Find the average value of the function over the given interval. (Round your answer to three decimal places.)
f(x) = 6e^x, [−3, 3]
=
Find all values of x in the interval for which the function equals its average value. (Enter your answers as a comma-separated list. Round your answers to three decimal places.) x=

4 times a number can be written as the sum of two equal addends because the value of the addends is equal to the value of the number divided by 4.

To start with, let's define some terms. An addend is a number that is added to another number to form a sum. So, for example, in the equation 2 + 3 = 5, 2 and 3 are addends, and 5 is the sum.

Now, let's consider the statement that 4 times a number can be written as the sum of two equal addends. In mathematical terms, we can write this as:

4x = 2y + 2y

Here, x represents the number we're starting with, and y represents the addends we're trying to find. We're saying that if we multiply x by 4, we can express that product as the sum of two equal addends, each equal to y.

To see why this is true, let's simplify the equation:

4x = 2y + 2y

4x = 4y

We can divide both sides of the equation by 4 to get:

x = y

This tells us that the value of x (the number we started with) is equal to the value of y (each of the addends). So, if we take y and add it to itself, and then multiply the result by 2, we get the same value as if we had multiplied x by 4.

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To conclusively test the hypothesis "among these four people, those above age 30 definitely have feature a", you need to gather age and feature A information for Anne, Bob, Charlie, and Daniel.

You have the ages of Anne (35) and Bob (24) and feature A status of Charlie (has feature A) and Daniel (doesn't have feature A). You need to ask Anne and Bob about their feature A status and Charlie and Daniel about their ages.


1. Ask Anne if she has feature A.
2. Ask Bob if he has feature A.
3. Ask Charlie his age.
4. Ask Daniel his age.

After obtaining the missing information, compare it with the hypothesis to check if it holds true. The hypothesis will be true if both people above 30 years of age have feature A, and the others do not.

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The surface area of the given figure is 215 square centimeters.

We know that the Area of a Rectangle with Length 'L and Width 'W' will be -

Area = L*W

For Rectangle II and IV, length is 5 cm and 3.5 cm.

Area of each II and IV = 5*3.5 = 17.5 square cm.

For Rectangle I and III, length is 3.5 cm and width is 15 cm.

Area of each I and III = 3.5*15 = 52.5 square cm.

Length and Width of Rectangle V are 15 cm and 5 cm respectively.

Area of Rectangle V = 15*5 = 75 square cm.

So total surface area of the figure is = (2*17.5) + (2*52.5) + 75 = 215 square cm.

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The question is incomplete. The complete question will be -

The appropriate null hypothesis for this scenario is option b. The null hypothesis is a statement of no difference or no effect, and it assumes that any observed difference is due to chance. In this case, the null hypothesis would be that the average age of customers who buy BMX bicycles is less than or equal to 47. This can be written as:

H0: µ ≤ 47

where µ is the population mean age of customers who buy BMX bicycles.

The alternative hypothesis, which is the statement that we are trying to prove, would be that the average age of customers who buy BMX bicycles is greater than 47. This can be written as:

Ha: µ > 47

where Ha is the alternative hypothesis.

In summary, the appropriate null hypothesis for this scenario is:

H0: µ ≤ 4

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Radioactive particles 2000  the sample have 80 years from today.

We have the information:

There is 2000 radioactive particles in a sample.

and,  your grandmother measured 4000 radioactive particles in it 80 years ago.

We have to find the samples of radioactive particles present it in 80 years from today.

By the definition of Half line,  The half line of the item is 80 years.

=> 80 years from now, means that another half life is achieved means, 2000 will reduced to half on decay.

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Answer:

40 inches.

Step-by-step explanation:

If you want to measure how much fence you need to enclose a rectangle, you have to add up the lengths of all its sides. That's called the perimeter. The easiest way to find it is to use this handy-dandy formula:

Perimeter = 2 (length + width) units

where length and width are how long the rectangle's sides are.

For example, let's say you have a rectangle that is 12 inches long and 8 inches wide. To find its perimeter, you just plug in those numbers into the formula:

Perimeter = 2 (12 + 8) inches

Perimeter = 2 (20) inches

Perimeter = 40 inches

Ta-da! The perimeter of the rectangle is 40 inches.

The vertical height of pyramid is 3.6 cm.

Given that,

The base area of the pyramid is 5 cm², or 25 cm²

The formula V = (1/3) base area height can be used to determine the volume of a pyramid,

Where the solid's total height equals the sum of the cuboid and pyramid heights. T.

Let's abbreviate the pyramid's vertical height "h"

The height of the cuboid would thus be,

=  (1000g - 104.17gh)/(200g) or (1000g - (8g (25 h)/3))/(8 (5)²).

Therefore,

V = (1/3) 25cm2 h + 5cm (1000g - 104.17gh)/(8g/cm³ * (5cm)²)

It represents the solid's overall volume.

Given that we are aware that the solid has a mass of 1 kg (1000 g),

Adjust density times volume to equal mass:

8g/cm³ * V = 1000g

When we solve for V,

⇒V = 125cm3.

Solve for h by adding V to the previous equation and getting the following result:

h = (3(125cm³ 8g/cm³ - 5cm 1000g)/(25cm² 8g/cm³)

which can be expressed as h = 3.6 cm.

As a result, the pyramid has a 3.6 cm vertical height.

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Option c is correct.

The gradual shifting or movement of a time series to relatively higher or lower values over a longer period of time is called a trend. Therefore, option c. "a trend" is the correct answer.

Periodicity refers to the tendency of a time series to exhibit patterns that repeat at regular intervals, such as daily, weekly, or seasonal cycles.

Regression refers to the statistical method used to analyze the relationship between two or more variables.

A cycle refers to a repeating pattern of fluctuations in a time series that are not necessarily regular or periodic.

Hence, option c is correct.

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Using percentages the number of people is who watch the fireworks is 647.

A percentage is a ratio out of hundred.

Since we have 492 people who choose to watch the fireworks from the castle and this is 76% of the people who watch the fireworks at all, how many people watch the fireworks altogether?

Now, percentage = number/total × 100 %

Now, we require the total number. Making the total number subject of the formula, we have that

total = number × 100 %/percentage

Since

So, substituting the values of the variables into the equation, we have that

total = number × 100 %/percentage

total = 492 × 100 %/76%

= 6.474 × 100 %

= 647.4

≅ 647

So, the number of people is 647.

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The negative slope of the linear equation indicates that the number of violent crimes has decreased over the years. The intercept of the line at (0, 1,773,900) represents the number of violent crimes in 1993.

The scatter plot shows a negative linear relationship between the number of violent crimes and the years since 1993. As the number of years increases, the number of violent crimes decreases. The linear equation that best models this relationship is y = -31,256x + 1,773,900. This means that for every one-year increase since 1993, the number of violent crimes decreases by 31,256.

The y-intercept of 1,773,900 represents the number of violent crimes in 1993, the starting year of the data set. The slope of -31,256 indicates that the decrease in violent crimes over time is quite significant.

It is important to note that while the linear equation provides a good model for this data set, it does not necessarily mean that it can accurately predict the number of violent crimes in future years. Other factors not accounted for in the data set may influence the number of violent crimes in the future.

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To identify the correct scatter plot for variables y1 and x1, you need to analyze the data and look for a clear pattern in the relationship between the two variables.

What is scatter plots?

A scatter plot is a type of data visualization that displays the relationship between two variables.

In a scatter plot, each point represents a pair of values, one for each of two variables. The horizontal axis represents the values of one variable (in this case, x1), and the vertical axis represents the values of the other variable (y1).

The scatter plot can show the relationship between the two variables. If there is a positive correlation, the points will tend to cluster in a line that slopes up and to the right. If there is a negative correlation, the points will cluster in a line that slopes down and to the right. If there is no correlation, the points will be scattered randomly.

To determine which scatter plot is correct, you need to examine the data and see which plot matches the pattern of the data. If there is a clear positive or negative correlation, the correct plot will show a line sloping up or down. If there is no correlation, the correct plot will show a scatter of points with no clear pattern.

In summary, to identify the correct scatter plot for variables y1 and x1, you need to analyze the data and look for a clear pattern in the relationship between the two variables.

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The length of OP is √(3) and the Pythagorean Theorem is used to find it.

A theorem is a statement that has been proven to be true based on rigorous mathematical reasoning and evidence. It is a fundamental concept in mathematics and plays a central role in building the structure of mathematical knowledge. Theorems are often used as a basis for further mathematical analysis and the development of new theories and applications.

The length of OP can be found using the Pythagorean Theorem.

By the Pythagorean Theorem, we know that:

MN² + NO² = MO²

Since angle MNP is congruent to angle ONP, we know that triangles MNP and ONP are similar. Therefore, we can set up a proportion:

MN/NP = ON/NP

Simplifying this proportion, we get:

MN = ON

Substituting this into the equation for MO², we get:

MN²  + NO²  = MO²

2(MN² ) = MO²

Since MP is given as 2, we can use the Pythagorean Theorem in triangle MOP to find OP:

MO² = MP² + OP²

2(MN²) = MP² + OP²

2(MN²) - MP² = OP²

2(2²) - 1² = OP²

3 = OP²

OP = √3

Therefore,

The length of OP is √(3) and the Pythagorean Theorem is used to find it.

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To order the sides of ΔABC from shortest to largest, we need to measure each side and compare them. The shortest side of the triangle will be the one with the smallest length, while the largest side will be the one with the greatest length. The middle side will be the one that is neither the shortest nor the largest.

To measure the sides of ΔABC, we can use a ruler or a measuring tape. Once we have measured each side, we can compare them to determine which is the shortest, which is the middle, and which is the largest.

For example, if the lengths of the sides of ΔABC are 3 cm, 4 cm, and 5 cm, then the shortest side is 3 cm, the middle side is 4 cm, and the largest side is 5 cm.

In summary, to order the sides of ΔABC from shortest to largest, we need to measure each side and compare them. The side with the smallest length will be the shortest side, the side with the greatest length will be the largest side, and the remaining side will be the middle side.

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Answer: $2.09

Step-by-step explanation: 0.76/2=0.38 so each pound of apples costs $0.38. Since we want to find how much 5.5 pounds cost, we multiply by 5.5. $0.38*5.5=$2.09 so our answer is $2.09

Answer:

[tex] \frac{2}{7} = \frac{4}{x} [/tex]

[tex]x = 14[/tex]

Approximately 9.5% of the racers did not complete the marathon.

The total of those who quit and those who were disqualified represents the number of runners who did not finish the marathon

Number who did not complete = 12 + 8 = 20

To find the percentage of racers who did not complete the marathon, we need to divide this number by the total number of racers who started the marathon (210) and multiply by 100

Percentage who did not complete is

= (20 / 210) x 100

= 9.523%

Rounding this to the nearest tenth of a percent gives us a final answer of 9.5%. Therefore, approximately 9.5% of the racers did not complete the marathon.

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The greatest common divisor of each of these pairs of integers are

a) 1    b)  1    c) 12   d)  1     e)  3   g)  1     h)  2    i) 1  

f) 0 is not a positive integer, so it doesn't have a prime factorization.

The greatest common divisor (GCD), also known as the highest common factor (HCF), of two or more integers is the largest positive integer that divides each of the integers without a remainder.

To find the GCD we need to write each number as a product of prime factorization and then we need to find out the common factors which is greatest among them.

a) 10 = 2 × 5,

11 is prime

GCD(10, 11) = 1 (no common factors)

b) 21 = 3 × 7,

44 = 2 × 2 × 11

GCD(21, 44) = 1 (no common factors)

c) 36 = 2 × 2× 3 × 3

48 = 2 × 2 × 2 × 2 × 3

GCD(36, 48) = 2 × 2 × 3 = 12

d) 34 = 2  × 17,

55 = 5 × 11

GCD(34, 55) = 1 (no common factors)

e) 117 = 3 × 3 × 13,

213 = 3 × 71

GCD(117, 213) = 3

f) 0 is not a positive integer, so it doesn't have a prime factorization.

g) 123 = 3 × 41, 2347 is prime

GCD(123, 2347) = 1 (no common factors)

h) 3454 = 2 × 1727,

4666 = 2 × 2333

GCD(3454, 4666) = 2

i) 9999 = 3 × 3 × 11 × 101,

11111 = 41 × 271

GCD(9999, 11111) = 1 (no common factors)

Therefore,

The greatest common divisor of each of these pairs of integers are

a) 1    b)  1    c) 12   d)  1     e)  3   g)  1     h)  2    i) 1  

f) 0 is not a positive integer, so it doesn't have a prime factorization.

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For a sample of adults from country A, related to their unconfident that the food they eat in country A is safe, the point estimate of population proportions p and q are equals 0.202 and 0.798 respectively.

One sample proportion test is conducted to check whether the population proportion (P) shows a significant difference from the hypothesized value (p)or not. Sample proportion

[tex](\hat p)[/tex] can be defined as the ratio of number of successes in the sample and the size of the sample. We have a sample survey of 1562 adults from country A, in which, 316 were not confident that the food they eat in country A is safe.

So, Sample size (n)= 1562

The number of successes (x) = 316

Let's consider X be the number of adults that were not confident with the food that they eat in country A is safe. The point estimate of the population proportion (p) is written as below,

[tex]\hat p=\frac{x}{n}[/tex] [tex] = \frac{316}{1562}[/tex]

= 0.2023047

≈ 0.202

Therefore, the point estimate of the population proportion is 0.202. The point estimate for q is [tex]\hat q = 1 - \hat p[/tex] = 1 -0.202

= 0.797695 ≈ 0.798

Therefore, the point estimate for q is 0.798.

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For a sample of adults from country A, related to their unconfident that the food they eat in country A is safe, the point estimate of population proportions p and q are equals 0.202 and 0.798 respectively.

One sample proportion test is conducted to check whether the population proportion (P) shows a significant difference from the hypothesized value (p)or not. Sample proportion

can be defined as the ratio of number of successes in the sample and the size of the sample. We have a sample survey of 1562 adults from country A, in which, 316 were not confident that the food they eat in country A is safe.

So, Sample size (n)= 1562

The number of successes (x) = 316

Let's consider X be the number of adults that were not confident with the food that they eat in country A is safe. The point estimate of the population proportion (p) is written as below,

= 0.2023047

≈ 0.202

Therefore, the point estimate of the population proportion is 0.202. The point estimate for q is  = 1 -0.202

= 0.797695 ≈ 0.798

Therefore, the point estimate for q is 0.798.

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The function that relates the distance traveled, d, to the time, t is d(t) = 120t

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

Speed = s

Time = t0

The distance is calculated as

d = s * t

So, we have

d = st

Given that the train travels 360 kilometers in 3 hours.

We have

d = 360/3 * t

So, we have

d(t) = 120t

Hence, the function is d(t) = 120t and the graph is added as an attachment

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Complete question

A train leaves the station at time t0. Traveling at a constant​ speed, the train travels kilometers in hours.

Traveling at a constant speed, the train travels 360 kilometers in 3 hours.

(a) Write a function that relates the distance traveled, d, to the time, t.

(b). Graph it

Answer:

3:1

Step-by-step explanation:

If the height of the card is 2.5 cm., the height of the actual card that Monica is purchasing is 3.75 inches.

Since the scale of the drawing is 1 cm: 1.5 in, we can use this proportion to find the actual height of the card.

Let h be the height of the actual card in inches. Then, we can set up the proportion:

1 cm / 1.5 in = 2.5 cm / h

Cross-multiplying, we get:

1 cm * h = 1.5 in * 2.5 cm

Simplifying, we get:

h = (1.5 in * 2.5 cm) / 1 cm

h = 3.75 inches

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Complete question is:

Monica is looking at a catalog to find thank you cards. In the scale drawing of the one she likes, 1 centimeter represents 1.5 inches. The height of the card in the scale drawing is shown below.

What is the height of the actual card she is purchasing?

12 boys scored 8 points in the exam.

It is given that the table shows the midterm exam results for boys and girls in a class.

As we can see clearly in the table in the first column we have the number of boys, in the second column we have their exam score and in the last column, we have number of the girls.

As we can see 12 is encircled in the first column and in the same row and second column there is 8 present which means 12 boys scored 8 points in the exam.

Therefore, 12 boys scored 8 points in the exam.

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Given question is incomplete, the complete question is given below:

the midterm exam scores obtained by boys and girls in a class are listed in the table below:

what does the circled section represent?

The present value of the ordinary annuity is 29569 and the periodic payment R is 669.

What is the present value?

In economics and finance, present value, also known as present discounted value, is the value of an expected income stream determined as of the date of valuation.

Here, we have

Given: Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of 9%/year compounded monthly. If the future value of the annuity after 10 years is $50,000.

We have to find the present value of the ordinary annuity.

We use the ordinary annuity formula here

A = P{(1+r)ⁿ-1}/r

P = 1200

n = 16

r = 5.5% = 0.055

A = 1200{(1+0.055)¹⁶-1}/0.055

A = 29569

Future value of annuity = 50000

Interest rate per period = interest rate per annum/no.of payment per annum

= 9%/12 = 0.75%

Number of period = 120

Each deposit = FVA/{(1+r)ⁿ-1}/r

=  50000/{(1+0.75%)¹²⁰-1}/0.75%

= 258

Now, P = 18,000, r = 10, x = 6, n = 6

Periodic payment (R) = P(r/n)×(1+r/n)ⁿˣ/(1+r/n)ⁿˣ-1

= 18000(0.167)(1+0.167)³⁶/(1+0.167)³⁶-1

R = 669

Hence, the present value of the ordinary annuity is 29569 and the periodic payment R is 669.

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The probability that a junior is taking physics, given they are in Algebra II, is approximately 0.3846 or 38.46%.

1. First, let's find the number of juniors taking only Algebra II and not physics. We do this by subtracting the number of juniors taking both courses from the total number of juniors taking Algebra II:
  468 (Algebra II) - 180 (both courses) = 288 (only Algebra II)

2. Now, we have two groups of juniors enrolled in Algebra II:
  - 288 juniors taking only Algebra II
  - 180 juniors taking both Algebra II and physics

3. Since we want to find the probability that a junior is taking physics, given they are in Algebra II, we'll focus on the group taking both courses.

4. To calculate the probability, divide the number of juniors taking both courses by the total number of juniors taking Algebra II:
  Probability = (number of juniors taking both courses) / (total number of juniors in Algebra II)
  Probability = 180 / (288 + 180)
  Probability = 180 / 468
  Probability ≈ 0.3846 or 38.46%

So, if one of the 650 juniors was picked at random and we know they are in Algebra II, the probability that they are also taking physics is approximately 38.46%.

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The value of the double integral of x² over the region bounded by the ellipse 9x² + 4y² = 36 using the given transformation is π/4.

First, let's define the given transformation. We are given that x=2u and y=3v, which means that we are transforming our original x-y plane into a u-v plane.

We know that the region is bounded by the ellipse 9x² + 4y² = 36. Substituting the given transformations into this equation, we get:

9(2u)² + 4(3v)² = 36 36u² + 36v² = 36 u² + v² = 1

Now, we can use the transformation formula to evaluate the double integral of x² over this region. The transformation formula tells us that:

∬R f(x,y) dA = ∬S f(u,v) |J| dA

where R is the region in the x-y plane, S is the region in the u-v plane, f(x,y) is the integrand in the x-y plane, f(u,v) is the integrand in the u-v plane, |J| is the Jacobian determinant of the transformation (which we will find shortly), and dA is the area element in the respective planes.

In our case, f(x,y) = x² and f(u,v) = (2u)² = 4u². The area element dA in the x-y plane is dx dy, while in the u-v plane it is |6| du dv (since |J| = |d(x,y)/d(u,v)| = |6|). Thus, our integral becomes:

∬R x² dA = ∬S (4u²) (|6|) du dv

Integrating this expression over the limits -1 to 1 for both u and v, we get:

∬S (4u²) (|6|) du dv = 48 ∫∫S u² du dv

where S is the unit circle in the u-v plane. To evaluate the double integral ∫∫S u² du dv, we can use polar coordinates, where u = r cos θ and v = r sin θ. Then, the integral becomes:

[tex]\int _{\theta =0} ^{2\pi} \int_{r=0} ^{ 1}[/tex] (r² cos² θ) r dr dθ

Evaluating this integral using standard techniques, we get:

[tex]\int _{\theta =0} ^{2\pi} \int_{r=0} ^{ 1}[/tex] (r³ cos² θ) dr dθ

[tex]\int _{\theta =0} ^{2\pi} \int_{r=0} ^{ 1}[/tex] (cos² θ)/4 dθ

Simplifying this expression, we get:

[tex]\int _{\theta =0} ^{2\pi}[/tex] (cos² θ)/4 dθ = ∫θ=0 to 2π (1 + cos 2θ)/8 dθ

Using the fact that ∫ cos 2θ dθ = 0 and ∫ dθ = 2π, we get:

[tex]\int _{\theta =0} ^{2\pi}[/tex] (cos² θ)/4 dθ = (1/8) [tex]\int _{\theta =0} ^{2\pi}[/tex]dθ + (1/8) [tex]\int _{\theta =0} ^{2\pi}[/tex] cos 2θ dθ

Simplifying further, we get:

[tex]\int _{\theta =0} ^{2\pi}[/tex](cos² θ)/4 dθ = π/4

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The situation of popcorn scooping at Pineville County Fair can be modeled as a linear relationship.

A linear relationship is a type of relationship between two variables that can be represented by a straight line. In this case, the two variables are the amount of popcorn in the container and the number of bags filled. As Vivian scoops popcorn from the container, the amount of popcorn decreases, and the number of bags filled increases. This creates a linear relationship between the two variables.

To model this relationship, we can use the equation of a straight line, which is y = mx + b, where y represents the dependent variable (in this case, the number of bags filled), x represents the independent variable (the amount of popcorn in the container), m represents the slope of the line, and b represents the y-intercept.

In this situation, the slope represents the rate at which Vivian is filling bags with popcorn, and the y-intercept represents the initial number of bags filled when the container is full. By analyzing the data, we can estimate the slope and y-intercept and use the linear model to make predictions about the amount of popcorn remaining in the container and the number of bags that can still be filled.

Overall, the linear relationship provides a useful tool for understanding and managing the popcorn scooping process at the Pineville County Fair.

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The function F = ∇f is obtained by taking the partial derivatives is (3 + 2xy) i + (x² − 3y²) j where C is a constant. The line integral ∫ F. Dr is evaluated by plugging in the parametric equations of the given curve C into the function F is 38.36.

To find a function F = ∇f, we need to find the gradient of f, which is a vector-valued function that has F as its gradient. Therefore, we need to find the partial derivatives of f with respect to x and y.

∂f/∂x = 3 + 2xy

∂f/∂y = x² − 3y²

Integrating the first equation with respect to x, we get

f(x, y) = 3x + x²y + g(y)

where g(y) is an arbitrary function of y that depends only on y.

Taking the partial derivative of f with respect to y, we get:

∂f/∂y = x² + g'(y)

Comparing this to the second equation, we get

g'(y) = −3y²

Integrating g'(y) with respect to y, we get

g(y) = −y³ + C

where C is a constant of integration.

Substituting this into our expression for f(x, y), we get:

f(x, y) = 3x + x²y − y³ + C

Therefore, F = ∇f is given by

F = ∇f = (3 + 2xy) i + (x² − 3y²) j

To evaluate the line integral ∫ F·dr, we need to first parameterize the curve C using t as the parameter.

[tex]r(t) = e^t sin(t) i + e^t cos(t)[/tex] j, 0 ≤ t ≤ π

The velocity vector of the curve is given by

r'(t) = [tex]e^t[/tex] (cos(t) i − sin(t) j)

Using the formula for the line integral, we get

∫ F·dr = ∫ F(r(t)) · r'(t) dt

Substituting in the expressions for F and r'(t), we get

∫ [(3 + 2xy) i + (x² − 3y²) j] · [[tex]e^t[/tex] (cos(t) i − sin(t) j)] dt

=[tex]\int\limits [3e^t cos(t) + 2e^{2t} sin(t) cos(t)] dt - \int\limits [3e^{2t} sin^{2t} - e^{2t} cos^{2t}] dt[/tex]

Evaluating these integrals, we get:

= [tex][3e^t sin(t) + 2e^{2t} sin^2(t)] + [3/2 e^{2t} sin^2(t) - 1/2 e^{2t} cos^2(t)][/tex] |_0^π

= [tex]3e^{\pi} - 3 + 7/2 e^{2\pi} - 7/2[/tex]

Therefore, the value of the line integral ∫ F·dr is approximately 38.36.

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The bus company will have maximum revenue when there are 120 people on the charter bus.

What is the quadratic equation?

The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.

The revenue R for the bus company as a function of n can be expressed as follows:

For n < 80: R = 0 (since the company requires at least 80 people to operate a charter bus)

For n a 80: R = n(8 - 0.05(n - 80))

Simplifying the expression:

R = 8n - 0.05n² + 4n

R = -0.05n² + 12n

Therefore, the revenue of the bus company is a quadratic function of the number of people, with a negative coefficient for the quadratic term, indicating that the function has a maximum value.

This maximum value occurs at the vertex of the parabola, which can be found using the formula -b/2a, where a = -0.05 and b = 12:

n = -b/2a = -12/(2*(-0.05)) = 120

Thus, the bus company will have maximum revenue when there are 120 people on the charter bus.

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