Determine the null and alternative hypotheses. The principal of a middle school claims that the seventh grade test scores at her school vary less than the test scores of seventh-graders at neighboring schools, which have variation described by σ=14.7

.

A.H0:σ=14.7;Ha:σ<14.7
B.H0:σ=14.7;Ha:σ>14.7
C.H0:σ=14.7;Ha:σ≤14.7
D.H0:σ=14.7;Ha:σ≠14.7
E.H0:σ=14.7;Ha:σ≥14.7

The value of sin 2Ф is  (√3 )/ 2 ,  cos 2Ф is 1/2 and tan 2Ф is √3 by application of Trigonometric formulas of Sine, Cosine and Tangent.

We have, sin Ф= 1/2

Applying Trigonometric formulas of Sine, Cosine and Tangent, we get

We know,

sin² Ф + cos² Ф = 1

⇒ (1/2)² + cos² Ф = 1

⇒ cos² Ф = 1 - (1/4) = 3/4

Rooting both sides we get,

⇒ cos Ф = (√3 )/ 2

Therefore, sin 2Ф = 2{sin Ф} {cos Ф}

⇒ sin 2Ф = 2 (1/2){(√3 )/ 2}

= (√3 )/ 2

Therefore, cos 2Ф = 2{ cos² Ф} - 1

⇒ cos 2Ф = 2 ( 3/4) - 1

= 1/2

Therefore, tan 2Ф = [tex]\frac{sin (theta)}{cos(theta)}[/tex] = [tex]\frac{1/2}{(\sqrt{3} )/ 2}[/tex] = 1/ (√3 )

So, tan 2Ф = [tex]\frac{2 tan (theta)}{ 1 - tan^{2} (theta) }[/tex] = [tex]\frac{2(1/\sqrt{3}) }{1 - (1/\sqrt{3})^{2} }[/tex] = √3

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The solution to the equation cos θ = 7/16, rounded to the nearest tenth, is θ = 64.05 degrees.

Equations with trigonometric functions that hold true for all of the variables in the equation are known as trigonometric identities.
These identities are used to solve trigonometric equations and simplify trigonometric expressions.

To solve for θ when cos θ = 7/16, we need to use the inverse cosine function (also called the arc-cosine function), which is denoted as cos⁻¹

Using this function, we can write:

cos θ = 7/16

θ = cos⁻¹(7/16)

To find the value of θ in degrees, we can use a calculator or a table of trigonometric values. Using a calculator, we get:

θ = 64.05°

Therefore, the solution to the equation cos θ = 7/16, rounded to the nearest tenth, is θ = 64.05 degrees.

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(a)

(i) To find the pdf of X, we differentiate the cdf:

f_x(x) = d/dx (1 - e^(-bx)) = b e^(-bx), x >= 0

0, x < 0

(ii) To find the pdf of Y = sqrt(X), we use the transformation method:

f_y(y) = f_x(x) / |dy/dx|, where x = y^2

dy/dx = 1 / (2 sqrt(x)) = 1 / (2y)

f_y(y) = f_x(y^2) / (2y) = b e^(-b y^2) / (2y), y >= 0

(b)

(i) We want to find g(U) such that Y = g(U) has the desired pdf:

f_y(y) = f_u(g(y)) |dg/dy|

Since U is a standard uniform random variable, its pdf is f_u(u) = 1, 0 <= u <= 1.

We want to find g(U) such that Y = g(U) has the pdf:

f_y(y) = 3y^2, 0 < y < 1

0, otherwise

From the above equation, we can see that:

f_u(u) = 1 = f_y(g(u)) |dg/du|

So, we need to find a function g(U) that satisfies:

f_y(y) = 3y^2 = |dg/dy|, 0 < y < 1

We can solve this differential equation by integrating both sides:

∫_0^y 3u^2 du = ∫_0^g(u) |dg/dy| dy

y^3 = g(u)

So, the required function is:

g(u) = u^(1/3)

(ii) If the student has a random variable X in the interval (2,5), then we need to modify the function g(U) to map the interval [0,1] of the standard uniform random variable U to the interval [2,5] of X.

Let X have the pdf f_x(x). Then, the cdf of X is given by:

F_x(x) = ∫_2^x f_x(t) dt

We can use this cdf to transform U to X:

F_x(g(u)) = u, 0 <= u <= 1

Solving for g(u), we get:

g(u) = F_x^(-1)(u) = ∫_2^x f_x(t) dt)^(-1)

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Answer:

Step-by-step explanation:

To solve by completing the square, we need to have the equation in the form:

(x - h)^2 = k

where h and k are constants. To get the equation in this form, we need to move the constant term to the right side and group the x terms together. So let's start by moving 9 to the right side:

x^2 + 4x = 9

Next, we need to add and subtract a constant term that will allow us to complete the square. The term we need to add is (b/2a)^2, where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = 1 and b = 4, so (b/2a)^2 = (4/2)^2 = 4. So we add and subtract 4:

x^2 + 4x + 4 - 4 = 9

Now we can group the first three terms and simplify:

(x + 2)^2 - 4 = 9

Add 4 to both sides:

(x + 2)^2 = 13

So the answer is not given in any of the options provided.

The vertex form of the equation f f(x) is: B. f(x) = −(x + 1)² + 25.

In this exercise, you are required to determine the vertex form of a quadratic function h(x) that is written in standard form. Mathematically, the vertex form of a quadratic equation is given by this formula:

f(x) = a(x - h)² + k

Where:

Since the quadratic function f(x) has roots of 4 and -6, and it passes through the point (1, 21), the leading coefficient can be determined as follows;

f(x) = a(x + 6)(x - 4)

f(x) = a(x² + 2x - 24).

21 = a(1 + 2 - 24)

-21a = 21

a = -1.

For the x-coordinate of the vertex, we have:

x = -b/2a = 2/-2 = -1.

For the y-coordinate of the vertex, we have:

y = -(b² - 4ac)/4a

y = -((-2)² - 4(-1)(24))/-4

y = 25.

Therefore, the vertex form of the quadratic function is given by:

f(x) = a(x - h)² + k

f(x) = −(x + 1)² + 25

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The warranty period should be 37.6 months to ensure that the company doesn't have to replace more than 2% of the total machines sold due to malfunctioning.

To solve this problem, we need to use the concept of deviation again. We know that the standard deviation of the PYQ machines is 8 months. Therefore, we can use the z-score formula to calculate the number of standard deviations from the mean that corresponds to the 2% malfunction rate. The z-score is calculated as:

z = (x - μ) / σ

where x is the value we want to find, μ is the mean, and σ is the standard deviation. In this case, we want to find the warranty period (x) that corresponds to a malfunction rate of 2%. The mean is 54 months, and the standard deviation is 8 months. Therefore, we can rearrange the formula as:

z = (x - 54) / 8

We want to find the z-score that corresponds to the 2% malfunction rate. We can use a standard normal distribution table to find the z-score that corresponds to a cumulative probability of 0.02. The z-score turns out to be -2.05.

Now that we know the z-score, we can solve for x:

-2.05 = (x - 54) / 8

-16.4 = x - 54

x = 37.6

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Answer:

yes, 2 divided by 4 is 1/2.

Step-by-step explanation:

4 only goes into 2 half a time, therefore 2 divided by 4 is 1/2.

Answer: I say yes

Step-by-step explanation: I guess like when dividing 2 into 4 it goes into 4, 2 times, but like if you were to look up the answer, it comes up as 0.5 which us basically considered 1/2 imo.

Answer:

B. Tenths

Step-by-step explanation:

Nine is one spot to the left of the decimal. this means that it is a decimal. It is in the first spot, otherwise known as the tenths place value. This means nine is in the tenths place value.

The earthquake in California with a magnitude of 4.9 had an amplitude 7943.28 times stronger than the baseline amplitude Ao.

The formula relating the Richter scale,  magnitude M to the amplitude A of an earthquake is

M = log10(A/Ao)

Solving for A, we get

A = Ao * 10^(M)

For the earthquake in California with a magnitude of 4.9, we can use this formula to find the ratio of its amplitude to the baseline amplitude Ao

4.9 = log10(A/Ao)

10^4.9 = A/Ao

A/Ao = 10^4.9

A/Ao = 7943.28

This means that the wave amplitude A of the earthquake in California was 7943.28 times stronger than the baseline amplitude Ao.

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--The given question is incomplete, the complete question is given

"An earthquake in california measured 4. 9 on richter scale use formula r=log(a/ao) to determine how many times stronger the wave amplitude a of the earthquake was than ao."--

The indicated derivative of the function is -6(x + 1) -4 (option b).

The second derivative of f(x) is 0.

Let's look at the first problem. We are asked to find the derivative of the function f(x) = 6x⁵ - 3x² - 5x + 1. The derivative of a function is written as f'(x) or dy/dx. To find the derivative of this function, we need to use the power rule and the constant multiple rule.

The power rule states that if we have a term of the form xⁿ, then its derivative is nxⁿ⁻¹. The constant multiple rule states that if we have a constant c multiplied by a function f(x), then the derivative of cf(x) is c times the derivative of f(x).

Using these rules, we can find that the derivative of f(x) is f'(x) = 30x⁴ - 6x - 5. This is the rate at which the function is changing at each point x.

The answer choices given are all in different forms, but we can see that choice B, 480x + 6, matches our answer if we simplify it.

Hence the correct option is (b).

Given function is f(x) = x + 1, and we are asked to find its second derivative f''(x). The first derivative of f(x) is found by differentiating the equation f(x) = x + 1 with respect to x:

f'(x) = d/dx(x + 1) = 1

Now, we can find the second derivative of f(x) by differentiating the equation for f'(x) with respect to x:

f''(x) = d/dx(1) = 0

Therefore, the second derivative of f(x) is 0, which means that the slope of the tangent line to the graph of f(x) is constant at all points.

Therefore, the correct answer is none of the above.

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Answer:

the difference quotient of f(x) = 2x^2 + x - 3 is:

(f(x + h) - f(x))/h = (4x + 2h - 1) + (2/h), where h is not equal to zero.

Step-by-step explanation:

To find the difference quotient of f(x) = 2x^2 + x - 3, we need to evaluate the expression (f(x + h) - f(x))/h, where h is not equal to zero.

f(x + h) = 2(x + h)^2 + (x + h) - 3 = 2x^2 + (4h + 1)x + 2h^2 - h - 1

f(x) = 2x^2 + x - 3

Therefore,

f(x + h) - f(x) = (2x^2 + (4h + 1)x + 2h^2 - h - 1) - (2x^2 + x - 3)

= 4hx + 2h^2 - h + 2

Dividing by h, we get:

(f(x + h) - f(x))/h = (4x + 2h - 1) + (2/h)

As h approaches 0, the second term (2/h) approaches infinity, so the difference quotient is not defined at h = 0.

Therefore, the difference quotient of f(x) = 2x^2 + x - 3 is:

(f(x + h) - f(x))/h = (4x + 2h - 1) + (2/h), where h is not equal to zero.

The following parts can be answered by the concept from Sets.

(a) No, ɸ is not an element of the power set P(S).

(b) Yes, a is an element of the power set P(S).

(a) The power set P(S) of a set S is the set of all possible subsets of S, including the empty set ɸ and the set S itself. However, ɸ is not an element of P(S) because it is not a subset of S. A subset must have at least one element, but ɸ has no elements, so it cannot be a subset of any set, including S. Therefore, ɸ is not an element of P(S).

(b) The element 'a' is an element of set S, and therefore it can be a subset of S. Since every element of S is also a subset of S, 'a' is also a subset of S. Hence, 'a' is an element of P(S) because P(S) includes all possible subsets of S. Therefore, 'a' is an element of P(S).

Therefore,

(a) ɸ is not an element of P(S).

(b) 'a' is an element of P(S)

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Derivatives play a crucial role in understanding the position, velocity, and acceleration of a moving object.

Position can be represented as a function of time, and its derivative with respect to time is velocity. Velocity, in turn, can be represented as a function of time, and its derivative with respect to time is acceleration. Therefore, by taking derivatives of position with respect to time, we can obtain the velocity of the moving object, and by taking the derivative of velocity with respect to time, we can obtain the acceleration of the moving object. In summary, derivatives help us to analyze and quantify the motion of a moving object by providing us with information about its position, velocity, and acceleration.

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On solving the provided query we have As a result, we may infer from the information provided that 39 pupils in Phil's school's seventh grade would select football as their favourite sport.

It is possible to multiply, divide, add, or subtract in mathematics. The following is how an expression is put together: Number, expression, and mathematical operator The components of a mathematical expression (such as addition, subtraction, multiplication or division, etc.) include numbers, variables, and functions. It is possible to contrast expressions and phrases. An expression, often known as an algebraic expression, is any mathematical statement that contains variables, numbers, and an arithmetic operation between them. For instance, the word m in the given equation is separated from the terms 4m and 5 by the arithmetic symbol +, as does the variable m in the expression 4m + 5.

39 is equal to 0.2N * (number of students who choose soccer/total number of students polled).

When we simplify this equation, we obtain:

The percentage of students who choose football was 39 * (the total number of students surveyed/0.2N).

By realising that the entire number of respondents to the survey is equal to 20% of all seventh-graders, we can further reduce this equation:

Students who choose football: 39 * (0.2N/0.2N) = 39

As a result, we may infer from the information provided that 39 pupils in Phil's school's seventh grade would select football as their favourite sport.

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Answer:

[tex]x + 2 = - 3x[/tex]

[tex] - 4x = 2[/tex]

[tex]x = - \frac{1}{2} [/tex]

[tex] - 3( - \frac{1}{2} ) = \frac{3}{2} = 1 \frac{1}{2} [/tex]

So the lines intersect at (-1/2, 1 1/2), or

(-.5, 1.5).

The rate of change in temperature being recorded by the weather balloon at time t = 1 is 1°C per hour.

I understand that you have a question about atmospheric temperature and its dependence on position and time. I'll address each part of your question using the provided terms.

(a) In the context of atmospheric temperature, the expressions x, y, z, and t represent the position and time variables. Specifically, x, y, and z are the spatial coordinates (measured in kilometres) that denote the position in 3D space, and t (measured in hours) denotes the time variable.

(b) According to the given function, T(x, y, z, t) = 1.(1+t), and the position of the weather balloon is represented by x = 1, y = 2t, and z = 1. To find the temperature as a function of t only, substitute these values into the temperature function:

T(1, 2t, 1, t) = 1.(1+t)

(c) To find the rate of change of the temperature being recorded by the weather balloon at t = 1, we first differentiate the temperature function T(t) with respect to time t:

dT/dt = d(1.(1+t))/dt = 1

Now, evaluate the rate of change at t = 1:

dT/dt |_(t=1) = 1

So, the rate of change in temperature being recorded by the weather balloon at time t = 1 is 1°C per hour.

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The midline of the graph is y = -3

The midline of a graph is a horizontal line that divides the graph into two equal parts. It is used in various fields such as mathematics, physics, and economics to represent the average or equilibrium value of a function or data set.

midline = (highest point + lowest point) / 2

midline = (2 + -8) / 2

midline = (2 - 8) / 2

midline = -6 / 2

midline = -3

Therefore, the midline of the attached graph is at y = -3.

If the graph is not symmetrical, the midline may not accurately represent the average or equilibrium value.

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To minimize the amount of tin used, the height of the can must be 4 inches (option D).

The volume of a cylindrical tin can is given by the formula V = πr²h, where V is the volume, r is the radius, and h is the height. To minimize the amount of tin used, we need to minimize the surface area, which is given by the formula A = 2πrh + 2πr².

Given the volume is 16π cubic inches, we have:

16π = πr²h

Now, we can find the relationship between r and h:

h = 16/r²

Now, substitute this into the surface area formula:

A = 2πr(16/r²) + 2πr²
A = 32π/r + 2πr²

To minimize the surface area, we can take the derivative with respect to r and set it to 0:

dA/dr = -32π/r² + 4πr

0 = -32π/r² + 4πr

Solving for r:

r³ = 8
r = 2 (since r > 0)

Now, substituting r back into the relationship between r and h:

h = 16/(2²)
h = 16/4
h = 4

Therefore, the height of the can must be 4 inches (option D) to minimize the amount of tin used.

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The population for this study is all hotel managers in hotels of size 200 to 500 rooms in Chicago and Detroit.

The major shortcomings in the obtained data are the small sample size (only 100 out of 400 managers responded) and the potential for response bias (managers who are more satisfied or dissatisfied with their computer systems may be more likely to respond to the survey).

Confidence interval for a population mean with a known standard deviation:

[tex]CI = \bar x \± z\times (\sigma/\sqrt n)[/tex]

[tex]\bar x[/tex] is the sample mean, [tex]\sigma[/tex] is the population standard deviation (assumed to be 1.75), n is the sample size (100), and z is the z-score corresponding to the desired level of confidence (95% corresponds to a z-score of 1.96).

Substituting the given values, we have:

[tex]CI = 5.396 \± 1.96\times (1.75/\sqrt 100)[/tex]

[tex]CI = 5.396 \± 0.34[/tex]

[tex]CI = (5.056, 5.736)[/tex]

The 95% confident that the true population mean for satisfaction with ease of use of computer systems for hotel managers in hotels of size 200 to 500 rooms in Chicago and Detroit is between 5.056 and 5.736.

To repeat this survey many times and construct a 95% confidence interval based on each sample, about 95% of the intervals would contain the true population mean for satisfaction with ease of use of computer systems.

The same formula as in part B, but with a z-score of 2.58 (corresponding to 99% confidence), we have:

[tex]CI = 4.398 \± 2.58\times (1.75/\sqrt 100)[/tex]

[tex]CI = 4.398 \± 0.45[/tex]

[tex]CI = (3.948, 4.848)[/tex]

The 99% confident that the true population mean for satisfaction with level of computer training for hotel managers in hotels of size 200 to 500 rooms in Chicago and Detroit is between 3.948 and 4.848.

To repeat this survey many times and construct a 99% confidence interval based on each sample, about 99% of the intervals would contain the true population mean for satisfaction with level of computer training.

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Based on the breach of these implied terms, Eric may have a valid claim to return the car and seek a refund. However, Peter's offer to replace the windscreen and compensate for the mileage discrepancy may be considered as an attempt to remedy the situation. It is ultimately up to Eric to decide whether to accept Peter's proposed solutions or pursue legal action to return the car and obtain a refund.

Based on the implied terms under the Sales of Goods Ordinance (Cap 26) ("SOGO"), Eric has the right to return the car to Peter and get a refund. One of the implied terms is that the goods sold must be of satisfactory quality, which includes being free from defects and safe to use. The scratch on the windscreen makes the car unsafe to drive and therefore, not of satisfactory quality.

Additionally, the inspection report revealing that the mileage meter had been adjusted also breaches the implied term that the goods sold must match their description. The sign board on the windscreen stated that the car had a mileage of 12,200km, but the actual mileage is 13,420km. This is a material difference that would have influenced Eric's decision to purchase the car.

Therefore, Eric can reject the car and get a refund under the implied term that the goods sold must be of satisfactory quality and match their description. The compensation offered by Peter for the understatement of the mileage reading does not address the issue of the scratch on the windscreen, which is a breach of an implied term.
Under the Sales of Goods Ordinance (Cap 26) ("SOGO"), there are implied terms concerning the quality and fitness of goods being sold. In Eric's case, two key implied terms are relevant: the satisfactory quality of the goods and the accuracy of the description provided.

1. Satisfactory quality (Section 16 of SOGO): Goods sold should be of satisfactory quality, which includes appearance, freedom from minor defects, and safety. The scratch on the windscreen makes the car unsafe and illegal to drive, which means the car is not of satisfactory quality. Peter's offer to replace the windscreen may address this issue, but it does not automatically negate Eric's right to reject the car.

2. Accuracy of description (Section 15 of SOGO): When the seller provides a description of the goods, the goods should correspond to the description. In this case, the mileage stated on the signboard (12,200km) differs from the actual mileage (13,420km). This misrepresentation could entitle Eric to reject the car.

Based on the breach of these implied terms, Eric may have a valid claim to return the car and seek a refund. However, Peter's offer to replace the windscreen and compensate for the mileage discrepancy may be considered as an attempt to remedy the situation. It is ultimately up to Eric to decide whether to accept Peter's proposed solutions or pursue legal action to return the car and obtain a refund.

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The expression sinâ¹(â3/2) is undefined. The value of cosâ¹(2/2) = 0 radians.

In the expression sinâ¹(â3/2), Since the sine function is only defined for angles between -π/2 and π/2, we cannot find an angle with a sine of -â3/2. Therefore, the expression is undefined.

In the expression cosâ¹(2/2), since the cosine of an angle is equal to the adjacent side over the hypotenuse in a right triangle, we can draw a right triangle with adjacent side 2 and hypotenuse 2. Using the Pythagorean theorem, we find that the opposite side has length 0.

Therefore, we have a right triangle with adjacent side 2, opposite side 0, and hypotenuse 2. This means that the angle we are looking for is a zero-degree angle, or 0 radians. Therefore, cosâ¹(2/2) = 0 radians.

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The equation of the tangent line is 6.033333333.

We have,

We have f(x) = √x, so f'(x) = 1/2√x

At x=36, the slope of the tangent line is f'(36) = 1/2√36 = 1/12.

The equation of the tangent line is y - f(36) = f'(36) (x-36), or

y - 6 =  1/12 x (x - 36)

which simplifies to y = x/12 + 3.

To approximate √36.4 using the tangent line,

We plug in x = 36.4 to get y = 1/12 x (36.4) + 3 = 6.03333333....

Rounding to 9 significant figures, we get

= 6.033333333.

Thus,

The equation of the tangent line is 6.033333333.

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The magnitude of the magnetic field inside the solenoid is 0.094 T.

A solenoid is a long coil of wire wrapped around a cylinder or other object. It is used to generate a nearly uniform magnetic field within the cylinder.

To determine the magnetic field inside a solenoid, the following equation can be used:

B = μ0nI

Where B is the magnetic field inside the solenoid, μ0 is the permeability of free space (4π × 10-7 T m/A),n is the number of turns per unit length, and I is the current in the solenoid.

Substitute the given values into the formula and solve for the magnetic field:

μ_0nI= (4π × 10^-7 T m/A) × (2100 turns/m) × (0.140 A)= 0.094 T[/tex]

Therefore, the magnitude of the magnetic field inside the solenoid is 0.094 T.

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complete question:

a solenoid of length 2.50 m and radius 1.80 cm carries a current of 0.140 a. determine the magnitude of the magnetic field inside if the solenoid consists of 2100 turns of wire.

The ratio that the odds used for determining the payoff is 23 : 1 (option d).

The odds used for determining the payoff in this game of chance can be found by dividing the total amount that will be paid out (including the original bet) by the amount of the bet. In this case, the total amount that will be paid out is $115, and the amount of the bet is $5. Therefore, the odds can be calculated as follows:

Odds = Total amount paid out : Amount of bet

Odds = $115 : $5

To simplify this ratio, we can divide both sides by $5:

Odds = $23 : $1

This means that for every $1 bet, the payout will be $23 if the event occurs.

Therefore, the answer to this question is option (d) 23:1.

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The value of the given integral is 390.

Given that, we need to use Fundamental Theorem of Calculus to evaluate [tex]\int\limits^{27}_8 {(x+3)} \, dx[/tex]

So,

[tex]\int\limits^{27}_8 {(x+3)} \, dx\\\\= \int\limits^{27}_8 {x} \, dx + \int\limits^{27}_8 {3} dx\\\\[/tex]

= [x²/2]²⁷₈ + 3[x]²⁷₈

= 332.5 + 57

= 389.5

≈ 390

Hence, the value of the given integral is 390.

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The integral values are:

1) x^4/4 - x^10/60.

2) x - x^3/18 + x^5/600.

We have,

For both problems, we can use Taylor series expansions to approximate the integrands to the desired degree of accuracy.

Then we integrate the Taylor series term by term to obtain a polynomial approximation for the integral.

We have:

sin t³ = t³ - (1/3!) t^9 + (1/5!) t^15 - ...

Using only the first two terms, we get:

sin t³ ≈ t³ - (1/3!) t^9

Integrating from 0 to x, we get:

f(x) ≈ ∫(0 to x) t^3 - (1/3!) t^9 dt

= x^4/4 - x^10/60

The error is bounded by the absolute value of the next term in the Taylor series, which is (1/5!) x^15.

Since 1/5! is less than 10^-3, this error is smaller than 10^-3 throughout the interval [0, 1].

Therefore, the answer is x^4/4 - x^10/60.

We have:

sin t/t = 1 - (1/3!) t² + (1/5!) t^4 - ...

Using only the first two terms, we get:

sin t/t ≈ 1 - (1/3!) t²

Integrating from 0 to x, we get:

f(x) ≈ ∫(0 to x) (1 - (1/3!) t²) dt

= x - x³/18

The error is bounded by the absolute value of the next term in the Taylor series, which is (1/5!) x^4.

Since 1/5! is less than 10^-3, this error is smaller than 10^-3 throughout the interval [0, 1].

Therefore, the answer is (b) x - x^3/18 + x^5/600.

Thus,

1) x^4/4 - x^10/60.

2) x - x^3/18 + x^5/600.

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Answer:

Therefore, the domain of the function r(t) = vt+1i+t-3j is (-∞, ∞).

Step-by-step explanation:

To find the domain of the following vector-valued function, r(t) = vt+1i+t-3j, we need to determine the range of values that t can take. Since t appears in both terms of the function, we need to ensure that both terms are defined for all possible values of t.

The domain of t is typically all real numbers, unless there are specific restrictions given in the problem. In this case, there are no such restrictions, so the domain of t is all real numbers, or (-∞, ∞).

Therefore, the domain of the function r(t) = vt+1i+t-3j is (-∞, ∞).

The probability that the sample mean x will be greater than 11 is approximately 9.01%.

To find the probability that the sample mean (x) will be greater than 11 given a sample size of 45 drawn from a population with a mean of 10 and a standard deviation of 5, follow these steps:

Step 1: Calculate the standard error (SE) of the sample mean.

The standard error is given by the formula:

SE = population standard deviation / √sample size.

In this case, SE = 5 / √45 ≈ 0.745.

Step 2: Calculate the z-score corresponding to the given value of x.

The z-score is given by the formula:

z = (x - population mean) / SE.

In this case, z = (11 - 10) / 0.745 ≈ 1.34.

Step 3: Use the z-score to find the probability that x will be greater than 11.

The probability can be found using a standard normal (z) table or a calculator.

A z-score of 1.34 corresponds to a probability of approximately 0.0901 (9.01%).

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The alternative hypothesis (Ha) is that the mean mortgage amount in

2016 is greater than that of 2015: Ha: µ2 > µ1

Hypothesis testing involves setting up a null hypothesis (H0) and an

alternative hypothesis (Ha) to test whether there is significant evidence

to reject the null hypothesis in favor of the alternative hypothesis.

The null hypothesis is a statement that assumes there is no significant

difference between two populations or samples being compared, while

the alternative hypothesis is the opposite of the null hypothesis.

In this case, we want to test whether the mean mortgage amount in

2016  is greater than that of 2015. Let us denote the population mean of

mortgage amounts in 2015 as µ1 and the population mean of mortgage

amounts in 2016 as µ2.

The null hypothesis (H0) is that there is no significant difference in the

mean mortgage amounts between the two populations:

H0: µ2 ≤ µ1

The alternative hypothesis (Ha) is that the mean mortgage amount in

2016 is greater than that of 2015:

Ha: µ2 > µ1

We will use a one-tailed test with a significance level of 0.05 (5%) since

we are interested in testing whether the mean mortgage amount in 2016

is greater than that of 2015. that since the standard deviations of both

populations are known, we can use a z-test to compare the means.

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