what is the volume of the prism

We can use the trigonometric identity sin^2(x) = (1 - cos(2x))/2 and simplify sin^4(x) as (sin^2(x))^2 = [(1 - cos(2x))/2]^2.

So, the integral becomes:

∫9sin^4(x)cos(x) dx = ∫9[(1-cos(2x))/2]^2cos(x) dx

Expanding the square and distributing the 9, we get:

= (9/4) ∫[1 - 2cos(2x) + cos^2(2x)]cos(x) dx

Now, we can simplify cos^2(2x) as (1 + cos(4x))/2:

= (9/4) ∫[1 - 2cos(2x) + (1 + cos(4x))/2]cos(x) dx

= (9/4) ∫(cos(x) - 2cos(x)cos(2x) + (1/2)cos(x) + (1/2)cos(x)cos(4x)) dx

Integrating term by term, we get:

= (9/4) [sin(x) - sin(2x) + (1/2)sin(x) + (1/8)sin(4x)] + C

where C is the constant of integration.

Therefore,

∫9sin^4(x)cos(x) dx = (9/4) [sin(x) - sin(2x) + (1/2)sin(x) + (1/8)sin(4x)] + C.

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After 25 years of continuous compounding at a 3% interest rate the painting will be worth  $42340

To calculate the future value of the painting after 25 years with continuous compounding, we can use the formula:

[tex]A = P \times e^(^r^t^)[/tex]

Where:

A = future value

P = initial value (present value)

e = base of natural logarithm (approximately 2.71828)

r = interest rate (as a decimal)

t = time (in years)

P is $20,000, the interest rate r is 3% (or 0.03 as a decimal), and the time t is 25 years.

Substituting the values into the future value formula

[tex]A = 20000 \times e^(^0^.^0^3^\times ^2^5^)[/tex]

A=20000×2.117

A = $42340

Therefore, the painting will be worth  $42340 after 25 years of continuous compounding at a 3% interest rate.

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Answer:

Step-by-step explanation:

You want the price of each kind of ticket if these ticket purchases were made:

The two purchases can be represented by the equations ...

  32s +8a = 164

  27s +4a = 120.50

We can solve these equations using elimination. Subtracting the first equation from twice the second eliminates the y-variable:

  2(27s +4a) -(32s +8a) = 2(120.50) -(164)

  22s = 77 . . . . . . . simplify

  s = 3.5 . . . . . . . divide by 22

  4a = 120.50 -27(3.5) = 26

  a = 26/4 = 6.5

Student tickets cost $3.50 each, and adult tickets cost $6.50 each.

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Answer:

given figure divide two parts.

area=length ×width

area=(3cm×1cm)+(3cm×1cm)

area=3cm^2+3cm^2=6cm^2

and

perimeter=1+3+1+1+3+1+3+1=14cm

The approximate distance from Mars to the Sun can be determined by subtracting the distance from Earth to Mars from the distance from the Sun to Earth. The answer is option O 4.409 x 10^7 miles.

To find the approximate distance from Mars to the Sun, we subtract the distance from Earth to Mars (4.881 x 10^7 miles) from the distance from the Sun to Earth (9.29 x 10^7 miles). By subtracting these values, we get approximately 4.409 x 10^7 miles, which corresponds to option O 4.409 x 10^7 miles.

This calculation accounts for the fact that the distance from the Sun to Earth and the distance from Earth to Mars are given separately. By subtracting the known distance between Earth and Mars from the total distance between the Sun and Earth, we can estimate the distance from Mars to the Sun.

Therefore, the approximate distance from Mars to the Sun is 4.409 x 10^7 miles, as indicated by option O 4.409 x 10^7 miles.

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The correct completion of the statement is B. A correlation of zero between two quantitative variables means that there is no linear association between the two variables.

Correlation is a measure of the strength and direction of the linear relationship between two quantitative variables.

The value of correlation coefficient 'r' ranges from -1 to 1, where 0 indicates no linear association between the variables.

A correlation of zero does not mean that there is no association at all between the variables. It only means that the variables do not show any linear trend or pattern.

There could be other types of relationships between the variables, such as non-linear, curvilinear, or categorical.

Therefore, re-expressing the data may not necessarily guarantee a linear association between the two variables.

It is also important to note that a correlation coefficient of zero does not necessarily indicate that the calculation of r is incorrect. It simply implies that there is no linear relationship between the variables.

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In all these stories, the authors use suspense, ambiguity, and unexpected plot twists to keep readers on edge and guessing what comes next. While the stories share some similarities in style and structure, they differ in terms of the specific themes and subject matter.

3. Summary of the story: Click Clack the Rattle Bag by Neil Gaiman is a spooky short story about a man walking his young granddaughter home from a party late one night. The young girl asks her grandfather to tell her a scary story to keep her distracted from the creepy noises and the darkness that surrounded them. The story is about an old man who goes to visit his neighbor's house to collect eggs. The neighbor gives him the eggs and warns him not to pay attention to the rattling bag in the corner of the room.4. The story unfolds gradually, and the author maintains an air of suspense by withholding key details about the story, such as who or what is inside the rattling bag. Gaiman uses this technique to keep the reader engaged, allowing them to imagine all kinds of potential horrors and keeps them guessing until the end.

5. Neil Gaiman's tone during his reading of Click Clack the Rattle Bag is calm, ominous, and measured, which adds to the suspense and fear factor of the story. The audience reacts with anticipation, fear, and wonder, while the reader feels a sense of foreboding and fear. At the end of the story, the reader is left with a sense of unease and discomfort.6. Gaiman's Click Clack the Rattle Bag is influenced by the stories we have read previously in this unit, such as Edgar Allan Poe's The Tell-Tale Heart, and The Monkey's Paw by W.W. Jacobs. In all these stories, the authors use suspense, ambiguity, and unexpected plot twists to keep readers on edge and guessing what comes next. While the stories share some similarities in style and structure, they differ in terms of the specific themes and subject matter.

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The correct order of decreasing acid strength is:
1. Oxalic acid (Ka = 5.4 x 10-2)
2. Benzoic acid (Ka = 6.3 x 10-5)
3. Hydrosulfuric acid (Ka = 1.1 x 10-7)
4. Boric acid (Ka = 5.8 x 10-10)

The acid strength of a compound is determined by its dissociation constant (Ka), which is the equilibrium constant for the reaction of an acid with water to produce its conjugate base and H+ ions. The smaller the Ka value, the weaker the acid, as it indicates that the acid is less likely to dissociate and donate H+ ions in solution.

Oxalic acid has the highest Ka value, indicating it is the strongest acid in the group.

Benzoic acid has a higher Ka value than hydrosulfuric acid, making it a stronger acid.

Hydrosulfuric acid is a stronger acid than boric acid, which has the smallest Ka value, indicating it is the weakest acid in the group.

Therefore, the correct answer is option 4.

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The correct answer is (a) standard deviation. The measure of central tendency that is NOT included among the given options is the standard deviation (a).

Measures of central tendency are statistical measures that represent the central or average value of a dataset. They provide insight into the typical or central value around which the data tends to cluster. The three commonly used measures of central tendency are the mean, median, and mode.

a. Standard deviation is not a measure of central tendency. It is a measure of dispersion or variability in a dataset. It quantifies how spread out the data points are from the mean. Standard deviation provides information about the spread or scatter of the data rather than representing a central value.

b. Median is a measure of central tendency that represents the middle value in a dataset when the data points are arranged in ascending or descending order. It divides the data into two equal halves.

c. Mean is a measure of central tendency that represents the arithmetic average of a dataset. It is calculated by summing all the data points and dividing by the total number of observations.

d. Mode is a measure of central tendency that represents the most frequently occurring value or values in a dataset. It identifies the value(s) that appear(s) with the highest frequency.

Therefore, the standard deviation (a) is the measure of central tendency that is NOT included among the given options.

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To find the area of the region rr bounded by the graphs of y = 2x and y = 4x - x^2, we need to determine the points of intersection between the two curves. Answer :  4/3 square units.

Setting the equations equal to each other, we have:

2x = 4x - x^2

Simplifying, we get:

x^2 - 2x = 0

Factoring out x, we have:

x(x - 2) = 0

So, x = 0 or x = 2.

Now, we can integrate the difference of the two curves between x = 0 and x = 2 to find the area:

Area = ∫[0, 2] (4x - x^2 - 2x) dx

Simplifying, we have:

Area = ∫[0, 2] (2x - x^2) dx

Integrating, we get:

Area = [x^2 - (x^3)/3] evaluated from 0 to 2

Evaluating at the limits, we have:

Area = (2^2 - (2^3)/3) - (0^2 - (0^3)/3)

Area = (4 - 8/3) - (0 - 0)

Area = (12/3 - 8/3) - 0

Area = 4/3

Therefore, the area of the region rr bounded by the curves y = 2x and y = 4x - x^2 is 4/3 square units.

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Answer:

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

Divide the number of players by the number of rackets.

In this case, there are 15 players and 12 rackets.

So the rate of players per racket would be:

1. We can estimate the population mean with 90% confidence to be between 9.1775 and 10.8225.

b. We can estimate the population mean with 90% confidence to be between 8.289 and 11.711.

c. We can estimate the population mean with 90% confidence to be between 7.09 and 12.91.

d. As the sample size decreases, the confidence interval becomes wider because the standard error of the mean (σ/√n) increases.

a For a 90% confidence level, the critical value is 1.645 (found using a z-table or a calculator). Therefore, plugging in the given values, we get:

CI = 10 ± 1.645 * (5/√100)

CI = 10 ± 0.8225

CI = (9.1775, 10.8225)

b. CI = 10 ± 1.711 * (5/√25)

CI = 10 ± 1.711

CI = (8.289, 11.711)

c. CI = 10 ± 1.833 * (5/√10)

CI = 10 ± 2.91

CI = (7.09, 12.91)

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The statements that we can see that are true from the statements that we have are B and E

A coefficient in mathematics is a number or constant multiplied by a variable or phrase in an algebraic statement. It displays the variable's scale or proportionate relationship to the rest of the expression.

Coefficients are commonly expressed in algebraic formulas by letters or numbers that are multiplied by variables. Coefficients can be zero, positive, or negative. A positive coefficient denotes an additive term that raises the expression's value, whereas a negative coefficient denotes a subtractive element. A zero coefficient indicates that the term is not present in the equation.

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a. The cost of tuition at the college in 2017 is $15,000.

b. The annual percentage rate at which the tuition grows is 7%.

c. Assuming the same growth rate before and after 2017, the tuition in 2000 was $10,000.

d. The tuition in 2010 was $12,754.

e. The tuition when you graduate from high school will depend on the specific year of graduation and can be calculated using the given function.

a. The cost of tuition in 2017 can be found by substituting x = 0 into the function f(x) = 15. (1.07)*, resulting in f(0) = 15. Therefore, the tuition cost in 2017 is $15,000.

b. The annual percentage rate of tuition growth can be determined from the given function. In the expression (1.07), the coefficient 1 represents 100%, and the exponent 0.07 represents 7%. Therefore, the tuition grows at an annual rate of 7%.

c. To find the tuition in 2000, we need to calculate the number of years from 2000 to 2017 and substitute it into the function. The difference between 2017 and 2000 is 17 years. Substituting x = -17 into the function f(x) = 15. (1.07)* gives f(-17) = 10. Therefore, the tuition in 2000 was $10,000.

d. Similar to the previous calculation, we need to find the number of years from 2010 to 2017 and substitute it into the function. The difference is 7 years, so substituting x = -7 into f(x) = 15. (1.07)* gives f(-7) = 12.754. Thus, the tuition in 2010 was $12,754.

e. To determine the tuition when you graduate from high school, you need to know the specific year of your graduation. You can substitute the number of years since 2017 into the function f(x) = 15. (1.07)* to calculate the corresponding tuition cost.

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This is the fourth harmonic and the wavelength of the wave is 40.67 cm.

For a standing wave on a guitar string, the length of the string (L) and the number of antinodes (n) determine the wavelength (λ) of the wave according to the formula:

λ = 2L/n

In this case, the length of the guitar string is 61 cm and the number of antinodes is 3. Therefore, the wavelength of the standing wave is:

λ = 2(61 cm)/3 = 40.67 cm

The harmonic number (i.e., the number of half-wavelengths that fit onto the string) for this standing wave can be determined by the formula:

n = (2L/λ) + 1

Plugging in the values of L and λ, we get:

n = (2(61 cm)/(40.67 cm)) + 1 = 4

Therefore, this standing wave has the fourth harmonic.

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The total number of integers from 1 through 999 that do not have any repeated digits is 9 + 81 + 648 = 738.

The total number of integers from 1 through 999 is 999-1+1=999.

To count the number of integers that do not have any repeated digits, we can break it down into cases based on the number of digits each integer has.

For one-digit integers, there are obviously no repeated digits, so there are 9 of them (1 through 9).

For two-digit integers, the first digit can be any of the 9 digits (excluding 0) and the second digit can be any of the remaining 9 digits (excluding the first digit). So there are 9x9=81 two-digit integers that do not have any repeated digits.

For three-digit integers, the first digit can be any of the 9 digits (excluding 0), the second digit can be any of the remaining 9 digits (excluding the first digit), and the third digit can be any of the remaining 8 digits (excluding the first two digits). So there are 9x9x8=648 three-digit integers that do not have any repeated digits.

Therefore, the total number of integers from 1 through 999 that do not have any repeated digits is 9 + 81 + 648 = 738.

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To prove that c = 13d and 1/3 = one-third, we can start with the given equation 24d = 1/3 (c - d) and perform algebraic manipulations to isolate and solve for c.

Starting with the equation 24d = 1/3 (c - d), we can begin by distributing the 1/3 to both terms inside the parentheses: 24d = 1/3 * c - 1/3 * d. Simplifying this further, we have 24d = c/3 - d/3.

Next, we can add d/3 to both sides of the equation to isolate c: 24d + d/3 = c/3. To combine the terms on the left side, we need to have a common denominator. Multiplying d by 3/3, we get 72d/3 + d/3 = c/3, which simplifies to 73d/3 = c/3.

To remove the fraction on both sides, we can multiply both sides by 3. This gives us 3 * (73d/3) = 3 * (c/3), which simplifies to 73d = c.

Therefore, we have proven that c = 13d. As for the statement 1/3 = one-third, it is a straightforward observation that both expressions represent the same value, where one-third is a fraction written in words and 1/3 is the corresponding numerical representation.

In conclusion, using algebraic manipulations, we have shown that c = 13d based on the given equation, and it is evident that 1/3 and one-third are equivalent representations.

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The critical value for testing if the correlation is significant at α = 0.01 is 2.576.

To determine the critical value for a correlation coefficient at a significance level of α = 0.01, we need to use a table of critical values. The table we use depends on the sample size and the significance level.

Assuming a two-tailed test, we can use the following steps to find the critical value:

Determine the sample size: Since the sample size is not given, we assume that it is large enough (i.e., n > 30) to use the normal distribution approximation for the correlation coefficient.

Find the degrees of freedom: The degrees of freedom for a correlation coefficient with n observations is df = n - 2.

Determine the critical value from the table: Using a table of critical values for the normal distribution, with α = 0.01 and df = n - 2, we can find the critical value. For df = n - 2 = ∞ - 2 = ∞, the critical value is approximately 2.576.

Therefore, the critical value for testing if the correlation is significant at α = 0.01 is 2.576.

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To test if the correlation between cost and distance is significant at α=0.01, we need to find the critical value. We can use the critical value table for a two-tailed test at α=0.01 and degrees of freedom (df) equal to n-2, where n is the sample size.


1. Determine the sample size (n). The sample size is not provided in your question, so I'll assume it's given elsewhere.

2. Calculate the degrees of freedom (df). To do this, use the formula: df = n - 2.

3. Refer to a critical value table for Pearson's correlation coefficient (r) using the degrees of freedom (df) and the significance level α=.01.

Here's the exact value from the critical value table:

Critical Value = r(df, α)

Once you have the critical value, compare it to the given correlation coefficient (0.842). If the correlation coefficient is greater than the critical value, the correlation is considered significant at α=.01.

Please provide the sample size (n) to complete the calculation and determine the critical value.

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The decision-making is a fundamental aspect of the planning, directing, and controlling functions within an organization -True.

Decision-making is a crucial component of the planning, directing, and controlling functions in any organization.

Planning involves setting goals, identifying potential strategies, and determining the resources needed to achieve those goals.

During this process, decision-making is required to evaluate the options and select the most appropriate course of action.
In the directing function, managers must make decisions about how to allocate resources and motivate employees to achieve the goals set during the planning phase.

This requires the ability to make sound decisions based on available information and data.
Finally, the controlling function involves monitoring performance and making adjustments as needed to keep the organization on track.

Effective decision-making is essential in this process to ensure that corrective actions are taken promptly and that resources are allocated efficiently.
Overall, decision-making plays an integral role in the success of an organization.

Managers who are skilled in making decisions that are based on sound analysis and evaluation are more likely to achieve their goals and maintain a competitive edge in today's fast-paced business environment.

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True. Decision-making is a crucial part of the planning, directing, and controlling functions in any organization. In the planning stage, decisions are made on the goals and objectives to be achieved, the resources required, and the timeline to be followed.

In the directing stage, decisions are made on how to allocate resources, delegate responsibilities, and ensure that the work is being carried out effectively. In the controlling stage, decisions are made on how to monitor progress, identify any deviations from the plan, and take corrective action. Therefore, decision-making is an integral part of these functions as it determines the success of an organization in achieving its goals and objectives.
True. Decision-making is an integral part of the planning, directing, and controlling functions. In the planning phase, managers make decisions regarding goal setting, resource allocation, and action plans. Directing involves making decisions to guide and motivate employees towards achieving the organization's objectives. Controlling involves monitoring performance, comparing results to goals, and making corrective decisions to ensure desired outcomes are achieved. Each of these functions requires effective decision-making to ensure the organization operates efficiently and meets its objectives.

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Tom and Zara's dog walking business is an example of a partnership. A partnership is a business entity in which two or more people share ownership, control, and profits. In a partnership, the partners share the profits and losses of the business, as well as the responsibility for managing and operating it.

Tom and Zara's dog walking business is an example of a partnership. A partnership is a business entity in which two or more people share ownership, control, and profits. In a partnership, the partners share the profits and losses of the business, as well as the responsibility for managing and operating it.
The partnership agreement between Tom and Zara stipulates that they will share all the money they make equally. This means that they split the earnings from the dog walking business 50-50.
One of the advantages of a partnership is that each partner brings different skills, knowledge, and experience to the business. This can be beneficial for the business, as it allows it to tap into the strengths and expertise of both partners.
However, a partnership also has its challenges. For example, the partners may have different opinions and ideas about how the business should be run, which can lead to disagreements. It is important for Tom and Zara to communicate effectively and work together to ensure that the business is successful.
In addition, the customer is the most important aspect of the business. Tom and Zara should make sure that they provide a high-quality service to their customers, as this will help them to attract and retain customers. They should also listen to their customers' feedback and take steps to address any concerns or complaints.
In conclusion, Tom and Zara's dog walking business is a partnership in which they share ownership, control, and profits equally. To ensure the success of their business, they should communicate effectively, work together, and provide a high-quality service to their customers.

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The system with like terms aligned is:-4x - 0.4y = -0.8;6x + 0.4y = 4.2;-4x + 0.4y = 0.8;6x + 0.4y = 4.2;-4x + 0.4y = -0.8;6x - 0.4y = 4.2.The above system has like terms aligned.

In the given system of equations, the system with like terms aligned is: -4x - 0.4y

= -0.8; 6x + 0.4y

= 4.2; -4x + 0.4y

= 0.8; 6x + 0.4y

= 4.2; -4x + 0.4y

= -0.8; 6x - 0.4y

= 4.2.

We know that like terms are the terms having the same variable(s) with same power(s) (if any).

In the given system of equations, we have the following terms : x, y. The coefficient of x in each equation is:

-4, 6, -4, 6, -4, 6.

The coefficient of y in each equation is:

0.4, 0.4, 0.4, 0.4, 0.4, -0.4.

Therefore, the system with like terms aligned is:

-4x - 0.4y

= -0.8;6x + 0.4y

= 4.2;-4x + 0.4y

= 0.8;6x + 0.4y

= 4.2;-4x + 0.4y

= -0.8;6x - 0.4y

= 4.2.

The above system has like terms aligned.

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A parallelogram is a polygon with four sides, where opposite sides are parallel and equal in length. ABCD is a parallelogram, which means that AB is parallel to DC and AD is parallel to BC.

Let's consider some of the properties of parallelograms. Firstly, opposite sides of a parallelogram are equal in length. This means that

AB = DC and AD = BC.

Secondly, opposite angles of a parallelogram are equal in measure. Therefore, angle

A = angle C and angle B = angle D.

Based on these properties, we can make some conclusions about ABCD.

Since AB = DC and AD = BC,

we can say that ABCD is a rectangle if all angles are right angles. If one angle is not a right angle, but all sides are still equal, then ABCD is a rhombus. If ABCD has no right angles,

but opposite sides and angles are equal, then ABCD is a kite.Furthermore, the area of a parallelogram can be found by multiplying the base by the height. The height is the perpendicular distance between a side and its opposite parallel side. The base can be any of the sides of the parallelogram. Therefore,

the area of ABCD can be found by multiplying the length of a base by the height of the parallelogram. Finally, it's worth noting that a parallelogram can be divided into two congruent triangles by drawing a diagonal. In ABCD, diagonal AC divides ABCD into two triangles, ABC and CDA.

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The length of the curve is exactly 10 units.

To find the length of the curve, we need to use the arc length formula:

L = ∫[tex](a to b) √[dx/dt]^2 + [dy/dt]^2 dt[/tex]

where a and b are the limits of integration.

Let's start by finding the derivatives of x and y with respect to t:

dx/dt = -5 sin(t) + 5 sin(5t)

dy/dt = 5 cos(t) - 5 cos(5t)

Now we can plug these derivatives into the arc length formula:

L = [tex]∫(0 to 2π) √[(-5 sin(t) + 5 sin(5t))^2 + (5 cos(t) - 5 cos(5t))^2] dt[/tex]

Simplifying this expression, we get:

L =[tex]∫(0 to 2π) √(50 - 50 cos(4t)) dt[/tex]

Next, we can use the trigonometric identity [tex]cos(2θ) = 2cos^2(θ)[/tex] - 1 to simplify the expression under the square root:

cos(4t) = [tex]2cos^2(2t) - 1[/tex]

cos(4t) =[tex]2(1 - sin^2(2t)) - 1[/tex]

cos(4t) = [tex]1 - 2sin^2(2t)[/tex]

Now we can substitute this expression back into the integral:

L = [tex]∫(0 to 2π) √(50 - 50(1 - 2sin^2(2t))) dt[/tex]

L =[tex]∫(0 to 2π) 10|sin(2t)| dt[/tex]

Since the integrand is an even function, we can simplify further:

L =[tex]2∫(0 to π) 10sin(2t) dt[/tex]

L = [tex][-5cos(2t)](0 to π)[/tex]

L = 10

Therefore, the length of the curve is exactly 10 units.

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The calculated exact length of the curve is 49.13 units

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

x = 5 cos(t) − cos(5t)

y = 5 sin(t) − sin(5t)

Differentiate the functions

So, we have

x' = 5 sin(5t) − 5sin(t)

y' = 5 cos(t) − 5cos(5t)

The length is then calculated as

L = ∫x'² + y'² dt

So, we have

L = ∫(5 sin(5t) − 5sin(t))² + (5 cos(t) − 5cos(5t))² dt

Integrate

L = 50t - 12.5sin(4t)

The interval is given as 0 ≤ t ≤ 1

So, we have

L = 50(1) - 12.5sin(4 * 1)  - [50(0) - 12.5sin(4 * 0)]

Evaluate

L = 49.13

Hence, the exact length of the curve is 49.13 units

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The actual selling price of the phone is 360000 Francs.

In order to find the actual selling price of a phone that was marked down by 10% at a kiosk, given the old price of 400000 Francs, let's use the following formula:

Actual selling price = Old price - (Marked down percentage * Old price)

In this case, the marked down percentage is 10%, which can be written as 0.1 in decimal form.

Substituting the given values, we get:

Actual selling price = 400000 Francs - (0.1 * 400000 Francs)

Simplifying the expression on the right side of the equation:

Actual selling price = 400000 Francs - 40000 Francs

Therefore, the actual selling price of the phone is:

Actual selling price = 360000 Francs

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Antiderivative is a fundamental concept in calculus. It refers to finding a function that, when differentiated, results in a given function. In other words, if F(x) is an antiderivative of f(x), then F'(x) = f(x). It is also known as the indefinite integral.

A.) The antiderivative of f(t) = [tex]e^{7t}[/tex] is given by F(t) = (1/7)[tex]e^{7t}[/tex] + C, where C is the constant of integration. This is because the derivative of (1/7[tex]e^{7t}[/tex] + C is (1/7)[tex]e^{7t}[/tex].
B.) The antiderivative of f(z) = 2019 is given by F(z) = 2019z + C, where C is the constant of integration. This is because the derivative of 2019z + C is 2019.
C.) The antiderivative of f(x) = 2x + e is given by F(x) = x² + ex + C, where C is the constant of integration. This is because the derivative of x² + ex + C is 2x + e.
D.) The most general form of the antiderivative of f(x) = [tex]xe ^ {x^2[/tex] is F(x) = (1/2) [tex]e^{x^2[/tex] + C, where C is the constant of integration. To see why, we can use the substitution u = x², du/dx = 2x, to rewrite the integral as (1/2) ∫ [tex]e^u[/tex] du. This integrates to (1/2) [tex]e^u[/tex] + C, which we can substitute back as (1/2) [tex]e^{x^2[/tex] + C to get the most general form of the antiderivative.

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The position vector from (3, − 8,0) to ( − 9, − 7, − 6) is  (-12, 1, -6).

To find the position vector, we need to add this vector to the initial point (3, -8, 0). This gives us:
(3, -8, 0) + (-12, 1, -6) = (-9, -7, -6)
In mathematics, position vector refers to a vector that describes the position of a point relative to an origin point. In this question, we are asked to find the position vector that is equal to the vector from (3, -8, 0) to (-9, -7, -6).

To find the position vector, we need to add the vector from the initial point to the final point to the initial point itself. This gives us the endpoint of the vector, relative to the origin. The position vector is important in many applications, such as physics, engineering, and computer graphics, where it is used to describe the position of an object or point in space.

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The equation you provided is:

x² + 4x + 4 + y² - 6y + 9 = 5 + 4 + 9

Simplifying both sides of the equation, we have:

x² + 4x + y² - 6y + 13 = 18

Combining like terms, we get:

x² + 4x + y² - 6y - 5 = 0

This is the simplified form of the equation.

Answer:

Step-by-step explanation:

[tex]\int\limits^a_b {x} \, dx i \lim_{n \to \infty} a_n \\\\\\.......\\..\\\\solving:\\\\x^{2}+y^{2} + 4x-6y = 5[/tex]

To solve this differential equation, we first need to separate the variables by multiplying both sides by dy and dividing by (x+2y^2):

dy/(x+2y^2) = dx/y

Next, we can integrate both sides. On the left side, we can use the substitution u = y^2, du/dy = 2y, and dy = du/2y to get:

∫(1/(x+2y^2)) dy = (1/2)∫(1/(x+u)) du
= (1/2)ln|x+u| + C
= (1/2)ln|x+y^2| + C

On the right side, we have:

∫(dx/y) = ln|y| + D

Putting it all together, we have:

(1/2)ln|x+y^2| + C = ln|y| + D

Simplifying and exponentiating both sides, we get:

|x+y^2|^(1/2) = e^(2(D-C)) * |y|

Taking the positive and negative square roots separately, we get two solutions:

x + y^2 = e^(2(D-C)) * y^2
and
x + y^2 = -e^(2(D-C)) * y^2

So the general solution to the differential equation is:

x + y^2 = Ce^(2D) * y^2  or  x + y^2 = -Ce^(2D) * y^2
where C and D are arbitrary constants.

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A) This graph shifts right 4 units and up 2 units. B) This graph shifts left 3 units and down 8 units.C) This graph shifts right 3 units.D) This graph shifts left 4 units.E) This graph shifts left 5 units and up 6 units.F) This graph shifts right 3 units and up 1 unit.

Quadratic functions are one of the most common types of functions that are used in algebra. In order to describe the movement of the quadratic function, we need to know the shape of the graph of the function and how it opens. We also need to know if there is any horizontal or vertical movement. Let's have a look at each of the given quadratic functions:

A) y=-3(x-4) +2The graph of this function opens downwards. It is because the coefficient of x² is negative (-3). Also, it is shifted 4 units rightward and 2 units upward. So, this graph shifts right 4 units and up 2 units.

B) y=2(x+3)² – 8The graph of this function opens upwards. It is because the coefficient of x² is positive (+2). There is no horizontal movement as there is no addition or subtraction to x. However, the graph is shifted 3 units leftward and 8 units downward. So, this graph shifts left 3 units and down 8 units.

C) y=x²-3The graph of this function opens upwards. It is because the coefficient of x² is positive (+1). There is no horizontal movement as there is no addition or subtraction to x. However, the graph is shifted 3 units rightward. So, this graph shifts right 3 units.

D) y=(x+4)²The graph of this function opens upwards. It is because the coefficient of x² is positive (+1). There is no horizontal movement as there is no addition or subtraction to x. However, the graph is shifted 4 units leftward. So, this graph shifts left 4 units.

E) y=-(x+5)² +6The graph of this function opens downwards. It is because the coefficient of x² is negative (-1). There is no horizontal movement as there is no addition or subtraction to x. However, the graph is shifted 5 units leftward and 6 units upward. So, this graph shifts left 5 units and up 6 units.

F) y=7(x-3)² +1The graph of this function opens upwards. It is because the coefficient of x² is positive (+7). There is no horizontal movement as there is no addition or subtraction to x. However, the graph is shifted 3 units rightward and 1 unit upward. So, this graph shifts right 3 units and up 1 unit.

In conclusion, we have analyzed each of the given quadratic functions and described how they open and if there is any horizontal or vertical movement. We have also stated how many spaces they move.

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To solve the initial value problem y′ 5y=t3e−5t, y(2)=0, we can use the method of integrating factors.

First, we need to identify the integrating factor, which is given by e^(∫5dt) = e^(5t).

Multiplying both sides of the differential equation by the integrating factor, we get:

e^(5t) y′ - 5e^(5t) y = t^3 e^(-t)

Using the product rule, we can rewrite the left-hand side as:

(d/dt)(e^(5t) y) = t^3 e^(-t)

Integrating both sides with respect to t, we get:

e^(5t) y = -t^3 e^(-t) - 3t^2 e^(-t) - 6t e^(-t) - 6 e^(-t) + C

where C is the constant of integration.

Using the initial condition y(2) = 0, we can solve for C:

e^(10) * 0 = -8e^(-10) + C

C = 8e^(-10)

Therefore, the solution to the initial value problem is:

y = (-t^3 - 3t^2 - 6t - 6)e^(-5t) + 8e^(-10)

and it satisfies the initial condition y(2) = 0.

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