4,707,270 / 328,802

Population density = 14.31 (rounded to the nearest hundredth)

Therefore, the population density of Norway is 14.31 people per square kilometer.

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Answer:

630 count each 180 and the add them upp

If the integral has a "finite-number" as a solution, then it is convergent, the correct option is (a).

In calculus, the convergence or divergence of an integral refers to whether the result of integral is a finite or infinite value when it is evaluated.

An integral is said to be convergent if its value is finite, and divergent if its value is infinite or does not exist.

If an integral has a finite solution, then it is convergent. This means that the area under the curve of the integrand is finite over the interval of integration.

Therefore, Option(a) is correct.

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The given question is incomplete, the complete question is

If the integral has a finite number as a solution than it is ______ .

(a) Convergent

(b) Divergent.

There are 30 distinct five-digit positive integers that can be made using the digits 1, 1, 1, 3, 8.

There are also 30 distinct five-digit positive integers that can be made using the digits 1, 1, 1, 3, 3.

Let's consider the first question: how many five-digit positive integers consist of the digits 1, 1, 1, 3, 8? Since we have five digits to work with and three of them are the same, we need to figure out how many distinct arrangements we can make with the digits. We can do this by using the permutation formula, which is n! / (n-r)!, where n is the total number of items and r is the number of items we are selecting.

In this case, we have five digits to choose from, so n = 5. However, we only have three distinct digits since there are three 1's, so r = 3. Using the permutation formula, we get:

5! / (5-3)! = 5! / 2! = 60 / 2 = 30

In this case, we have the same number of digits and the same number of repeating digits as in the first question, but the repeating digits are different.

Using the same permutation formula, we get:

5! / (5-3)! = 5! / 2! = 60 / 2 = 30

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The function f(x) = (x-1)/(x+1)^2 has 1 asymptote because the vertical asymptote occurs at x = -1, where the denominator (x+1)^2 is equal to zero. There are no horizontal asymptotes in this function. option c

The function f(x) = (x-1)/(x+1)^2 has only one asymptote. The denominator (x+1)^2 becomes zero at x = -1, which means that there is a vertical asymptote at x = -1. This means that it cannot be bigger than the value of 1 hence option b,c and are not correct.

However, the degree of the numerator is less than the degree of the denominator by 1. Therefore, there is no horizontal asymptote or slant asymptote. Hence, the correct option is c)1.

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For an automobile service center with average of 12 cars per hour, the probability of the service center being totally empty in a given hour is equals to 0.000335.

The Poisson Probability Distribution is use to determine the probability for the number of events that occur in a period when the average number of events is known. Formula is written as following :[tex]P( \lambda,x) = \frac{e^{ -\lambda } \lambda^{x}}{x!}[/tex], where

Now, we have an automobile service center take care of 12 cars per hour.

Rate on average of car survice hour ,[tex] \lambda[/tex] = 8

We have to determine the value of probability of the service center when it being totally empty in a hour, P(8, 0). So,

[tex]P( 8, 0) = \frac{e^{ -8 } 8^{0}}{0!}[/tex]

= [tex]e^{ -8 } [/tex]

= (2.7182)⁻⁸

= 0.000335

Hence, required probability value is 0.000335.

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The first and second derivative of the function are g'(x) = -24x² + 56x + 6 and g''(x) = -48x + 56

The first derivative of a function g(x) is denoted as g'(x) or dy/dx. To find the first derivative of the function g(x) = -8x³ + 28x² + 6x - 59, we need to apply the power rule of differentiation, which states that the derivative of xⁿ is nxⁿ⁻¹. Applying this rule, we get:

g'(x) = -24x² + 56x + 6

g''(x) = -48x + 56

This is the first derivative of the function g(x). It tells us the rate at which the function is changing at any given point x.

The second derivative of a function is denoted as g''(x) or d²y/dx². To find the second derivative of the function g(x) = -8x³ + 28x² + 6x - 59, we need to take the derivative of the first derivative. Applying the power rule of differentiation again, we get:

g''(x) = -48x + 56

This is the second derivative of the function g(x). It tells us the rate at which the rate of change of the function is changing at any given point x.

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The new dimensions of the photograph will be 5 inches by 7.5 inches.

What is measurement?

Measurement is the process of assigning numerical values to physical quantities, such as length, mass, time, temperature, and volume, in order to describe and quantify the properties of objects and phenomena.

If the photograph is reproduced 25% larger, then each dimension will be increased by 25% of its original value.

The new width will be:

4 inches + (25% of 4 inches) = 4 inches + 1 inch = 5 inches

The new length will be:

6 inches + (25% of 6 inches) = 6 inches + 1.5 inches = 7.5 inches

Therefore, the new dimensions of the photograph will be 5 inches by 7.5 inches.

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The second derivative of the function is zero.

The given function is -

y = 4 cos(x) sin(x)

We can write the first derivative as -

y' = dy/dx = 4 {cos(x) cos(x) - sin(x) sin(x)}

y' = dy/dx = = 4{cos²(x) - sin²(x)}

We can write the second derivative as -

y'' = d²y/dx² = 4{-2cos(x)sin(x) + 2sin(x)cos(x)}

y" = d²y/dx² = 4 x 0

y" = 0

So, the second derivative of the function is zero.

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The differences between a sampling distribution and its corresponding population distribution can be observed in terms of shape, outliers, center, and spread. These differences tend to decrease as the sample size increases, providing a more accurate representation of the population.

A sampling distribution can differ from the distribution of the population it has been drawn from in several ways.

(a) Shape: The shape of a sampling distribution may differ from the shape of the population distribution, particularly when the sample size is small. As the sample size increases, the sampling distribution tends to resemble the shape of the population distribution more closely, ultimately approaching a normal distribution according to the Central Limit Theorem.

(b) Outliers: In a sampling distribution, outliers might be less prevalent or more extreme than in the population distribution due to the smaller sample size. This is because the sample may not accurately represent the full range of values present in the population.

(c) Center: The center of a sampling distribution, which can be represented by the sample mean or median, may differ from the population mean or median. However, as the sample size increases, the sample mean converges towards the population mean, reducing the sampling error.

(d) Spread: The spread of a sampling distribution, as indicated by its standard deviation or variance, is generally smaller than the spread of the population distribution. This is because the sampling distribution represents the variability of sample means or medians rather than individual data points. As the sample size increases, the spread of the sampling distribution narrows, reflecting a more precise estimation of the population parameter.

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If the prοfit per yard οf denim is increased frοm $3.0 tο $4.0, the οptimal sοlutiοn will change. The new οptimal sοlutiοn will have x = 1,000 and y = 450, with a maximum prοfit οf $4,350.

Statistics is a branch οf mathematics that deals with the cοllectiοn, analysis, interpretatiοn, presentatiοn, and οrganizatiοn οf numerical data.

(a) Let x and y denοte the number οf yards οf denim and cοrdurοy prοduced, respectively. Then the οbjective functiοn tο be maximized is:

Prοfit = $3x + $4y

subject tο the fοllοwing cοnstraints:

Raw cοttοn: 5.5x + 8y ≤ 6,500 pοunds

Prοcessing time: 2.8x + 3.6y ≤ 3,000 hοurs

Demand fοr cοrdurοy: y ≤ 600

Nοn-negativity: x ≥ 0, y ≥ 0

(b) Using a linear prοgramming sοftware, the οptimal sοlutiοn is x = 887.50 and y = 600, with a maximum prοfit οf $4,150.

(c) At the οptimal sοlutiοn, the amοunt οf extra cοttοn left οver is 337.5 pοunds, the prοcessing time left οver is 125 hοurs, and the demand fοr cοrdurοy is met exactly.

(d) Tο identify the sensitivity ranges οf the prοfits οf denim and cοrdurοy, we perfοrm sensitivity analysis οn the οbjective functiοn cοefficients. The allοwable increase in the prοfit per yard οf denim is $0.50, and the allοwable increase in the prοfit per yard οf cοrdurοy is $1.00. The allοwable decrease in bοth prοfits is unlimited.

(e) If the prοfit per yard οf denim is increased frοm $3.0 tο $4.0, the οptimal sοlutiοn will change. The new οptimal sοlutiοn will have x = 1,000 and y = 450, with a maximum prοfit οf $4,350. The extra cοttοn left οver will be 3,500 pοunds, and the prοcessing time left οver will be 75 hοurs.

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To verify that f(x) = 3x³ + x² + 7x - 1 satisfies the requirements of the Mean Value Theorem (MVT) on the interval (1, 6], we need to check if f(x) is continuous on [1, 6] and differentiable on (1, 6).

f(x) is a polynomial, and polynomials are both continuous and differentiable everywhere, so it satisfies the MVT requirements.

To find the values of c that satisfy the MVT equation, first compute the derivative f'(x): f'(x) = 9x² + 2x + 7.

Now, compute f(b) - f(a) / (b - a): [f(6) - f(1)] / (6 - 1) = [(3(6³) + (6²) + 7(6) - 1) - (3(1³) + (1²) + 7(1) - 1)] / 5 = 714/5.

Set the derivative equal to the difference quotient: 9x² + 2x + 7 = 714/5. Solve for x to find the values of c.

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Since the function g(x) is a shift of 2 down and 5 to the right from the function f(x), the function g(x) is g(x) = √(x - 5) - 1.

In Mathematics, the translation a geometric figure or graph to the left simply means subtracting a digit from the value on the x-coordinate of the pre-image;

g(x) = f(x + N)

In Mathematics and Geometry, the translation a geometric figure downward simply means subtracting a digit from the value on the negative y-coordinate (y-axis) of the pre-image;

g(x) = f(x) - N

Since the parent function f(x) was translated 2 units downward and 5 units right, we have the following transformed function;

g(x) = f(x + 5) - 2

g(x) = √(x - 10 + 5) + 1 - 2

g(x) = √(x - 5) - 1

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The integral is evaluated and the polar coordinates are solved

Given data ,

To evaluate the given integral by changing to polar coordinates, we first need to express the region R in polar coordinates.

The region R is enclosed by the circle x² + y² = 4 and the lines x = 0 and y = x, in the first quadrant.

In polar coordinates, we have x = r cos(theta) and y = r sin(theta), where r is the radial distance and theta is the angle measured from the positive x-axis.

Since x = 0 represents the y-axis, which is also the polar axis in polar coordinates, we can have 0 <= theta <= pi/2, as we are considering only the first quadrant.

The circle x² + y² = 4 can be expressed in polar coordinates as:

(r cos(theta))² + (r sin(theta))² = 4

Simplifying, we get:

r² (cos²(theta) + sin²(theta)) = 4

r² = 4

r = 2

So, in polar coordinates, r varies from 0 to 2, and theta varies from 0 to pi/2.

Now, let's express the given integral SSR (2x - y) da in polar coordinates:

SSR (2x - y) da = ∫∫ (2r cos(theta) - r sin(theta)) r dr d(theta)

Integrating with respect to r from 0 to 2, and with respect to theta from 0 to pi/2, we get:

∫[0 to pi/2] ∫[0 to 2] (2r² cos(theta) - r³ sin(theta)) dr d(theta)

Now we can evaluate the above integral using the limits of integration for r and theta, as well as the appropriate trigonometric identities for cos(theta) and sin(theta) in the given region R.

Hence , the integral is solved by changing to polar coordinates

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The value of function f' (5) is,

⇒ f' (5) = - 26

And, The value of function f' (2) is,

⇒ f' (2) = 4

We have to given that;

1) Function is,

⇒ f(x) = - 3x² + 4x - 7

Derivative find as;

⇒ f '(x) = - 6x + 4

Put x = 5;

⇒ f' (5) = - 6 × 5 + 4

⇒ f' (5) = - 30 + 4

⇒ f' (5) = - 26

2) Function is,

⇒ F (x) = 4x + 3

Derivative find as;

⇒ f '(x) = 4

Put x = 2;

⇒ f' (2) = 4

Thus, The value of function f' (5) is,

⇒ f' (5) = - 26

And, The value of function f' (2) is,

⇒ f' (2) = 4

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The probability of both events happening together (drawing a dime and then drawing a penny) is equal to the product of their individual probabilities: (4/10) * (3/10) = 12/100 = 6/50 = 3/25.

Explain the term probability

Probability is a measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain to occur.

Explain the term events

An event is any outcome or set of outcomes of an experiment or situation. In probability, an event can be a simple event (a single outcome) or a compound event (a combination of outcomes).

According to the given information

The probability of choosing a dime is 4/10 because there are 4 dimes in the pocket and 10 coins in total.

The probability of choosing a penny is 3/10 because there are 3 pennies in the pocket and 10 coins in total. Since we are replacing the dime after we choose it, we can assume that we have 4 dimes and 10 coins again for the second draw.

Therefore, the probability of drawing a penny after drawing a dime is also 3/10.

The probability of both events happening together (drawing a dime and then drawing a penny) is equal to the product of their individual probabilities: (4/10) * (3/10) = 12/100 = 6/50 = 3/25.

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We are 95% confident that the population mean test score falls within the interval of 78.64 to 86.36.

To construct a 95% confidence interval for the population mean, we can use the formula:

CI = x ± tα/2 × (s/√n)

where x is the sample mean (82.5), s is the sample standard deviation (9.2), n is the sample size (28), tα/2 is the t-value from the t-distribution table with a degrees of freedom of n-1 and a level of significance of 0.05/2 = 0.025 (since we want a two-tailed test for a 95% confidence interval).

Looking up the t-value with 27 degrees of freedom and a level of significance of 0.025, we get t0.025,27 = 2.048.

Plugging in the values, we get:

CI = 82.5 ± 2.048 × (9.2/√28)
CI = 82.5 ± 3.86
CI = (78.64, 86.36)

Therefore, we are 95% confident that the population mean test score falls within the interval of 78.64 to 86.36.

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The 18th term of the arithmetic sequence -13, -9, -5, -1, 3,... is 55. Thus, the right answer is option A. which is A(18) = 55

Arithmetic Progression is a sequence of numbers in which the difference between two numbers in the series is a fixed definite value.

The specific number in the arithmetic progression is calculated by

[tex]a_n=a_1+(n-1)d[/tex]

where [tex]a_n[/tex] is the term in arithmetic progression at the nth term

[tex]a_1[/tex] is the initial term in the arithmetic progression

d is the difference between two consecutive terms

In the given question, [tex]a_1[/tex] = -13

d = -9 - (-13) = 4

Therefore, to calculate the 18th term,

[tex]a_{13}=a_1+(18-1)d[/tex]

= -13 + (17) 4

= -13 + 68

= 55

Hence, the 18th term of the above AP is 55

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The given statement "A manufacturing manager has developed a table that shows the average production volume which is an example of a numerical measure." is true because it represents central tendency.

The average production level is a numerical measure that represents the central tendency of the production volume for a given period of time. In this case, the manufacturing manager has calculated the average production volume for each day over the past three weeks.

Numerical measures are quantitative values that summarize or describe the data. They are used to provide insights into the characteristics of a dataset, such as the distribution, variability, and central tendency.

Common numerical measures include measures of central tendency, such as the mean, median, and mode, as well as measures of dispersion, such as variance and standard deviation.

The average production level, also known as the mean, is a commonly used measure of central tendency. It is calculated by adding up all the production volumes and dividing by the number of days. The resulting value represents the typical or average production level for the period of time in question.

Therefore, the statement that the average production level is an example of a numerical measure is true.

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Answer:

No--of the 52 cards, 13 are hearts. Of the 13 cards that are hearts, there is one card that is also an ace--the ace of hearts.

Answer:

The middle 95% of the players fall within two standard deviations of the mean. Using the empirical rule, we know that this corresponds to the interval (mean - 2*standard deviation, mean + 2*standard deviation), or (200 - 2*25, 200 + 2*25), which simplifies to (150, 250). Therefore, the minimum weight of the middle 95% of the players is 150 pounds.

The answer is d. 151.

The limit of (2x-1)(3-x)/(x-1)(x+3) as x approaches infinity is 0.

To find the limit of the function (2x-1)(3-x)/(x-1)(x+3) as x approaches infinity, we will divide both the numerator and denominator through the highest power of x. In this case, the highest power of x is x², so we can divide both the numerator & the denominator through x²:

[tex][(2x-1)/(x^2)] * [(3-x)/((x-1)/(x^2)(x+3))][/tex]

Now, as x approaches infinity, every of the fractions within the expression procedures zero except for (2x-1)/(x²). This fraction techniques 0 as x procedures infinity because the denominator grows quicker than the numerator. therefore, the limit of the expression as x strategies infinity is:

0 * 0 = 0

Consequently, When x gets closer to infinity, the limit of (2x-1)(3-x)/(x-1)(x+3) is 0.

Learn more about numerator and denominator:-https://brainly.com/question/20712359

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Answer: integers 0 <= x <= 60. Determine the mean and variance of X. This problem has been solved!

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Theory : If random variable Y fol

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Step-by-step explanation:

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