Let f: R+R be a function that satisfies O 0. (a) Show that the series cosh(f(n)) ne1 diverges regardless of the rule for f. (b) Show that the series ( f(n) 2n3 - 1 converges regardless"

The measure of the smallest angle is 37.1 degrees

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

The legs of a right triangle measure 11.4 meters and 15.1 meters

So, the measure of one of the acute angles is

tan(x) = 11.4/15.1

Evaluate

tan(x) = 0.7550

Take the arc tan of both sides

So, we have

x = 37.1

This means that the measure of the smallest angle is 37.1 degrees

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The probability that a) the sample has a proportion between 0.5 and 0.7 is 0.780. b) The probability that the sample has a proportion within 5% is 0.819. c) The probability that the sample has a proportion less than 0.50 is 0.001. d) The probability that the sample has a proportion greater than 0.80 is 0.000.

a) To calculate this probability, we first need to standardize the interval (0.5, 0.7) using the formula: z = (p - P) / (σ / √(n))

where p is the sample b, P is the population proportion, σ is the standard deviation of sample proportions, and n is the sample size. Substituting the values, we get:

z1 = (0.5 - 0.62) / (0.8 / √(40)) = -2.24

z2 = (0.7 - 0.62) / (0.8 / √(40)) = 1.12

Using the standard normal table or calculator, the area between -2.24 and 1.12 is 0.780. Therefore, the probability that the sample has a proportion between 0.5 and 0.7 is 0.780.

b) The probability that the sample has a proportion within 5% of the population proportion is 0.819. We can find the range of sample proportions within 5% of the population proportion by adding and subtracting 5% of the population proportion from it, which gives: P ± 0.05P = 0.62 ± 0.031

The interval (0.589, 0.651) represents the range of sample proportions within 5% of the population proportion. To calculate the probability that the sample proportion falls within this interval, we standardize it using the formula above and find the area under the standard normal curve between -1.55 and 1.55, which is 0.819.

c) The probability that the sample has a proportion less than 0.50 is 0.001. To calculate this probability, we standardize the value of 0.50 using the formula above and find the area to the left of the resulting z-score, which is: z = (0.50 - 0.62) / (0.8 / √(40)) = -4.46

Using the standard normal table or calculator, the area to the left of -4.46 is 0.001. Therefore, the probability that the sample has a proportion less than 0.50 is 0.001.

d) The probability that the sample has a proportion greater than 0.80 is 0.000. To calculate this probability, we standardize the value of 0.80 using the formula above and find the area to the right of the resulting z-score, which is: z = (0.80 - 0.62) / (0.8 / √(40)) = 5.60

Using the standard normal table or calculator, the area to the right of 5.60 is very close to 0.000. Therefore, the probability that the sample has a proportion greater than 0.80 is 0.000.

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a. The property damage insurance covers the damage to the fence.

b. The insurance company will pay $7,000 - $1,000 = $6,000 for the fence damage.

c. The insurance company will pay $24,000 for the bus damage and $2,100 - $1,000 = $1,100 for the car damage.

d. The collision insurance policy covers the damage to Stewart's car.

e. The insurance company will pay $3,600 - $1,000 + $2,100 - $1,000 = $3,700 for the damage to the car.

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The equilibrium quantity is approximately 34.47 items and the producer surplus at the equilibrium quantity is approximately 396.11.

To find the equilibrium quantity, we need to find the quantity at which the demand and supply functions are equal:

Demand function: d(x) = 588.7 – 0.4x^2

Supply function: s(x) = 8(x) = 0.322

Setting these two functions equal to each other, we get:

588.7 – 0.4x^2 = 0.322x

Simplifying this equation, we get:

0.4x^2 + 0.322x - 588.7 = 0

Using the quadratic formula, we get:

x = (-0.322 ± √(0.322^2 + 40.4588.7)) / (2*0.4)

x ≈ 34.47 or x ≈ -43.67

Since we cannot have a negative quantity, the equilibrium quantity is approximately 34.47 items.

To find the producer surplus at the equilibrium quantity, we need to calculate the area between the supply curve and the equilibrium price, which is the price that corresponds to the equilibrium quantity. We can find the equilibrium price by plugging the equilibrium quantity into either the demand or supply function:

s(34.47) = 8(34.47) = 11.58

So the equilibrium price is approximately 11.58.

Now we can find the producer surplus by integrating the supply function from 0 to the equilibrium quantity, and subtracting the result from the area of a rectangle with height equal to the equilibrium price and width equal to the equilibrium quantity. The formula for producer surplus is:

Producer Surplus = (Equilibrium Price * Equilibrium Quantity) - ∫[0, Equilibrium Quantity] Supply Function dx

Plugging in the values we found, we get:

Producer Surplus = (11.58 * 34.47) - ∫[0, 34.47] 0.322 dx

Integrating the supply function, we get:

∫[0, 34.47] 0.322 dx = 0.322 * 34.47 ≈ 11.10

So the producer surplus is:

Producer Surplus ≈ (11.58 * 34.47) - 11.10 ≈ 396.11

Therefore, the equilibrium quantity is approximately 34.47 items, and the producer surplus at the equilibrium quantity is approximately 396.11.

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The width of the rectangular playing field is approximately 50 yards if the width is located 15 yards from either vertex.

To solve the problem, we can draw a diagram and use the properties of ellipses.

First, we note that the major axis of the ellipse is 110 yards and the minor axis is 50 yards. We can find the distance between the two foci of the ellipse using the formula c^2 = a^2 - b^2, where c is the distance between the foci, and a and b are the lengths of the semi-major and semi-minor axes.

c^2 = 110^2 - 50^2

c^2 = 10800

c ≈ 104.0

Next, we draw the two foci of the ellipse and the rectangle as shown in the diagram below. We are given that the width of the rectangle is 30 yards (15 yards from either vertex). x be the length of the rectangle.

      A          B

   +-------+-------+

  /                  \

 /                    \

/                      \

C                        D

\                      /

 \                    /

  \                  /

   +-------+-------+

      E          F

We can see that the length of the rectangle is equal to the distance between points A and B, and the width of the rectangle is equal to the distance between points C and D. Using the Pythagorean theorem, we can find the length of the rectangle.

AB^2 = AE^2 + EB^2

AB^2 = (a/2)^2 + (c - b/2)^2

AB^2 = (55)^2 + (104 - 15)^2

AB^2 = 3025 + 7225

AB = sqrt(10250)

AB ≈ 101.2

Therefore, the length of the rectangle is approximately 101.2 yards.

To find the width of the rectangle, we can use the fact that the distance between points C and D is equal to twice the distance between the center of the ellipse and the minor axis. The center of the ellipse is the midpoint of the major axis, and the distance from the center to the minor axis is 25 yards.

CD = 2 * 25 = 50

Therefore, the width of the rectangle is approximately 50 yards.

In summary, the width of the rectangular playing field is approximately 50 yards if the width is located 15 yards from either vertex.

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Answer:

The radius of the basketball is 20 cm.

Step-by-step explanation:

The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius.

We are given that the volume of the basketball is 11488.2 cm, so we can set up the equation:

Simplifying, we get:

Dividing both sides by (4/3)π, we get:

Taking the cube root of both sides, we get:

Rounding to the nearest whole number, the radius of the basketball is 20 cm.

The 95% confidence interval for the mean score for all bowlers in the league is option (E) (179.9, 192.1).

To construct a 95% confidence interval for the mean score for all bowlers in the league, we can use the formula:

CI = X ± z* (σ/√n)

where X is the sample mean, σ is the population standard deviation (unknown), n is the sample size, and z* is the critical value for the desired confidence level (95% in this case).

Since the sample size is 50, we can assume that the population standard deviation is approximately equal to the sample standard deviation, which is 22. The critical value for a 95% confidence interval with a two-tailed test is 1.96.

Substituting the values, we get:

CI = 186 ± 1.96 (22/√50)

  = 186 ± 6.44

  = (179.56, 192.44)

Therefore, the answer is (B) (177.66, 194.34), which is the closest to the calculated confidence interval.

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a) The median of the dataset is: 11.5

b) The mean of the dataset is: 12.5

c) The mode of the dataset is: 9 and 14

The term average mean is defined as the finding of the average of a sample data. Thus, the average is finding the central value in math, which tells us that mean is finding the central value in statistics.

The numbers arranged in ascending order is:

5, 7, 9, 9, 14, 14, 19, 23

a) The median is defined as the middle term of the distribution when arranged in ascending or descending order. Thus, the median here is:

(9 + 14)/2 = 11.5

b) The mean of the data is expressed as:

(5 + 7 + 9 + 9 + 14 + 14 + 19 + 23)/8

= 12.5

c) The mode is the most frequently occurring term in the data.

In this case, the mode is 9 and 14

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Answer: Therefore, the angle of the terminal side through the point p is 315.0 degrees (to the nearest tenth of a degree).

Step-by-step explanation:

The point p = (-√2/2,√2/2) lies on the unit circle, which is centered at the origin (0,0) and has a radius of 1. To find the angle of the terminal side through this point, we need to use the trigonometric ratios of sine and cosine.

Recall that cosine is the x-coordinate of a point on the unit circle, and sine is the y-coordinate. Therefore, we have:

cos(θ) = -√2/2

sin(θ) = √2/2

We can use the inverse trigonometric functions to solve for θ. Taking the inverse cosine of -√2/2, we get:

θ = cos⁻¹(-√2/2)

Using a calculator, we find that θ is approximately 135.0 degrees.

However, we need to ensure that the angle is between 0 and 360 degrees. Since the point lies in the second quadrant (i.e., x < 0 and y > 0), we need to add 180 degrees to the angle we found. This gives:

θ = 135.0 + 180 = 315.0 degrees

The angle of the terminal side through the point p is 315.0 degrees (to the nearest tenth of a degree).

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The volume of the block of wood is 78 cubic inches.

What is cube?

A cube is a three-dimensional geometric shape that has six equal square faces, 12 equal edges, and eight vertices (corners). All the angles between the faces and edges of a cube are right angles (90 degrees), and all the edges are of equal length. A cube is a special type of rectangular prism where all the sides are equal in length, making it a regular polyhedron.

To find the volume of the block of wood, you need to multiply its length, width, and height together.

Volume = length x width x height

Volume = 6.5 inches x 1.5 inches x 8 inches

Volume = 78 cubic inches

Therefore, the volume of the block of wood is 78 cubic inches.

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The correct option is D, the statement that best describes the expression 4+3x means "4 plus 3 times x".

An expression is a combination of numbers, symbols, and/or variables that represents a mathematical or logical statement. It can be as simple as a single number or letter, or as complex as a series of operations that involve multiple variables and functions. Expressions can be used to represent equations, inequalities, functions, and other mathematical concepts. They can be evaluated to produce a numerical value or a boolean value (true or false) depending on the values of the variables involved.

Expressions are used to represent calculations or logical conditions. They can be used to assign values to variables, manipulate data, and control the flow of a program. expressions are a fundamental concept in both mathematics and computer science, and play a critical role in solving problems and building complex systems.

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A definite integral is defined as the signed area under a function between certain limits (bounds) of integration.

An indefinite integral represents the family of antiderivatives of a function and is also known as its general integral or antiderivative.

An indefinite integral represents the family of antiderivatives of a function and is also known as its general integral or antiderivative. An indefinite integral does not have specific limits of integration; its result includes a constant of integration (usually denoted +C), which accounts for all possible constant shifts within its antiderivative.

A definite integral is defined as the signed area under a function between certain limits (bounds) of integration. The real number that represents its net area between it and x-axis during an interval.

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Willka would need approximately 0.2222 L of paint to cover 1 m² and approximately 2.2222 L of paint to cover 10 m².

Willka can cover 13.5 m² with 3 liters of paint. To find equivalent ratios, we can determine how much paint is needed to cover 1 m² and then use that to find how much paint is required for other areas.

To find the amount of paint needed for 1 m², divide the area covered by the paint used:
1 m² = (13.5 m²)/(3 L) = 4.5 m²/L

Now, we can use this ratio to complete the table:

Area covered (m²) - Paint (L)
13.5               - 3
1                  - (1/4.5) = 0.2222 L (approximately)
10                 - (10/4.5) = 2.2222 L (approximately)

So, the completed table is:

Area covered (m²) - Paint (L)
13.5               - 3
1                  - 0.2222
10                 - 2.2222

Using equivalent ratios, Willka would need approximately 0.2222 L of paint to cover 1 m² and approximately 2.2222 L of paint to cover 10 m².

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Step-by-step explanation:

[tex] - \infty \: and \: 0[/tex]

or

[tex]x \leqslant 1[/tex]

The dimensions of the poster with an area of 392 square inches is equal to 14 inches and 28 inches.

Area of rectangular poster to print = 392 square inches

Let us assume that dimensions of the posters are,

Width of the poster is x inches and the length of the poster is y inches.

Area of the rectangular poster is,

xy = 392

Add 2 inches to the top and bottom margins for a total of 4 inches

And 1 inch to the left and right margins for a total of 2 inches.

Total area of the poster including the margins using the following equation,

Total area = (x + 2) × (y + 4)

Minimize the total area of the poster while still satisfying the area constraint.

Use the first equation to solve for one variable

And substitute it into the second equation,

y = 392/x

Total area = (x + 2) × (392/x + 4)

⇒ Total area = 4x + 392 +784/x + 8

⇒Total area = 4x + 400 +784/x

Minimize the total area, take the derivative of this expression with respect to x and set it equal to 0,

d/dx (4x + 400 +784/x ) = 0

⇒  4 + 0 - 784/x² = 0

⇒ x² = 784 /4

⇒ x = 14

Substituting this value of x back into the equation for y, we get,

y = 392/14

  = 28

Therefore, the dimensions of the poster should be 14 inches by 28 inches.

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Corbin's account will have a higher balance than Victoria's account at the end of 3 years" is true.

To calculate the balance at the end of 3 years, we can use the simple interest formula for Victoria's account and the compound interest formula for Corbin's account.

For Victoria's account:

Simple interest = P * r * t

= 2000 * 0.05 * 3

= $300

Balance after 3 years = P + Simple interest

= 2000 + 300

= $2300

For Corbin's account:

Balance after 3 years = [tex]P * (1 + r)^t[/tex]

= 1800 * (1 + 0.09)³

= $2401.40

Therefore, the statement "Corbin's account will have a higher balance than Victoria's account at the end of 3 years" is true.

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Answer:

29

Step-by-step explanation:

To solve this we have to add corresponding line segments and make them equal to each other.

We can see XZ is broken into XA and AZ.

We can also see that WY is broken into WA and AY.

We are given:

XA=12

AY=14

WA=3+3x

AZ=4x+1

So, we combine and make them equal to each based on their whole line segments:

[tex]12+4x+1=3+3x+14[/tex]

combine like terms

[tex]13+4x=17+3x[/tex]

subtract 13 from both sides

[tex]4x=4+3x[/tex]

subtract 3x from both sides

x=4

We aren't done yet, because the question is asking us to find XZ which is 12+4x+1:

substitute 4 for x

12+4(4)+1

multiply

12+16+1

=29

So, XZ is 29 units.

Hope this helps! :)

Out of 25 students in the class, 40% are male is: Descriptive statistics.

Descriptive statistics is the process of summarizing and organizing data from a sample or population in order to provide an overview of the main characteristics. In this case, the data provided tells us that out of 25 students in the class, 40% are male.

This information is a summary of the gender distribution within this specific class, rather than making any predictions or generalizations about a larger population.

In contrast, inferential statistics is the process of using data from a sample to make predictions or draw conclusions about a larger population. If we were given data about a sample of classes and asked to estimate the proportion of male students in all classes, that would be an example of inferential statistics.

To summarize, the situation you provided, which states that out of 25 students in the class, 40% are male, is an example of descriptive statistics as it only provides a summary of the data for that specific class.

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The translation of the graph from y = x² to y = (x - 2)² +1 is describe by - B.   shift of 2 units left and then shift of 1 unit up.

In geometry, a translation is a transfer that occurs either horizontally to a left or right as well as vertically up or down. It may also consist of a mix of the two.

Given data:

Parent function- y = x²

New function -  y = (x - 2)² +1

Thus, the translation of the graph from y = x² to y = (x - 2)² +1 is describe by - B.   shift of 2 units left and then shift of 1 unit up.

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Complete question:

1. Which of the following most accurately describes the translation of the graph from y = x² to y = (x - 2)² +1?

A.  shift of 2 units right and then shift of 1 unit up.

B.   shift of 2 units left and then shift of 1 unit up.

Answer:2,2

Step-by-step explanation:

The four ordered pairs relating the side length and the corresponding perimeter are (3,20), (4,24), (5,28), and (6,32).

The rule "Add 2" to the side length of a square means that if the original side length is x, the new side length will be x+2.

The rule "Add 8" to the square's perimeter means that if the original perimeter is 4x (since a square has four equal sides), the new perimeter will be 4(x+2), which simplifies to 4x+8.

To find four ordered pairs relating the side length and corresponding perimeter, we can plug in different values for x and use the above formulas to calculate the corresponding perimeters. For example, if we choose x=3, the new side length will be 3+2=5, and the new perimeter will be 4(3+2)=20. So, one ordered pair would be (3,20).

Similarly, if we choose x=4, the new side length will be 4+2=6, and the new perimeter will be 4(4+2)=24. So, another ordered pair would be (4,24).

By choosing different values for x, we can find four ordered pairs that relate the side length and corresponding perimeter. These ordered pairs are (3,20), (4,24), (5,28), and (6,32).

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Dexter has a better chance of winning in the long run with an expected value of $4/3 per game, while Thanh has an expected loss of $1/3 per game.

This is a game of probability that involves calculating expected values. Let's first calculate the probability of drawing a black chip:

Probability of drawing a black chip = (number of black chips) / (total number of chips) = 10 / 15 = 2/3

Similarly, the probability of drawing a white chip can be calculated as:

Probability of drawing a white chip = (number of white chips) / (total number of chips) = 5 / 15 = 1/3

Now, we can calculate the expected value of winning for each player.

For Dexter:

- If he draws a black chip, he will win $7 with probability 2/3

- If he draws a white chip, he will lose $10 with probability 1/3

Expected value of winning for Dexter = (7 x 2/3) + (-10 x 1/3) = 4/3

This means that on average, Dexter will win $4/3 per game.

For Thanh:

- If Dexter draws a black chip, Thanh will lose $7 with probability 2/3

- If Dexter draws a white chip, Thanh will win $10 with probability 1/3

Expected value of winning for Thanh = (-7 x 2/3) + (10 x 1/3) = 1/3

This means that on average, Thanh will win $1/3 per game.

So, based on these calculations, Dexter has a better chance of winning in the long run with an expected value of $4/3 per game, while Thanh has an expected loss of $1/3 per game.

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The area of the wrapping paper would be = 72.5in².

To calculate the area of the wrapping paper, the figure is first divided into two leading to the formation of a triangle and a rectangle.

For the triangle, the formula use to calculate it's area is given as follows;

Area = 1/2 base × height

base = 15-10 = 5 in

height = 9-4 = 5 in

area = 1/2×5 × 5

= 25/2 = 12.5 in²

Area of a rectangle = length× width

width = 4 in

length = 15 in

area = 4×15 = 60in²

Therefore the area of the wrapping paper = 12.5+60 = 72.5in²

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The height of the house by the given data is 4000ft.

We are given that;

Number of BTUs for heating= 52000

Now,

The time from minutes to hours by dividing by 60:

t=606.24​ hr

t≈0.104 hr

Then, we can plug in the values into the heat loss formula and solve for A, which is the surface area of the house:

Q=UAΔTt

52,000=0.25A×50×0.104

A=0.25×50×0.10452,000​ ft2

A≈4000 ft2

Therefore, by the algebra the answer will be 4000 ft.

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Answer:3.75 BTUs/ft

height=18 ft

Step-by-step explanation:

A relation is a set of ordered pairs that define the relationship between two sets. And, a function is a relation in which each element of the domain is connected to a single element of the codomain. The evaluated function is -10.

To find the expression (f(7) - f(3))/ 7 -3, we need to first find f(7) and f(3).

Using the given function F(X) = - 8 - x^2, we can find:

f(7) = -8 - 7^2 = -57

f(3) = -8 - 3^2 = -17

Now, we can substitute these values into the expression:

(f(7) - f(3))/ 7 -3 = (-57 - (-17))/ (7-3) = -40/4 = -10

Therefore, the answer is -10.
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The possible set of values for a, b, and c could be: a=2, b=4, c=5.

Here is a possible set of values for a, b, and c,

- a = 2 (since there are 7 houses with 1-2 bedrooms and 8 houses in total, we know that there must be at least 1 more house with 1-2 bedrooms, which could be house a)
- b = 4 (since the median number of bedrooms is 3 and there are 7+1+1=9 houses total with either 1, 2, or 3 bedrooms, we know that the median house must have either 3 or 4 bedrooms. Since b must be different from a and c, we can assign it to 4)
- c = 5 (since there are only 3 houses left and we need to assign one to each remaining number of bedrooms, we can assign c to 5)

Therefore, a possible set of values for a, b, and c could be: a=2, b=4, c=5.

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The 95% confidence interval for the population proportion of city dwellers who would like to live at the beach is estimated to be between 0.56 and 0.62.

A statistical inference  is a range of values within which the true value of a population parameter, such as the proportion of city dwellers who would like to live at the beach, is likely to fall with a certain level of confidence. In this case, the chamber of commerce for a beach town asked a random sample of city dwellers whether they would like to live at the beach, and based on the survey results, they constructed a 95% confidence interval for the population proportion.

The 95% confidence interval they obtained was (0.56, 0.62). This means that if they were to repeat their survey many times and construct a confidence interval each time, approximately 95% of those intervals would contain the true value of the population proportion.

In practical terms, this means that the chamber of commerce can be reasonably confident that the true proportion of city dwellers who would like to live at the beach falls somewhere between 0.56 and 0.62. It also suggests that the proportion of city dwellers who would like to live at the beach is relatively high, with more than half of the sample expressing a desire to do so. However, it is important to keep in mind that this confidence interval is based on a sample of city dwellers, and the true population proportion could differ from this estimate.

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The time taken by the younger sister to finish the same work by herself is 12 hours.

To solve the problem of how long it will take the younger sister to finish the job by herself, let's use the following terms:

1. Older sister's work rate = O
2. Younger sister's work rate = Y
3. Time taken by the younger sister alone = T

Given that the older sister works 3 times faster than the younger sister, we have:  O = 3Y.

Also, the sisters together can finish the job in 3 hours. Therefore, their combined work rate is equal to completing 1/3 of the job per hour. So,

O+Y=1/3.

Now, we can substitute O with 3Y:  3Y+Y=1/3. Combine the terms and simplify:

4Y=1/3

Now, solve for Y:

Y=1/12

Since Y is the work rate of the younger sister, to find the time it takes for her to complete the job alone (T), we can use the following formula:

Work rate × Time = 1 job.

So, Y × T = 1.

Substitute Y with  1/12:

[tex]\frac{1}{12} \times T=1[/tex]

Now, solve for T:

T = 12.

Therefore, it will take the younger sister 12 hours to finish the job by herself.

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r has negative values when 2 cos θ > 1, and positive values otherwise.

r has negative values when 3θ is in the second or third quadrant, and positive values otherwise.

r has negative values when sin θ > 1/5, and positive values otherwise.

To find the intervals of positive and negative r values, we need to look at the cosine function. Since the cosine function has a maximum value of 1, we have r = 1 - 2 cos θ ≥ -1. Solving for cos θ, we get 2 cos θ ≤ 2, which means that r is negative when 2 cos θ > 1 and positive otherwise.

We can rewrite the polar equation r = 5 sin (3θ) as r = 5(sin θ)(cos^2 θ)(3)^(1/2). This equation is negative when sin θ is negative, which happens in the second and third quadrants. Therefore, r is negative when 3θ is in the second or third quadrant and positive otherwise.

Similarly, we can rewrite the polar equation r = 1 - 5 sin θ as r = 5(cos θ)(sin(π/2 - θ)). This equation is negative when sin(π/2 - θ) is negative, which happens when θ is in the second and third quadrants. Therefore, r is negative when sin θ > 1/5, and positive otherwise.

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1. The price-demand equation for the trumpets is:

   x = 238.18 - 1.09p

2. The manufacturer should set the price of each trumpet at $296.50  to break even        

3. The manufacturer should set the price of each trumpet at $138.63 to maximize profit.

In this problem, the manufacturer has conducted a customer survey and found out that the price of each trumpet affects the demand for it. We need to analyze this data and come up with a price-demand equation that helps the manufacturer set the price of each trumpet to maximize profit.

To start with, we need to assume a linear model, where the demand for the trumpets is directly proportional to the price. We can represent the demand as "x" and the price as "p". Using the data from the survey, we can form two linear equations:

110 = ap + b     (1)

128 = cp + d    (2)

Here, a, b, c, and d are constants that we need to find. We can solve these equations simultaneously to get the values of a, b, c, and d.

Subtracting equation (2) from equation (1), we get:

-18 = (a-c)p + (b-d)   (3)

Dividing both sides of equation (3) by -18, we get:

p = (d-b)/(c-a)          (4)

Using equation (4), we can find the value of p, which is the price at which the demand for trumpets is equal to the values obtained from the survey. Substituting the values from either equation (1) or (2) into equation (4), we get:

p = ($160.74 x 110 - $220.18 x 128)/(-18 x 110 + 18 x 128)

  = $186.46

Therefore, the price-demand equation for the trumpets is:

x = 238.18 - 1.09p

To answer the second question, we need to find the price of each trumpet at which the manufacturer will break even. In other words, the revenue earned from selling the trumpets should be equal to the total cost incurred in making and selling them.

We know that the one-time cost of buying equipment is $3274.78, and each trumpet costs $90.05 to make. Let's represent the break-even price as "[tex]P_{be}[/tex]". Then we can form the following equation:

110[tex]P_{be}[/tex] = 3274.78 + 110 x 90.05

Solving for [tex]P_{be}[/tex], we get:

[tex]P_{be}[/tex]= $296.50

Therefore, the manufacturer should set the price of each trumpet at $296.50 to break even.

To answer the third question, we need to find the price of each trumpet that maximizes the profit for the manufacturer.

The profit is given by the revenue earned minus the total cost incurred. Let's represent the profit as "P" and the price as "p". Then the profit equation becomes:

P = xp - (3274.78 + 90.05x)

To find the price that maximizes profit, we need to take the derivative of the profit equation with respect to p and equate it to zero.

dP/dp = x - 90.05 = 0

Solving for x, we get:

x = 90.05

Substituting this value of x into the price-demand equation, we get:

p = $138.63

Therefore, the manufacturer should set the price of each trumpet at $138.63 to maximize profit.

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