About 24 randomly selected people were asked how long they slept at night. The mean time was 6 hours, and the standard deviation was 1.3 hour. Calculate the 80% confidence interval of the mean time by assuming that the variable is normally distributed. Provide only the value required below. Express your answer in 3 decimal places.

The given series 3n!/2ⁿ is divergent.

To determine if the sequence a_n = 3n!/2ⁿ converges or diverges, we can use the ratio test.

Taking the limit of a_(n+1)/a_n as n approaches infinity, we get:

lim [(3(n+1)!/2ⁿ⁺¹) / (3n!/2ⁿ)]
= lim [3(n+1)!/2ⁿ⁺¹ * 2ⁿ/3n!]
= lim [3(n+1)/2]
= infinity

Since the limit is greater than 1, the sequence diverges.

The ratio test is a way to determine the convergence or divergence of a series by taking the limit of the ratio of consecutive terms. If the limit is less than 1, the series converges.

If the limit is greater than 1, the series diverges. In this case, we applied the ratio test to the sequence a_n = 3n!/2ⁿ and found that the limit is infinity, indicating that the sequence diverges. This means that the terms of the sequence do not approach a finite limit as n approaches infinity, but instead grow without bound.

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The slope of the tangent line to the curve is equal to 122 when x is approximately 24.17.

To find the value of x where the slope of the tangent line to the curve y = 3x^2 - 23x + 10 is equal to 122, we first need to find the derivative of y with respect to x. The derivative represents the slope of the tangent line at any given point.

Using the power rule, the derivative of y with respect to x (denoted as y') is:
y' = 6x - 23

Now we need to set y' equal to 122 and solve for x:
6x - 23 = 122

Add 23 to both sides:
6x = 145

Divide by 6:
x ≈ 24.17 (rounded to two decimal places)

So, the slope of the tangent line to the curve is equal to 122 when x is approximately 24.17.

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The probability that the sample mean compressive strength exceeds 4985 psi is approximately 1.

We need to find the probability

Calculate the standard error of the sample mean.

Standard Error (SE) = σ / √n = 100 / √9

                               = 100 / 3

                               = 33.33 psi

Calculate the z-score of the sample mean.

z = (sample mean - μ) / SE = (4985 - 5500) / 33.33

                                           = -515 / 33.33

                                           = -15.45

Find the probability using the z-score.

Since the z-score is -15.45, which is very far from the mean in the left tail, the probability of the sample mean

compressive strength exceeding 4985 psi is almost 1.

So, the probability that the sample mean compressive strength exceeds 4985 psi is approximately 1.

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lim (1 + 2 + 4x²)⁴/ˣ = 256 as x approaches infinity.

To evaluate the limit, you can follow the steps given below:

Step 1: Substitute the value of x in the expression.

lim (1 + 2 + 4x²)⁴/x = lim (1 + 2 + 4(x)²)⁴/x as x approaches some value.

Step 2: Simplify the expression inside the limit.

The expression inside the limit can be simplified by adding the terms inside the parentheses.

lim (1 + 2 + 4(x)²)⁴/x = lim (4x² + 3)⁴/x

Step 3: Use the limit law of constant multiples.

The limit law of constant multiples states that the limit of a constant multiple of a function is equal to the constant multiple of the limit of the function. In this case, we can apply this law to simplify the expression.

lim (4x² + 3)⁴/x = 4⁴ lim (x² + 3/4²)⁴/ˣ

Step 4: Apply the power rule of limits.

The power rule of limits states that the limit of a function raised to a power is equal to the limit of the function raised to that power. In this case, we can apply the power rule to simplify the expression further.

4⁴ lim (x² + 3/4²)⁴/ˣ = 4⁴ lim (x^2 + 3/4²)^(⁴/ˣ)

Step 5: Evaluate the limit.

As x approaches infinity, the expression inside the limit approaches 1, and 4 raised to any power remains finite. Hence the limit of the expression is equal to 4⁴ = 256.

lim (1 + 2 + 4x²)⁴/ˣ = 256 as x approaches infinity.

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The number of elements in the sample space of the situation is given as follows:

36.

The Fundamental Counting Theorem states that if there are m ways to do one thing and n ways to do another, then there are m x n ways to do both.

This can be extended to more than two events, where the number of ways to do all the events is the product of the number of ways to do each individual event

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

For this problem, we have two events, in which:

Hence the total number of outcomes is given as follows:

6 x 6 = 36.

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Answer:

Step-by-step explanation:

First, let's calculate the volume of the glass sphere:

V_sphere = (4/3) * pi * r^3

V_sphere = (4/3) * 3.14 * 12^3

V_sphere ≈ 7238.23 cm^3

Next, let's find the volume of the cylinder:

V_cylinder = pi * r^2 * h

V_cylinder = 3.14 * 30^2 * 100

V_cylinder ≈ 282,600 cm^3

Since the glass sphere is completely submerged in water, the volume of the water in the cylinder will be the difference between the volume of the cylinder and the volume of the sphere:

V_water = V_cylinder - V_sphere

V_water ≈ 275,362.77 cm^3

Therefore, the approximate volume of the water contained in the cylinder is approximately 275,365 cm^3, which is closest to option D.

The probability of having 63 or more successes in the sample is approximately 0.0266, or 2.66%.

To use the normal approximation to find the probability P(X ≥ 63) for a sample size of n = 90 and population proportion of successes p = 0.6, follow these steps:

Step 1: Calculate the mean (μ) and standard deviation (σ) for the binomial distribution.
[tex]μ = n * p[/tex] = 90 * 0.6 = 54
[tex]σ = \sqrt{(n * p * (1 - p))} = \sqrt{(90 * 0.6 * 0.4) }[/tex]= √21.6 ≈ 4.65

Step 2: Use the normal approximation.
To find P(X ≥ 63), first convert X to a z-score:
z = [tex](X - μ) / σ[/tex] = (63 - 54) / 4.65 ≈ 1.93

Step 3: Find the probability using a z-table or calculator.
Using a z-table or calculator, find the probability of a z-score less than 1.93 (since we want P(X ≥ 63), we need to find the area to the right of the z-score):
P(Z ≤ 1.93) ≈ 0.9734

Step 4: Calculate the complement probability.
Since we want P(X ≥ 63), we need to find the complement probability (1 - P(Z ≤ 1.93)):
P(X ≥ 63) = 1 - 0.9734 = 0.0266

So, the probability of having 63 or more successes in the sample is approximately 0.0266, or 2.66%.

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Answer:

i want to say A, 9.

Step-by-step explanation:

The answer is (A) 9.

If Sonia removes 1 tile from the pile of 10 tiles, there will be 9 tiles left.If Roberto removes 1 tile, then there will be 8 tiles left. Sonia can remove 2 tiles, leaving 6 tiles for Roberto.

If Roberto removes 1, 2, or 3 tiles, then there will be 5, 4, or 3 tiles left, respectively. Sonia can then remove enough tiles to leave Roberto with a multiple of 6 tiles, ensuring a win on her next turn.For example, if Roberto removes 3 tiles, then there will be 7 tiles left. Sonia can remove 2 tiles, leaving 5 tiles for Roberto. Then, regardless of how many tiles Roberto removes, Sonia can always remove enough tiles to leave Roberto with a multiple of 6 tiles on his turn, ensuring a win on her next turn.

Answer:

2×2×3<61+9÷4×3>2+6+36÷7÷8×9+94>2×1÷3÷34÷

In linear regression, unbiasedness of OLS estimator for $\beta$ is guaranteed

True.

In linear regression, unbiasedness of OLS estimator for $\beta$ is guaranteed when the following conditions hold:

The regression model is correctly specified and the true model is linear in the parameters.

The errors are homoscedastic and normally distributed with mean zero and constant variance.

The regressors are not perfectly collinear.

The expected value of the errors given the values of the regressors is zero.

Under these conditions, if the OLS residuals are uncorrelated with the regressors, then the estimate of $\beta$ obtained from OLS is unbiased.

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Answer:

The Correct answer is A

2x²√7y

a) z-scores for Mississauga-Erindale: Z = (143,361 - 102,639.28) / 21,855.384 ≈ 1.865 and for Parkdale-High Park: Z = (102,142 - 102,639.28) / 21,855.384 ≈ -0.023
b) The citizens might feel underrepresented in the House of Commons as their votes have less weight compared to those in smaller ridings.

a) To compare the z-scores for these ridings, we need to use the formula:

z-score = (x - μ) / σ

where x is the population of the riding, μ is the mean population of all ridings, and σ is the standard deviation of all ridings.

For Mississauga-Erindale:

z-score = (143,361 - 102,639.28) / 21,855.384 = 1.87

For Parkdale-High Park:

z-score = (102,142 - 102,639.28) / 21,855.384 = -0.02

Therefore, Mississauga-Erindale has a z-score of 1.87, which means its population is above the mean population of all ridings by 1.87 standard deviations. Parkdale-High Park has a z-score of -0.02, which means its population is almost exactly at the mean population of all ridings.

b) The citizens of Mississauga-Erindale could argue that their riding is overrepresented in the House of Commons. This is because their population is above the mean population of all ridings by almost 2 standard deviations, which means they have more political influence per person compared to other ridings. However, it's important to note that the electoral district boundaries are redrawn every 10 years based on population changes, so the population of each riding may change over time.
a) To compare the z-scores for Mississauga-Erindale and Parkdale-High Park, we need to calculate the z-scores for each riding using the given mean and standard deviation. The formula for calculating z-scores is:

Z = (X - μ) / σ

For Mississauga-Erindale:
Z = (143,361 - 102,639.28) / 21,855.384 ≈ 1.865

For Parkdale-High Park:
Z = (102,142 - 102,639.28) / 21,855.384 ≈ -0.023

b) The citizens of Mississauga-Erindale could argue that their representation in the House of Commons is unfair because their riding has a significantly larger population compared to the average riding size. The z-score of 1.865 indicates that the population of Mississauga-Erindale is approximately 1.865 standard deviations above the mean, meaning it is larger than a majority of other ridings. Consequently, the citizens might feel underrepresented in the House of Commons as their votes have less weight compared to those in smaller ridings.

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Answer: 10

Step-by-step explanation: Volume = length*width*depth. 180 = 6*3*depth. 6*3 = 18. We divide 180 by 18, to get 10. Depth = 10

Answer:

20 feet

Step-by-step explanation:

Let's start by using the formula for the volume of a rectangular box:

[tex]\sf\qquad\dashrightarrow V = lwh[/tex]

where:

We know that the box is half filled with grain, so the volume of the grain is:

[tex]\sf:\implies V_{grain} = 0.5V = 0.5lwh[/tex]

We also know that the volume of the grain is 180 cubic feet, so we can set up an equation:

[tex]\sf\qquad\dashrightarrow 0.5lwh = 180[/tex]

We are given that the box is 6 feet wide and 3 feet long, so we can substitute those values in:

[tex]\sf:\implies 0.5(3)(6)h = 180[/tex]

Simplifying:

[tex]\sf:\implies 9h = 180[/tex]

[tex]\sf:\implies \boxed{\bold{\:\:h = 20\:\:}}\:\:\:\green{\checkmark}[/tex]

Therefore, the depth of the box is 20 feet.

a. At a spending level of $16 million, the percentage of pollution removal is increasing at a rate of 0.97% per million dollars spent.

b. For spending levels below $150 million, the percentage of pollution removal is decreasing, while for spending levels above $150 million, the percentage of pollution removal is increasing.

a. To answer the first part of the question, we need to find the rate at which the percentage of pollution removal is changing when $16 million is spent. This is the derivative of the function P(x) evaluated at x = 16. Taking the derivative, we get P'(x) = 0.06x + 9/100.

Evaluating at x = 16, we get P'(16) = 0.06(16) + 9/100 = 0.969, or approximately 0.97%.

b. To determine for what values of x the function P(x) is increasing or decreasing, we need to find the sign of the derivative P'(x). Taking the derivative of P(x), we get P'(x) = 0.06x + 9/100. This derivative is positive for x > -9/0.06, or x > -150, and negative for x < -150.

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The expression 2+2+210-(5*2) equals 14.

The expression 2+2+2 x 10-(5 x 2) can be solved using the order of operations, also known as PEDMAS.

In the given expression, we don't have any parentheses or exponents, so we move to the multiplication and division step. We have 2 multiplications, 210 and 52. Since multiplication comes before addition and subtraction, we must perform these multiplications first. Therefore, we get:

2 + 2 + 20 - 10

Now, we have only addition and subtraction left. According to the order of operations, we perform these operations from left to right. Therefore, we get:

4 + 20 - 10

Now, we can perform the addition and subtraction, which gives us the final answer:

14

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For the given point (-0. 905) on number line is represented by point B.

A number line is a graphical depiction of numbers organised in a linear form, typically from left to right or right to left. It is a simple mathematical tool used to represent and illustrate the order and size of numbers, and it is often used in early mathematics education as a visual aid for teaching and comprehending basic arithmetic principles.

A number line is often made up of a straight line with equally spaced markers or ticks that represent individual numbers. These marks are typically identified with integers (positive and negative whole numbers), but may also include fractions or decimals depending on the context.

For locating (-0.905) on number line , first we determine where it will lie on number line.

(-0.905) > (-1) and (-0.905) < 0.hence it will lie between 0 and (-1).

In given figure both point A and B are located between 0 and (-1), out of these, considering each small spacing represent (-0.01) as per given image. Hence point A will lie at (-0.95) while point B will lie between (-0.91) and (-0.90) which corresponds to the location of point (-0.905) as well.

Thus, point (-0.905) is represented by point B on given number line.

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Complete Question:(refer image attached)

The slant height of the triangular pyramid is approximately 19.88 cm.

To find the slant height of the triangular pyramid, we can use the formula:

Slant height = sqrt(h^2 + (0.5b)^2)

where h is the height of the triangular pyramid and b is the base length.

First, we need to find the height of the triangular pyramid using the formula for the surface area of a triangular pyramid:

Surface area = 0.5 * Perimeter * Slant height + Base area

where the base area is 0.5 * b * h, and the perimeter is the sum of the lengths of the three sides of the base.

In this case, we know the surface area of the triangular pyramid (532 square cm), the base length (24 cm), and the hypotenuse of the base (25 cm).

The length of the other leg of the base can be found using the Pythagorean theorem:

a^2 + b^2 = c^2

where a and b are the two legs of the right triangle formed by the base, and c is the hypotenuse.

In this case, we have:

a^2 + 24^2 = 25^2

a^2 = 625 - 576

a^2 = 49

a = 7 cm

Therefore, the perimeter of the base is:

24 + 25 + 7 = 56 cm

Now, we can use the formula for the surface area to find the height:

532 = 0.5 * 56 * Slant height + 0.5 * 24 * h

532 = 28 * Slant height + 12 * h

We need to find the height h in terms of the slant height, so we can isolate h:

h = (532 - 28 * Slant height) / 12

Now, we substitute this expression for h into the formula for the height:

h^2 = 25^2 - (0.5 * 24)^2

h^2 = 625 - 144

h^2 = 481

h = sqrt(481)

h = 21.93 cm

Now, we substitute the expression for h in terms of the slant height into the formula for the slant height:

Slant height = sqrt(h^2 + (0.5b)^2)

Slant height = sqrt((532 - 28 * Slant height)^2 / 144 + 144)

Squaring both sides and simplifying, we get:

756 * Slant height^2 - 149984 * Slant height + 70624 = 0

Using the quadratic formula, we get:

Slant height = (149984 +/- sqrt(149984^2 - 4 * 756 * 70624)) / (2 * 756)

Slant height = (149984 +/- sqrt(141562496)) / 1512

Taking the positive root and simplifying, we get:

Slant height ≈ 19.88 cm

Therefore, the slant height of the triangular pyramid is approximately 19.88 cm.

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The function f(x) is a quadratic function in factored form, revealing the x-intercepts, vertex, and behavior of the function. X-intercepts are (0,0) and (-4,0), vertex is (-2,8), and y-intercept is (0,0). The graph opens upward.

A function is a relation between two sets of values, such that each input value maps to a unique output value. It describes a mathematical rule or relationship that links inputs to outputs.

According to the given information:

a. The function f(x) is in the form of a quadratic function, which is a second-degree polynomial function. The equation is written in factored form, revealing the x-intercepts (zeros) and the behavior of the function as it approaches the x-axis.

b. To find the x-intercepts, we set f(x) equal to zero and solve for x:

f(x) = (x^2)(x + 4) = 0

x^2 = 0 or x + 4 = 0

x = 0 or x = -4

So the x-intercepts are (0,0) and (-4,0).

To find the vertex, we can use the formula -b/2a to find the x-value of the vertex, where a and b are coefficients in the quadratic equation ax^2 + bx + c. In this case, a = 1 and b = 4, so the x-value of the vertex is -b/2a = -4/2 = -2. To find the y-value, we evaluate f(-2):

f(-2) = (-2)^2(-2+4) = 8

So the vertex is (-2, 8).

To find the y-intercept, we evaluate f(0):

f(0) = (0)^2(0+4) = 0

So the y-intercept is (0,0).

c. Here is a sketch of the graph of f(x):

      |          

      |          

      |          

  ___/ \___      

 /         \      

/           \    

/             \    

---------------  

     |     |      

    -4     4      

The graph has x-intercepts at (0,0) and (-4,0), a vertex at (-2,8), and a y-intercept at (0,0). It opens upward since the leading coefficient (coefficient of x^2) is positive

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This is an exercise in Newton's second law is one of the fundamental laws of physics and is used to describe the relationship between force, mass, and acceleration of an object. This law can be expressed mathematically as F = ma, where F is the net force applied to an object, m is its mass, and a is the acceleration produced. The law states that the acceleration experienced by an object is directly proportional to the net force acting on it and inversely proportional to its mass.

In other words, if the force acting on an object increases, its acceleration will also increase, and if the object's mass increases, its acceleration will decrease. This mathematical relationship is very useful in understanding how objects move and how forces affect their motion. Newton's second law is especially important for dynamics, the branch of physics that deals with the study of the movement of objects and its causes.

The law of force can be intuitively explained by observing how an object moves when a force is applied to it. If a force is applied to an object, such as pushing a box, the box will start to move in the direction of the applied force. If a larger force is applied, the box will move faster, and if a smaller force is applied, the box will move more slowly. If the box is heavier, it will take more force to move it at the same speed as a lighter box.

Newton's second law also states that the direction of the acceleration produced is the same as the direction of the applied force. For example, if a box is pushed to the right, the box will move to the right. If the box is pushed up, the box will move up. This relationship between the direction of force and the direction of acceleration is important in understanding how objects move in different situations.

In addition, Newton's second law is also important in understanding how forces are applied in different situations. For example, if a force is applied to a box at an angle, the box will move in a different direction than the applied force due to the breakdown of the force into horizontal and vertical components. The law of force can be used to calculate the components of force and determine how the object will move.

Newton's second law can also be used to understand the relationship between force and motion in nature. The law of force applies to all objects in the universe and is essential in understanding how planets, stars, and other celestial bodies move. For example, the gravitational force acting between two objects depends on the objects' mass and the distance between them, and this force determines how the objects move in space.

It tells us a model rocket has a mass of 0.1 kg, is pushed by the engine, and has an acceleration of 35 m/s².

We apply the formula F = m × a. We do not clear because it asks us to calculate the force, where:

F = Calculated force in Newton (N).

m = calculated mass in Kilograms (kg).

a = acceleration calculated in meters per second squared (m/s^2).

F = m  × a

F = 0.1 kg × 35 m/s²

F = 3.5 N

The amount of force that the motor exerts on the rocket is 3.5 Newtons.

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Answer:

112 cm²

Step-by-step explanation:

Area of shaded region =

(9 x 16) - (5 x 4) - 2(1/2 x 3 x 4) = 112 cm²

If the perimeter and two-sides of the triangle are 18c, 4cm and 5cm, then the unknown third-side is 9cm.

The "Peri-meter" of a triangle is defined as the sum of lengths of all 3 sides.

Let the unknown third side of the triangle be denoted as "x".

We know that, "first-side" of triangle is = 4 cm,

"Side-2" of triangle = 5 cm,

"Peri-meter" of the triangle is given to be 18 cm,

The Perimeter equation is written as:

⇒ Side 1 + Side 2 + Side 3 = Perimeter,

⇒ 5 + 4 + x = 18,

⇒ 9 + x = 18,

⇒ x = 18 - 9,

⇒ x = 9.

Therefore, the measure of the unknown third side of triangle is 9 cm.

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The given question is incomplete, the complete question is

The two sides of the triangle are 4cm and 5cm , the perimeter of the triangle is 18cm, find the measure of the unknown third side.

Answer: 35, 92

Step-by-step explanation: We see that the third number is the result of adding the first two numbers (9+13=22). Continuing with this pattering, we know that the fourth spot is 22 + x = 57. 57-22 is 35, so the fourth spot is 35. The sixth spot will be 57 + 35, or, 92.

Answer:

The next numbers in the sequence are 35, and 92.

Step-by-step explanation:

The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding numbers. So, the next numbers in the Fibonacci sequence would be:

Therefore, the next numbers in the sequence are 35, and 92, if the sequence is continued using the Fibonacci sequence.

The local maximum is 423 at x = 4 and local minimum is 207 at x = 10.

Given function is,

f(x) = 2x³ - 42x² + 240x + 7

f'(x) = 6x² - 84x + 240

Let f'(x) = 0.

6x² - 84x + 240 = 0

x² - 14x + 40 = 0

Using the factorization method,

(x - 4)(x - 10) = 0

x - 4 = 0 and x - 10 = 0

x = 4 and x = 10

Limiting points are x = 4 and x = 10.

The immediate points nearby x = 4 are {3, 5}

f'(3) = 42

f'(5) = -30

Using the first derivative theorem, since the derivative of the function is positive towards the left and negative towards the right of x = 4, the function has the local maximum at x = 4.

Local maximum = f(4) = 423

The  immediate points nearby x = 10 are {9, 11}

f'(9) = -30

f'(11) = 42

Since the derivative changes from negative to positive, the function has the local minimum at x = 10.

Local minimum = f(10) = 207

Hence the local maximum and minimum values are 423 and 207 at x = 4 and x = 10 respectively.

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The integrals we solved by substitution method.

∫sin²(x) cos³(x) dx =  -1/2 (sin(x) - 1/3 sin³(x)) + C

∫sin⁵ (2x) cos³ (2x)dx  = -1/6 (cos(2x) - 2/5 cos³(2x) + 1/7 cos⁵(2x)) + C

∫cos⁴ (2x)dx =1/4 (sin(2x) + 1/3 sin³(2x)) + C

∫√cos(x) sin³(x)dx =  4/3 cos(x)√cos(x) - 8/15 cos⁵(x) + C

∫sin²(1/x)/x² dx= -(1/2)(1/x - (1/2)sin(2/x)) + C

∫cot(x) cos²(x) dx = (1/2)(x + sin(x)cos(x))cot(x) + (1/6)sin³(x) + (1/2)xsin(x) +

We solve the integrals by using substitution method.

∫sin²(x) cos³(x) dx =  -1/2 (sin(x) - 1/3 sin³(x)) + C

∫sin⁵ (2x) cos³ (2x)dx  = -1/6 (cos(2x) - 2/5 cos³(2x) + 1/7 cos⁵(2x)) + C

∫cos⁴ (2x)dx =1/4 (sin(2x) + 1/3 sin³(2x)) + C

∫√cos(x) sin³(x)dx =  4/3 cos(x)√cos(x) - 8/15 cos⁵(x) + C

∫sin²(1/x)/x² dx= -(1/2)(1/x - (1/2)sin(2/x)) + C

∫cot(x) cos²(x) dx = (1/2)(x + sin(x)cos(x))cot(x) + (1/6)sin³(x) + (1/2)xsin(x) + C

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The roots of the given equation is real and equal.

Given , 5[tex]x^{2} \\[/tex] - 10x+ k=0

The quadratic equation is b² - 4ac = 0

Here, a= 5, b= -10, c= k

substitute in b² - 4ac = 0

(-10)² - 4 * 5* k =0

100 - 20k =0 , let this be equation (1)

100 = 20k

k = [tex]\frac{100}{20}[/tex]

k = 5.

now, substitute  k= 5 in equation (1)

100 -20k = 0

100 - 20*5 = 0

100 - 100 = 0

Therefore, the given equation is real and equal .

The correct question is 5x² - 10x + k =0

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The statement "If dy/dt=f(t)g(y), the equilibrium solutions can be obtained by finding the solutions to f(t)=0" is not entirely correct.

In a differential equation of the form dy/dt = f(t)g(y), the equilibrium solutions are the constant solutions where dy/dt = 0. These occur when g(y) = 0.

To find the equilibrium solutions, we need to solve g(y) = 0. Once we have found these solutions, we can determine their stability by analyzing the sign of f(t) near these equilibrium values. If f(t) is positive near an equilibrium value, the solution is unstable (i.e., solutions near the equilibrium will move away from it). If f(t) is negative near an equilibrium value, the solution is stable (i.e., solutions near the equilibrium will move towards it).

So, while f(t) = 0 may be useful in some cases for finding equilibrium values, it is not the correct approach for finding all equilibrium solutions in a differential equation of the form dy/dt = f(t)g(y). The equilibrium solutions are found by solving g(y) = 0.

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The expression that can be solved to give a value of  13 is 1/2 * ((12/2) - 2) + 11

From the question, we have the following parameters that can be used in our computation:

1/2 x 12 / 2 - 2 + 11

Next, we solve one after the other to get 13

Using the above as a guide, we have the following expressions:

12 /2 = 6

So, we have

6 - 2 = 4

Next, we have

1/2 * 4  = 2

Lastly, we have

2 + 11 = 13

When the above steps are combined, we have

1/2 * ((12/2) - 2) + 11

Hence, the expression is 1/2 * ((12/2) - 2) + 11

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The median of this distribution is 4.375.

To find the median of a distribution, we need to first arrange the values in order from smallest to largest along with their corresponding frequencies.

VALUE.....1....2...3...4....5

Freq.........4....3...2...6....8

There are a total of 23 observations in this distribution (4 + 3 + 2 + 6 + 8 = 23). Since the total frequency is an odd number, the median will be the value that is exactly in the middle when the observations are arranged in order.

To find this middle value, we need to first find the cumulative frequency for each value. The cumulative frequency is the sum of the frequencies up to and including that value.

VALUE.....1....2...3...4....5

Freq.........4....3...2...6....8

Cumulative Freq...4...7...9..15..23

The median will be the value that has a cumulative frequency of (23 + 1)/2 = 12. This value falls in the interval 4-5 since the cumulative frequency of 5 is greater than 12 and the cumulative frequency of 4 is less than 12.

To find the exact median value, we need to interpolate between the two values in the interval 4-5. We can use the following formula to find the median:

Median = L + [(n/2 - CF) / f] * w

where L is the lower limit of the interval, n is the total frequency, CF is the cumulative frequency up to the lower limit, f is the frequency of the interval, and w is the width of the interval.

For the interval 4-5, L = 4, n = 23, CF = 9, f = 8, and w = 1. The median can be calculated as:

Median = 4 + [(12 - 9) / 8] * 1 = 4.375

Therefore, the median of this distribution is 4.375.

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The solution to the homogeneous equation is z(x)=2/3x²e⁶ˣ-1/3x³e⁶ˣ.

Given that, z"(x)+z(x)=4e⁶ˣ;z(0)=0,z'(0)=0

The homogeneous equation is z''(x)+z(x)=0. The general solution to this equation is z(x)=Aeˣ+Be⁻ˣ, where A and B are constants.

Now, solving the non-homogeneous equation z''(x)+z(x)=4e⁶ˣ, using the method of Undetermined Coefficients, we make the Ansatz

z(x)=cx²e⁶ˣ+dx³e⁶ˣ.

Substituting this into the equation, we get

2c+d=0 and 12c+18d=4.

Solving this system of equations, we get c=2/3 and d=-1/3.

Therefore, the solution to the non-homogeneous equation is

z(x)=2/3x²e⁶ˣ-1/3x³e⁶ˣ.

Plugging in the boundary conditions, we get

z(0)=0=2/3(0)²e⁶⁽⁰⁾-1/3(0)³e⁶⁽⁰⁾

z'(0)=0=4/3(0)e⁶⁽⁰⁾-3/3(0)²e⁶⁽⁰⁾

Both these conditions are satisfied, so the solution is

Therefore, the solution to the homogeneous equation is z(x)=2/3x²e⁶ˣ-1/3x³e⁶ˣ.

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