Find all the first and second order partial derivatives of f(x,y)=−10sin(2x y)−3cos(x−y)

To evaluate this line integral, we will use Green's theorem, which states that for a closed curve C, oriented counterclockwise, and a region R bounded by C, the line integral of the vector field F along C is equal to the double integral of the curl of F over R:

∮c F ⋅ dr = ∬R (curl F) ⋅ dA
In this case, our vector field F is:
F = (5y - e×sin(x)) i + (8x - sin(y³)) j
And the curl of F is:
curl F = ∂(8x - sin(y³))/∂x - ∂(5y - e*sin(x))/∂y = 5e×cos(x) + 3y²×cos(y³)
Now, we need to find the region R bounded by C, which is the circle of radius 5 centered at (3,-5). This is simply the disk with center (3,-5) and radius 5.
Using polar coordinates, we can write the double integral as:
∬R (curl F) ⋅ dA = ∫θ=0..2π ∫r=0..5 (5e×cos(θ) + 3r²×cos(r³×sin(θ)³)) r dr dθ
Evaluating this integral, we get:
∬R (curl F) ⋅ dA = π×(625e + 375)
Therefore, by Green's theorem:
∮c F ⋅ dr = π×(625e + 375)
Substituting F and evaluating the integral, we get:
∮c (5y - e×sin(x)) dx + (8x - sin(y³)) dy = π×(625e + 375)
This is the value of the line integral.

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The graph of the solution set for the system of inequalities is on the image at the end.

Here we have a system of inequalities, it can be written as:

y - 5 ≥ -(x + 1)

y ≥ 3x - 2

To graph this, we can write both inequalities in linear form:

y  ≥ -(x + 1) + 5

y ≥ 3x - 2

We can see that in both cases we have the symbol "≥", so we need to graph both of these lines with solid lines, and then shade the region above of these lines, then the graph of the system is the one you can see at the end.

The region where the two shades intercept (dark blue one) is the set of solutions.

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Answer:

Step-by-step explanation:

folding in thirds the value is 1/3 of 90, therefore 90 : 3 = 30°

[tex]\textit{area of a circle}\\\\ A=\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=35 \end{cases}\implies A=\pi 35^2\implies A=\cfrac{22}{7}\cdot 35^2\implies A=3850[/tex]

Answer:

47/100

Step-by-step explanation:

2/10*10/10= 20/100

67/100-20/100= 47/100.

Answer:

(-6,9)

Step-by-step explanation:

We can use the midpoint formula to find the coordinates of C. The midpoint formula states that the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is:

((x1 + x2)/2, (y1 + y2)/2)

In this case, we know that the midpoint of CD is M(-1, 7) and one endpoint of CD is D(4, 5). Let C = (x, y) be the coordinates of the other endpoint of CD. Then we can use the midpoint formula to set up two equations:

(x + 4)/2 = -1

(y + 5)/2 = 7

Solving for x and y, we get:

x + 4 = -2

y + 5 = 14

x = -6

y = 9

Therefore, the coordinates of C are (-6, 9).

Since the p-value (0.1021) is greater than the significance level (0.05), we fail to reject the null hypothesis.

The null hypothesis is a type of hypothesis that describes the population parameter and is used to examine the validity of experimental results.

To test whether the manufacturer's defective rate is less than 5%, we can use a one-tailed hypothesis test with a significance level of 0.05.

Let p be the true proportion of defective batteries in the population. The null hypothesis is that p >= 0.05, and the alternative hypothesis is that p < 0.05.

We can use the normal approximation to the binomial distribution, since n = 100 and p₀ = 0.05 > 10. The test statistic is:

z = (x - np₀) / √(np₀(1-p₀))

where x is the number of defective batteries in the sample, n is the sample size, and p₀ is the hypothesized proportion under the null hypothesis.

Plugging in the values, we get:

z = (4 - 100*0.05) / √(100*0.05*0.95) = -1.2649

The p-value for this test is the probability of getting a test statistic as extreme as -1.2649 or less, assuming the null hypothesis is true. From a standard normal distribution table, the probability of getting a z-score of -1.2649 or less is 0.1021.

Since the p-value (0.1021) is greater than the significance level (0.05), we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that the manufacturer is not meeting their quality control standards.

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Answer:

(a) - [tex]V_{tank}\approx 2010.6 \ yd^3[/tex]

(b) - [tex]Total= 1005 \ \text{fish}[/tex]

Step-by-step explanation:

Given the cylindrical fish tank with a diameter of 16 yards and a height of 10 yards. We are asked to answer the following.

(a) - How much water can the tank hold, in other words what is the tank's volume?

(b) - Given that a specific kind of fish requires 2 yd³ of water per fish, find the total amount of fish that can occupy the tank.

Part (a):

Given:

[tex]d=16 \ yd\\h=10 \ yd[/tex]

Find:

[tex]V_{tank}= \ ?? \ yd^3[/tex]

For part (a) the question is essentially asking what the volume of the tank is, the tank is a cylinder. We can use the following formula to find the volume of a cylinder.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Volume of a Cylinder:}}\\\\V=\pi r^2h\end{array}\right}[/tex]

(1) Find the radius of the tank given that the diameter is 10 yards

[tex]r=\frac{d}{2}\\\\ \Longrightarrow r=\frac{(16 \ yd)}{2} \\\\\therefore \boxed{r=8 \ yd}[/tex]

(2) - Compute the volume of the tank using the formula from above and the value of the radius we just found.

[tex]V_{tank}=\pi r^2h\\\\\Longrightarrow V_{tank}=\pi (8 \ yd)^2(10 \ yd)\\\\ \Longrightarrow V_{tank}=\pi (64 \ yd^2)(10 \ yd)\\\\\Longrightarrow V_{tank}=640\pi\\\\\therefore \boxed{\boxed{V_{tank}\approx 2010.6 \ yd^3}}[/tex]

Thus, the volume of the tank is found.

Part (b):

Given:

[tex]V_{tank}=2010.6 \ yd^3\\V_{1 \ fish}= 2 \ yd^3[/tex]

Find:

[tex]\text{Total}= \ ?? \ \text{fish}[/tex]

To find the total amount of fish that can fit in the tank, simply divide the total volume of the tank by the volume one fish requires.

[tex]Total=\frac{V_{tank}}{V_{1 \ fish}} \\\\\Longrightarrow Total=\frac{2010.6 \ yd^3}{2 \ yd^3}\\\\\therefore \boxed{\boxed{Total= 1005 \ fish}}[/tex]

Thus, the tank can hold 1005 fish.

The total surface area of the cylinder 572 cm²is and it's volume is 924cm³

A cylinder is a three-dimensional shape consisting of two parallel circular bases, joined by a curved surface.

The total surface area of a cylinder is expressed as ;

SA = 2πr( r+h)

r is the radius and h is the height.

radius = 7cm

height = 6cm

SA = 2 × 3.14 × 7( 7+6)

= 44 × 13

= 572 cm²

The volume of a cylinder is expressed as

V = πr²h

= 3.14 × 7² × 6

= 154 × 6

= 924 cm²

Therefore the surface area and volume of the cylinder are 572cm² and 924cm²

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The block of metal with a 4 cubic cm volume and density of 19.58 g/cm³ has a mass of 78.32 g.

Density refers to a dimension that is it is the mass per unit volume. The density of an object is not affected by the volume or the mass of the object.

It is a unitless quantity. Density is affected by the nature of the substance as well as the temperature. It is expressed as below:

ρ = m/v

where ρ is the density

m is the mass

v is the volume

Given:

ρ = 19.58 g/cm³

v = 4 cm³

19.85 = m / 4

m = 19.58 * 4

= 78.32 g

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To solve the initial value problem, we first need to find the general solution of the given differential equation, which is a second-order linear differential equation with constant coefficients. The characteristic equation is r^2 - 8x^3 = 0, which has roots r = ±(2x)^(3/2). Therefore, the general solution of the differential equation is v(x) = c1(2x)^(3/2) + c2(-2x)^(3/2).

To find the values of the constants c1 and c2, we use the initial conditions v(8) = -48 and x0 = 2. Substituting x = 8 and v(x) = -48 into the general solution, we get -48 = c1(16)^(3/2) + c2(-16)^(3/2). Simplifying, we get c1 - c2 = -3.

To find c1 and c2, we differentiate the general solution with respect to x and substitute x = 2 and v'(2) = -8 into the resulting expression. We get v'(x) = 3x^(1/2)c1 - 3x^(1/2)c2, and substituting x = 2 and v'(2) = -8, we get -8 = 6c1 - 6c2. Simplifying, we get c1 + c2 = 4/3.

Solving the system of equations c1 - c2 = -3 and c1 + c2 = 4/3, we get c1 = 5/6 and c2 = -17/6. Therefore, the solution of the initial value problem is v(x) = (5/6)(2x)^(3/2) - (17/6)(-2x)^(3/2).

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The selected all the correct answers are  table 5 and table 6.

We are given that;

The tables of 4 options

Now,

If the first differences are not constant, but the second differences are constant, then the relationship is quadratic.

we can check each table and see which ones have constant second differences. Here are the results:

Table First Differences Second Differences Quadratic?

1 2, 2, 2 0, 0 No

2 -2, -4, -8 -2, -4 No

3 1, 1, 1 0, 0 No

4 -2, 0, 0 2, 0 No

5 1, 2, 4 1, 2 Yes

6 -8, 0, 8 8, 8 Yes

Therefore, by quadratic equation the answer will be table 5 and table 6.

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The probability that a single marble chosen at random from the jar is yellow or green is 7/10, or 0.7.

The probability that a single marble chosen at random from the jar is yellow or green can be found by adding the probability of choosing a yellow marble to the probability of choosing a green marble.

There are a total of 1 + 3 + 2 + 4 = 10 marbles in the jar. The probability of choosing a yellow marble is 4/10 = 2/5, and the probability of choosing a green marble is 3/10. Therefore, the probability of choosing a yellow or green marble is:

P(yellow or green) = P(yellow) + P(green) = 4/10 + 3/10 = 7/10

So the probability that a single marble chosen at random from the jar is yellow or green is 7/10, or 0.7.

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If a boat leaves port and follows a course of n77°e at 9 knots for 3 hr and 20 min. then, the boat changes to a new course ofs27°east 12knots for 4hr then the boat is approximately 67.9 nautical miles from the port.

To solve the problem, we can use the law of cosines to find the distance from the boat to the port.

Let A be the position of the boat after the first leg of the trip and B be the position after the second leg. Let x be the distance from A to B. Then, we have:

cos(77°) = (distance from port to A) / x

cos(27°) = (distance from port to B) / x

We can solve for the distances from the port to A and B using the given course and speed information:

(distance from port to A) = 9 knots x 3.33 hours = 29.97 nautical miles

(distance from port to B) = 12 knots x 4 hours = 48 nautical miles

Substituting into the law of cosines, we get:

x^2 = (29.97)^2 + (48)^2 - 2(29.97)(48)cos(130°)

Solving for x gives:

x ≈ 67.9 nautical miles

Therefore, the boat is approximately 67.9 nautical miles from the port.

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Answer: 7/12

Step-by-step explanation:

1. The remainder is 7/10.

2. He pays 1/6 of the remaining amount as rent; this is the rent he pays.

3. His usage total is 3/10 + 7/60 = 25/60, or 5/12.

The process described above satisfies the Markov property because the future state of the system only depends on the current state, and not on any previous states. The transition probability matrix is P =[tex]\left[\begin{array}{cccc}0&1&0&0\\1/2&0&1/2&0\\0&1/2&0&1/2\\0&0&1&0\end{array}\right][/tex]  

In other words, given the current state, the probability distribution of the next state depends only on the current state and not on any previous history.

Let Xn denote the state of the system after the nth step, where Xn = i means that the first urn contains i white balls and 3-i black balls. Since each step involves drawing one ball from each urn and switching them, the number of white balls in the first urn can only change by one after each step. Thus, the only possible transitions are:

If Xn = 0, then the next state Xn+1 can only be 1, with probability 1, since there is only one white ball in the first urn.

If Xn = 1, then the next state Xn+1 can be either 0 or 2, each with probability 1/2, since there are two white balls in the first urn and each ball is equally likely to be drawn.

If Xn = 2, then the next state Xn+1 can be either 1 or 3, each with probability 1/2, since there are two black balls in the first urn and each ball is equally likely to be drawn.

If Xn = 3, then the next state Xn+1 can only be 2, with probability 1, since there is only one black ball in the first urn.

Therefore, the transition probability matrix is:

P =[tex]\left[\begin{array}{cccc}0&1&0&0\\1/2&0&1/2&0\\0&1/2&0&1/2\\0&0&1&0\end{array}\right][/tex]  

where Pij is the probability of transitioning from state i to state j. For example, P12 = 1/2 means that the probability of transitioning from state 1 (one white ball in the first urn) to state 2 (two white balls in the first urn) is 1/2.

Correct Question :

Three white balls and three black balls are distributed in two urns in such a way that each contain three balls. We say that the system is in state i, i=0,1,2,3, if the first urn contains i white balls. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. Let Xn denote the state of the system after the nth step. Explain why Xn = 0, 1, 2, ......... is a Markov chain and calculate its transition probability matrix.

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65% is the probability that a randomly selected student is a senior or a student who drives to school.

First, let's calculate the probability of a student being a senior. We can do this by dividing the number of seniors by the total number of students:

Probability of being a senior = Number of seniors / Total number of students

Probability of being a senior = 25 / 100

Probability of being a senior = 0.25

Next, let's calculate the probability of a student driving to school. Again, we divide the number of students who drive by the total number of students:

Probability of driving to school = Number of students who drive / Total number of students

Probability of driving to school = (2 + 13 + 25) / 100

Probability of driving to school = 40 / 100

Probability of driving to school = 0.4

To find the probability of a student being a senior or driving to school (the union of these two events), we add their probabilities:

P(Senior U Drive to school) = P(Senior) + P(Drive to school)

P(Senior U Drive to school) = 0.25 + 0.4

P(Senior U Drive to school) = 0.65

Therefore, the probability that a randomly selected student is a senior or a student who drives to school is 0.65, or 65%.

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Answer:

Since the cost of producing one cell phone is $46, the cost of producing 10 cell phones is 10 times the cost of producing one, which is:

10 x $46 = $460Similarly, the cost of producing 20 cell phones is 20 times the cost of producing one, which is:

20 x $46 = $920Therefore, the cost to produce y cell phones is:

y x $46 = $46yTherefore, the correct answer is A. 46y.

The value of MD (mean difference) for the given repeated-measures research study data can be calculated by subtracting the scores of the first condition from the scores of the second condition for each subject, then calculating the mean of those differences.

The mean difference (MD) for the given data can be calculated as follows:

MD = ((15-10) + (8-4) + (5-7) + (11-6))/4

MD = (5 + 4 - 2 + 5)/4

MD = 3

Therefore, the value of MD for the given data is 3.

In a repeated-measures research study, the same group of subjects is measured on the same variable multiple times. MD represents the average difference between the scores of the same group of subjects on the same variable across two different conditions or time points.

To calculate MD, we need to subtract the scores of the first condition from the scores of the second condition for each subject, and then calculate the mean of those differences. In this case, the mean difference is 3, indicating that there is an average increase of 3 units from the first to the second condition. MD is a useful statistic in repeated-measures studies, as it provides information about the magnitude and direction of the change in the variable being measured.

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Answer: 24

Step-by-step explanation:

6 sides on the dice. 4 sections on the spinner.

6 times 4 = 24

24 possible outcomes.

The correct answer is c. marginal; average total; minimum. When plotting marginal and average cost curves, the marginal cost curve intersects the average total cost curve at its minimum point.

This is because the average total cost curve is U-shaped, with a downward-sloping portion at low levels of output and an upward-sloping portion at high levels of output. The marginal cost curve intersects the average total cost curve at the point where the upward-sloping portion of the average total cost curve starts, which is also the point where the average total cost curve is at its minimum. At this point, the marginal cost is equal to the average total cost.

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If the test statistic is -3.11, it means that the observed data is 3.11 standard deviations below the mean of the null distribution.

Since 2.36 is in the rejection region, it means that the critical value for the test is greater than 2.36 in absolute value. This implies that the null hypothesis would be rejected if the test statistic is either less than the negative critical value or greater than the positive critical value.

However, since the test statistic of -3.11 is smaller than the negative critical value, we would fail to reject the null hypothesis. This means that we would conclude that there is not enough evidence to support the alternative hypothesis and we accept the null hypothesis. Therefore, we would have made an incorrect decision if we rejected the null hypothesis based on the given information.

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The hypotheses of the research is:

Null Hypothesis: H₀: p = 0.15

Alternative Hypothesis: Hₐ: p < 0.15

The conclusion is;

Angela  fails to reject the null hypothesis because the p-value is greater than the significance value.

Null and alternative hypotheses are basically methods used in statistical hypothesis testing. The null hypothesis of a test is one that always predicts no effect or no relationship existing between variables. Meanwhile, the alternative hypothesis states the research prediction of an effect or relationship.

We are told that the proportion of engineering students at the local college who did not own a calculator was 0.15.

Now, Angelina believes the proportion of current population of engineering students who do not own a calculator is lower.

Thus, the hypotheses is defined as:

Null Hypothesis: H₀: p = 0.15

Alternative Hypothesis: Hₐ: p < 0.15

Now, the p-value is greater than the significance value and as such we fail to reject the null hypothesis

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You would pay approximately $1630.19 in interest for the amount you are able to borrow.

To calculate the amount you are able to borrow, we can use the formula for the present value of an annuity:

P = PMT ((1 - (1 + r)⁻ⁿ) / r)

Where:

P is the principal amount (the amount you are able to borrow)

PMT is the monthly payment you can afford ($300)

r is the monthly interest rate (3.8% divided by 100, then divided by 12)

n is the number of months (5 years multiplied by 12)

Let's calculate the amount you are able to borrow:

r = 3.8% / 100 / 12 = 0.0031667 (rounded to 7 decimal places)

n = 5 years x 12 = 60 months

P = $300 x ((1 - (1 + 0.0031667)⁻⁶⁰) / 0.0031667)

P ≈ $16369.81

Therefore, you are able to borrow approximately $16369.81.

To calculate the interest you would pay for the amount you are able to borrow, we can subtract the principal amount from the total amount paid over the 5-year period. The total amount paid can be calculated as:

Total Amount Paid = Monthly Payment x Number of Months

Total Amount Paid = $300 x 60 = $18,000

Interest Paid = Total Amount Paid - Principal Amount

Interest Paid = $18,000 - $16369.81 ≈ $1630.19

Therefore, you would pay approximately $1630.19 in interest for the amount you are able to borrow.

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Answer:

Step-by-step explanation:

a. 45mph

135miles ÷ 3hrs = 45mph

b. 44mph

22m x 2 = 44mph

c. 30mph

1/4 = 15 mins

7.5mins x 4 = 30mph

d. 50mph

100/3 miles = 2/3hr

xmiles (x = number of miles) = 1h

we use cross multiplying and use x to find the miles

2x  = 100 x 1

3        3

2x = 100

x = 50mph

e. 65mph

speed = 97.5 miles half an hour

The number of cups of cat food the feeder can hold is 12 cups.

The total volume of the feeder is calculated as follows;

total volume = volume of cylinder + volume of cone

The volume of the cylindrical part of the feeder is calculated as;

V = πr²h

where;

V = π(2.5 in)²(7.5 in)

V = 147.3 in³

The  volume of the cone part of the feeder is calculated as;

V = ¹/₃ πr²h

V = ¹/₃ π(2.5 in)²( 4 in )

V = 26.2 in³

Total volume =  147.3 in³ + 26.2 in³

Total volume = 173.5 in³

The number of cups of cat food the feeder can hold is calculated as follows;

n =  (173.5 in³) / 14.4 in³

n = 12 cups

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Participating in international trade can have both benefits and costs for a nation. One of the main benefits of international trade is the potential for increased economic growth and development.

By engaging in trade, countries can access larger markets for their goods and services, which can lead to increased sales and profits for businesses. This, in turn, can lead to increased investment and job creation, as well as increased tax revenues for the government.

Another benefit of international trade is the potential for increased consumer choice and lower prices for consumers.

By importing goods from other countries, consumers can access a wider variety of products than would be available domestically, and competition from foreign producers can help to drive down prices.

However, there are also costs associated with international trade. One potential cost is the risk of job losses in industries that face competition from imports.

When businesses in other countries can produce goods more efficiently or at lower cost than domestic producers, this can lead to job losses and a decline in certain industries.

Another potential cost of international trade is the risk of economic instability.

If a country becomes heavily dependent on exports to one or a few countries, a decline in demand from those countries can have a major impact on the economy.

Overall, the benefits and costs of international trade depend on a variety of factors, including the specific industries involved, the level of competition, and the economic policies of the countries involved.

While international trade can provide opportunities for economic growth and development, it is important to consider and manage the potential risks and costs.

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Answer:

715 miles

Step-by-step explanation:

We Know

Lashonda drove 495 miles in 9 hours.

495 / 9 = 55 miles per hour

At the same rate, how many miles would she drive in 13 hours?

We Take

55 x 13 = 715 miles

So, she drives 715 miles in 13 hours.

[tex]\begin{array}{ccll} miles&hours\\ \cline{1-2} 495 & 9\\ m& 13 \end{array} \implies \cfrac{495}{m}~~=~~\cfrac{9}{13} \\\\\\ (495)(13)=9m\implies \cfrac{(495)(13)}{9}=m\implies 715=m[/tex]

Answer:

  ∠U = 65°

Step-by-step explanation:

You want the measure of external angle U formed by two tangents that intersect a circle creating major arc TV of 245°.

The measure of the external angle is half the difference of the two arcs of the circle that the secants (tangents) intercept.

The major arc is shown as 245°. The minor arc TV completes the circle, so is 360° -245°.

Half the difference is ...

  (245° -(360°-245°))/2 = 245° -180° = 65°

Angle U is 65°.

__

Additional comment

You will notice the angle formed by the tangents is 180° less than the measure of the major arc. (It is also equal to the supplement of the minor arc.)

A tangent is a special case of a secant, where the two points of intersection with the circle merge to a single point. The relations applicable to secants also apply to tangents.