One of the most famous large fractures (cracks) in the earth's crust is the San Andreas fault in California. A geologist attempting to study the movement of the earth's crust at a particular location found many fractures in the local rock structure. In an attempt to determine the mean angle of the breaks, she sample 50 fractures and found the sample mean and standard deviation to be 39.8 degrees and 17.2 degrees respectively. estimate the mean angular direction of the fractures and find the standard error of the estimate

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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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 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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The 95% confidence interval estimate of the population mean μ is 97.53 < μ < 112.47.

We are required to find the 95% confidence interval estimate of the population mean μ, given the sample size n=18, mean x =105, and standard deviation s=15.

1. Determine the t-score for a 95% confidence interval with n-1 degrees of freedom. For a sample size of 18, you have 17 degrees of freedom.

Using a t-table or calculator, you find the t-score to be approximately 2.11.

2. Calculate the margin of error (E) using the formula:

E = t-score × (s / √n)

E = 2.11 × (15 / √18)

E ≈ 7.47

3. Calculate the lower and upper bounds of the confidence interval using the sample mean (x) and margin of error (E):

Lower bound: x - E = 105 - 7.47 = 97.53

Upper bound: x + E = 105 + 7.47 = 112.47

So, the 95% confidence interval estimate is 97.53 < μ < 112.47. Please note that these values are rounded to two decimal places.

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The particular antiderivative of X(t) that satisfies the given condition X(0) = 1 is:

[tex]X(t) = 6e^t - 6t + 1[/tex].

The given derivative is X(t) = 6e^t.

To find the particular antiderivative of X'(t), we need to integrate X'(t) with respect to t:

[tex]\int6e^t)dt = 6e^t + C[/tex]

where C is the constant of integration.

Now, we use the given condition X(0) = 1 to find the value of C:

X(0) = 6(1) - 6 + C = 1

Simplifying, we get:

C = 1 + 6 - 6 = 1

Therefore, the particular antiderivative of X(t) that satisfies the given condition X(0) = 1 is:

[tex]X(t) = 6e^t - 6t + 1[/tex].

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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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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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Answer:

Step-by-step explanation:

Mode is the one that occurs the most.

Stem is first number repeated for leaf

so your list of numbers is

10,12,12,22,23,24,30

The one that occurs the most is 12.  So your mode is 12.

If there are multiple numbers that are "most" then there is no mode.

Answer:

630 count each 180 and the add them upp

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.

The new coordinates are A'=(2, -10), B'=(12, -10), C'=(12, -2), D'=(2, -2), and Square ABCD and Square A'B'C'D' are similar, with a scale factor of 2.

What is the scale factor?

A scale factor is a number that represents the amount of magnification or reduction applied to an object, image, or geometrical shape.

Part A:

When Square ABCD is reflected across the x-axis, the y-coordinates of each vertex will change sign while the x-coordinates remain the same. Therefore, the new coordinates of the reflected square will be:

A=(1,-5), B=(6,-5), C=(6,-1), D=(1,-1)

When this reflected square is dilated by a scale factor of 2, each vertex is multiplied by a scalar value of 2, centered at the origin. This means that the new coordinates will be:

A'=(2, -10), B'=(12, -10), C'=(12, -2), D'=(2, -2)

Part B:

To determine whether Square ABCD is similar or congruent to Square A'B'C'D', we need to compare their corresponding side lengths and angles.

First, let's compare the side lengths:

AB = BC = CD = DA = 5 units

A'B' = B'C' = C'D' = D'A' = 10 units

We can see that the side lengths of Square A'B'C'D' are exactly twice as long as the corresponding side lengths of Square ABCD. This tells us that the two squares are similar, with a scale factor of 2.

Next, let's compare the angles:

Square ABCD has four right angles (90 degrees) at each vertex.

Square A'B'C'D' also has four right angles (90 degrees) at each vertex.

Since the corresponding angles of the two squares are congruent, this confirms that the two squares are similar.

Therefore, we can conclude that Square ABCD and Square A'B'C'D' are similar, with a scale factor of 2.

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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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Answer: Calculate

Step-by-step explanation: How to calculate the margin of error?

1. Get the population standard deviation (σ) and sample size (n).

2. Take the square root of your sample size and divide it into your population standard deviation.

3. Multiply the result by the z-score consistent with your desired confidence interval according to the following table:

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

1 answer

·

Top answer:

Theory : If random variable Y fol

Doesn’t include: < ‎6).

Step-by-step explanation:

Doesn’t include: < ‎6).

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 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 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 Maclaurin series are:

f(x) = x^2/(1-x^3) = x^2 + x^4 + x^6 + x^8 + ...

The interval of convergence can be found by testing the endpoint values of the series.

We can use the geometric series formula to find the Maclaurin series of f(x):

1/(1-x) = 1 + x + [tex]x^{2} +x^{3}[/tex]...

Differentiating both sides with respect to x gives:

[1/(1-x)]' = 1 + 2x + [tex]3x^{2} +4x^{3}[/tex]+ ...

Multiplying both sides by [tex]x^{2}[/tex] gives:

[tex]x^{2}[/tex]/(1-[tex]x^{3}[/tex]) = [tex]x^{2}[/tex] + [tex]x^{4}[/tex] + [tex]x^{6}[/tex] + [tex]x^{8}[/tex] + ...

Therefore, we have:

f(x) = [tex]x^{2}[/tex]/(1-[tex]x^{3}[/tex]) = [tex]x^{2} +x^{4} +x^{6} +x^{8} +...[/tex]

The interval of convergence can be found by testing the endpoint values of the series. When x = ±1, the series becomes:

1 - 1 + 1 - 1 + ...

This series does not converge, so the interval of convergence is -1 < x < 1.

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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.

So, an equivalent expression to [tex]-18 - 64x[/tex] is[tex]-2(9+32x)[/tex].

Expressions are considered equivalent when they have the same value, regardless of their form. There are different methods to determine whether expressions are equivalent depending on the type of expressions involved. Here are some examples:

Numeric expressions:

To determine whether two numeric expressions are equivalent, we can evaluate them and compare their results. For example, the expressions 3 + 4 and 7 have the same value, so they are equivalent.

Algebraic expressions:

To determine whether two algebraic expressions are equivalent, we can simplify both expressions using algebraic rules and compare the results. For example, the expressions 2x + 4 - x and x + 4 have the same value for any value of x, so they are equivalent.

Logical expressions:

To determine whether two logical expressions are equivalent, we can use truth tables to evaluate the expressions and compare the results. For example, the expressions (A ∧ B) ∨ C and (A ∨ C) ∧ (B ∨ C) have the same truth values for any values of A, B, and C, so they are equivalent.

The expression -18 - 64x can be simplified by factoring out a common factor of -2 from both terms. This gives:

[tex]-18 - 64x = -2(9 + 32x)[/tex]

So, an equivalent expression to -18 - 64x is -2(9 + 32x).

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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 limit of the given sequence is infinity. (e) an = tan(n). This sequence oscillates between -infinity and infinity, so it diverges.

(a) To determine if the sequence converges or diverges, we can use the limit comparison test. We compare the given sequence to a known sequence whose convergence/divergence we know. Let bn = 5n^2/n^3 = 5/n. Then, taking the limit as n approaches infinity of (an/bn), we get:

lim (an/bn) = lim [(3 + (5n^2)/tn^(3/n) + 2)/(5/n)]
= lim [(3n^(3/n) + 5n^2)/5]
= ∞

Since the limit is infinity, the sequence diverges.

(b) bn = 1/n. This is a p-series with p = 1, which diverges. Therefore, the given sequence also diverges.

(c) An = cos(n) + 1. The cosine function oscillates between -1 and 1, so the sequence oscillates between 0 and 2. However, since there is no limit to the oscillation, the sequence diverges.

(d) b) bn = e^(2n)/(n+2). To determine if this sequence converges or diverges, we can use the ratio test. Taking the limit as n approaches infinity of (bn+1/bn), we get:

lim (bn+1/bn) = lim (e^(2(n+1))/(n+3)) * (n+2)/e^(2n)
= lim (e^2/(n+3)) * (n+2)
= 0

Since the limit is less than 1, the sequence converges. To find the limit, we can use L'Hopital's rule to evaluate the limit of (e^(2n)/(n+2)) as n approaches infinity:

lim (e^(2n)/(n+2)) = lim (2e^(2n)/(1))
= ∞

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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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If Walmart is excluded from the list, the measure of center that would be more affected is the mean. The median, on the other hand, is less affected by extreme values, as it represents the middle value in the dataset.

If Walmart is excluded from the list, the measure of center that would be more affected is the mean. This is because Walmart is a large retailer and has a significant impact on the overall average of the data. Removing Walmart from the list would decrease the total sales and therefore decrease the mean. The median, on the other hand, would be less affected by the exclusion of Walmart as it only looks at the middle value of the data and is less sensitive to extreme values. The measure of center that would be more affected is the mean. Value like Walmart's revenue would significantly change the average. The median is less affected by extreme values, as it represents the middle value in the dataset.

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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.

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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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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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Expressing the regression equation in terms of the x variable instead of the y variable will cause the y-intercept and slope to change.

When the regression equation is expressed in terms of the y variable, it takes the form of: [tex]u=ax+b[/tex]                                where a is the y-intercept (the value of y when x = 0) and b is the slope (the rate at which y changes with respect to x).

If we express the same regression equation in terms of the x variable, it becomes:

[tex]x=\frac{y-a}{b}[/tex]

In this form, the y-intercept becomes [tex](0,\frac{a}{b} )[/tex], which is a point on the x-axis, and the slope becomes [tex]\frac{1}{b}[/tex] , which is the reciprocal of the slope in the original equation.

Therefore, expressing the regression equation in terms of the x variable instead of the y variable will cause the y-intercept and slope to change

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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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