The complete bipartite graph Km,n where m, n ≥ 1, has m red nodes, n blue nodes and has an edge between every red node and every blue node.Assume m > n, prove that the length of a longest path in Km,n is even.

Answer:

X = 34

Step-by-step explanation:

3x+44+x=180

4x+44=180

4x=136

x=34

hope this helps! :)

Answer:

-2

Step-by-step explanation:

-4 + (-1) - (-3) = ?

-4 - 1 + 3 = ?

-5 + 3 = ?

? = -2

Step-by-step explanation:

given

- 4 + (- 1) - ( - 3 )

= - 4 - 1 + 3

= - 5 + 3

= - 2

hope it helps

From the probability distribution for the number of defective items in a random sample, the expected value of X is 2.82.

To calculate the expected value of X, we need to multiply each possible value of X by its corresponding probability and sum them up.

The expected value of X, denoted as E(X) or μ, is calculated using the formula:

E(X) = ∑ (x * p(x))

where x represents each possible value of X and p(x) represents the corresponding probability.

In this case, the probability distribution for X is given as follows:

x: 0 1 2 3 4

p(x): 0.1 0.15 0.13 0.07 0.55

To calculate the expected value, we perform the following calculations:

E(X) = (0 * 0.1) + (1 * 0.15) + (2 * 0.13) + (3 * 0.07) + (4 * 0.55)

E(X) = 0 + 0.15 + 0.26 + 0.21 + 2.2

E(X) = 2.82

The expected value represents the average value or mean of the probability distribution. In this case, it represents the average number of defective items we expect to find in a random sample based on the given probabilities.

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The radius of the hot air balloon is increasing at a rate of approximately 0.00197 feet per minute when the radius is 16 feet.

To find how fast the radius of the hot air balloon is increasing when the radius is 16 feet, we need to use the formula for the volume of a sphere, which is V = (4/3)πr³.

Taking the derivative with respect to time, we get dV/dt = 4πr²(dr/dt). We are given that dV/dt = 10 cubic feet per minute and r = 16 feet.

Plugging in these values, we get 10 = 4π(16)²(dr/dt), which simplifies to dr/dt = 10/(4π(16)²) = 0.00197 feet per minute (approximately).

The problem involves finding the rate of change of the radius of a hot air balloon when the volume is increasing at a given rate. We use the formula for the volume of a sphere to relate the rate of change of volume to the rate of change of radius.

By taking the derivative of this formula with respect to time, we obtain an expression for the rate of change of volume in terms of the rate of change of radius. Substituting the given values and solving for dr/dt, we obtain the answer.

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To determine whether quality is associated with the source of the shipments, we need to test the competing hypotheses.

The competing hypotheses are as follows:

H0: Quality and source of shipment (vendor) are independent.
HA: Quality and source of shipment (vendor) are dependent.

To test these hypotheses, we can use a chi-square test for independence. The test statistic is calculated by comparing the observed frequencies with the expected frequencies under the assumption of independence.

b-1. To calculate the test statistic, we first need to create a contingency table with the observed frequencies of quality and source of shipment. Each cell in the table represents the count of shipments from a specific vendor with a specific quality.

For example, the table could look like this:

            | Vendor A | Vendor B | Vendor C
--------------------------------------------
Good Quality |   10     |   15     |   12
--------------------------------------------
Poor Quality |   20     |   25     |   18

Next, we calculate the expected frequencies assuming independence. The expected frequency for each cell is calculated by multiplying the row total by the column total and dividing by the total number of observations.

Finally, we calculate the chi-square test statistic by summing the squared differences between the observed and expected frequencies divided by the expected frequencies for each cell.

b-2. Once we have the test statistic, we can find the p-value associated with it. The p-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.

To find the p-value, we need to consult the chi-square table or use a statistical software. The p-value will indicate the strength of evidence against the null hypothesis. The smaller the p-value, the stronger the evidence against the null hypothesis.

Based on the given options, the p-value falls within the range of 0.01 ≤ p-value < 0.025. Therefore, we reject the null hypothesis and conclude that there is evidence to suggest that quality and source of shipment are dependent.

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Answer:

\(16x^2+25y^2-400=0\)

Step-by-step explanation:

Given:

There are 4 types of conic sections:

The definition of an ellipse is:

Therefore, the given definition is one for an ellipse.

\(\boxed{\begin{minipage}{7.4 cm}\underline{General equation of an ellipse}\\\\$\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$\\\\where:\\\phantom{ww}$\bullet$ $(h,k)$ is the center. \\ \phantom{ww}$\bullet$ $(h\pm a,k)$ or $(h,k\pm b)$ are the vertices. \\ \phantom{ww}$\bullet$ $(h\pm c,k)$ or $(h,k\pm c)$ where $c^2=a^2-b^2$.\\\end{minipage}}\)

As the given foci are (-3, 0) and (3, 0), they are on the x-axis. Therefore:

Therefore, the major axis is on the x-axis and the vertices are (h±a, k) and the co-vertices are (h, k±b).

The constant is the major axis which is 2a.  Given the constant is 10:

\(\implies 2a=10\)

\(\implies a=5\)

Since c² = a² - b²:

\(\implies 3^2=5^2-b^2\)

\(\implies b^2=5^2-3^2\)

\(\implies b^2=25-9\)

\(\implies b^2=16\)

Therefore, the equation for the conic section (ellipse) is:

\(\implies \dfrac{(x-0)^2}{5^2}+\dfrac{(y-0)^2}{16}=1\)

\(\implies \dfrac{x^2}{25}+\dfrac{y^2}{16}=1\)

In standard form:

\(\implies 25 \cdot \dfrac{x^2}{25}+25 \cdot \dfrac{y^2}{16}=25 \cdot 1\)

\(\implies x^2+\dfrac{25}{16}y^2=25\)

\(\implies 16 \cdot x^2+16 \cdot \dfrac{25}{16}y^2=16 \cdot 25\)

\(\implies 16x^2+25y^2=400\)

\(\implies 16x^2+25y^2-400=400-400\)

\(\implies 16x^2+25y^2-400=0\)

Answer:

sup!

the answer is

a point on (0,-1) and (1,3)

Step-by-step explanation:

hope this helps

keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above

\(7x-5y=-6\implies -5y=-7x-6\implies y=\cfrac{-7x-6}{-5} \\\\\\ y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{7}{5}}x+\cfrac{6}{5}\qquad \impliedby \qquad \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}\)

so we're really looking for the equation of a line whose slope is 7/5 and it passes through (-4 , -5)

\((\stackrel{x_1}{-4}~,~\stackrel{y_1}{-5})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{7}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-5)}=\stackrel{m}{ \cfrac{7}{5}}(x-\stackrel{x_1}{(-4)}) \implies y +5 = \cfrac{7}{5} ( x +4) \\\\\\ y+5=\cfrac{7}{5}x+\cfrac{28}{5}\implies y=\cfrac{7}{5}x+\cfrac{28}{5}-5\implies {\Large \begin{array}{llll} y=\cfrac{7}{5}x+\cfrac{3}{5} \end{array}}\)

Using binomial theorem we can expand the equation but We are not given the value of a or n, so we cannot determine c exactly.

Integers are real numbers that only comprise positive and negative whole integers as well as natural numbers. Because of rational and irrational numbers, real numbers may include fractions, whereas integers cannot.

A real number is a quantity in mathematics that may be expressed as an infinite decimal expansion. Real numbers, as opposed to natural numbers such as 1, 2, 3,..., which are generated from counting, are used in measures of continually changing quantities such as size and time.

by applying the binomial theorem:

\((1 + ax)^n = C(n, 0) + C(n, 1)(ax) + C(n, 2)(ax)^2 + C(n, 3)(ax)^3 + ...\)

where C(n, k) is the binomial coefficient, which equals\(n!/(k!(n-k)!).\)

The first few terms of this expansion are:

\((1 + ax)^n = 1 + nax + n(n-1)(a^2/2)x^2 + n(n-1)(n-2)(a^3/6)x^3 + ...\)

Comparing with the given expression [1 - 20x + 150x^2 + cx^3], we have:

\(1 - 20x + 150x^2 + cx^3 = 1 + nax + n(n-1)(a^2/2)x^2 + n(n-1)(n-2)(a^3/6)x^3 + ...\)

Equating coefficients of \(x^3\) on both sides, we get:

\(c = n(n-1)(n-2)(a^3/6)\)

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Answer:

42°

Step-by-step explanation:

angle b is complementary to 48° so their sum is 90°.

Answer:

42

Step-by-step explanation:

48 + b = 90

90 - 48 = 42

b = 42

Using the principle of common difference , we can observe that the distance moved per roll increases by a constant amount of centimeters . Hence, the distance moved in the 5th roll is 180 cm

Setting up a proportional relationship :

The common difference between each roll can be obtained thus :

d2 - d1 = d3 - d2 = d4 - d3

60 - 20 = 100 - 60 = 140 - 100

40 = 40 = 40

Hence, the common difference = 40

The distance obtained on the fifth roll :

d(5) = d(4) + common difference

d(5) = 140 + 40

d(5) = 180

Therefore, the distance obtained on the fifth roll is 180 cm

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The SAT has two sections, Mathematics and Reading, and the combined score is reported on a scale of 400 to 1600. The Z-score is (1600 - 1003) / 220 = 2.37.

The normal distribution is widely used in statistics and is a useful tool for analyzing many types of observations and data in nature and the sciences. Standard deviation is a measure of the spread of the distribution and can be used in computation of various probabilities.

To determine the proportion of SAT scores that were reported as 1600 or higher, we need to find the number of standard deviations above the mean a score of 1600 would be.

Using the normal distribution and Z-scores, we can calculate the Z-score for a score of 1600, given the mean of 1003 and standard deviation of 220.

The Z-score is (1600 - 1003) / 220 = 2.37.

This means that a score of 1600 is 2.37 standard deviations above the mean.

To find the proportion of scores that are higher than 1600, we can use a standard normal table to look up the area under the normal curve to the right of the Z-score of 2.37.

This proportion is approximately 0.9906,

meaning that approximately 99.06% of the SAT scores were lower than 1600, and only 0.94% of the scores were higher and reported as 1600.

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Answer: the answer is 47 degrees

Step-by-step explanation:

a. The value of the swap to firm XYZ is -$23,225.33.

b. This would help to protect the firm against an increase in interest rates.

a. Calculation of the value of the swap to firm XYZ

The value of the swap to firm XYZ can be calculated using the following formula:

Value of Swap = Value of Fixed Leg - Value of Floating Leg

Where Value of Fixed Leg = VFL = (Coupon Rate * Principal * (1 - (1 + r)-n / r))

Value of Floating Leg = VFL = (Interest Rate * Principal * (1 - (1 + r)-n / r))

Where, Coupon Rate = 4% p.a. Principal = $20 million

Interest Rate = LIBOR + Spread, Spread = 0, therefore,

Interest Rate = 4.5% p.a.

Time to Maturity, n = 15 months

Remaining Time to Next Payment = 6 - 2 = 4 months

Therefore, the current six-month LIBOR rate is required to calculate the value of the swap as follows:

Current six-month LIBOR rate = 4.5% * (4/6) + x% * (2/6)x = ((Value of Swap/Principal + (1 - (1 + r)-n / r)) * r) / (1 - (1 + r)-n / r) = 0.0251%

Value of Fixed Leg = VFL = (0.04 * 20,000,000 * (1 - (1 + 0.05/2)-30 / (0.05/2))) = $1,328,816.76

Value of Floating Leg = VFL = (0.0451 * 20,000,000 * (1 - (1 + 0.05/2)-10 / (0.05/2))) = $1,352,042.09

Value of Swap = $1,328,816.76 - $1,352,042.09 = -$23,225.33

Therefore, the value of the swap to firm XYZ is -$23,225.33.

b. Suggest a reason why XYZ entered into the swap in the first place

Firm XYZ might have entered into the swap in the first place to mitigate the risk of a rise in interest rates in the future. By entering into a fixed-for-floating interest rate swap, the firm has fixed its borrowing cost for the duration of the swap at 4%, regardless of the future interest rate changes.

Therefore, this would help to protect the firm against an increase in interest rates.

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Answer:

the anwser is b

Step-by-step explanation:

In order to determine the exact nature of the distribution, Professor X would need to analyze the data in more detail, for example, by creating a histogram or other visual representation of the grades, and conducting statistical tests to determine the nature of the distribution.

If Professor X finds that the grades on the first exam for their statistics class either did really well or really poorly, it suggests that the distribution of grades is bi-modal, meaning there are two distinct peaks in the distribution. This means that some students did very well on the exam and scored high grades, while others did very poorly and scored low grades, with very few students scoring grades in between these two groups.

A bi-modal distribution is not a common occurrence in a typical exam setting, as most exams tend to have a normal or bell-shaped distribution with a single peak, where the majority of students score around the average or mean grade. The bi-modal distribution found by Professor X could be due to a variety of factors, such as students having widely differing levels of prior knowledge, or students' varying levels of preparation for the exam.

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1. \( \frac{x}{15} = 3\)

2. 8x = 216

Answer:

1. 45

2. 27

(please correct me if im wrong :D)

The  equation of the line is y =  x + 6.

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

here, we have,

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given

4x + y = 7 ( subtract 4x from both sides )

y = - 4x + 7 ← in slope- intercept form

with slope m = - 4

given a line with slope m then the slope of a line perpendicular to it is

= 1/-m  =1/ 4  

then,

y =  x/4 + c ← is the partial equation of the perpendicular line

to find c substitute )- 4, 5 ) into the partial equation

5 = - 1 + c ⇒ c = 5 + 1 = 6

hence, y =  x/4 + 6 ← equation of perpendicular line.

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The height of the fence in design is 5 1/6 feet.

Given,

Posts: 6 3/4 feet in length.

Posts must be installed 1 1/2 feet below ground level.

We need to find out what will be the height of the fence in design.

A mixed fraction has three parts the whole number part, the numerator, and the denominator.

Example: 1 2/3

1 - whole number part

2 - Numerator part

3 - denominator part

We can convert mixed fractions into proper fractions.

a b/c = (cxa + b) / b

Find the height of the fence in design.

This means the height from the surface of the ground.

We have,

Post length = 6 2/3 feet

Height below the ground = 1 1/2 feet

Now,

= 6 2/3 - 1 1/2

= 20/3 - 3/2

= (40 - 9) / 6

= 31 / 6

= 5 1/6 feet

Thus the height of the fence in design is 5 1/6 feet.

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Answer:

y=-3x-10

Step-by-step explanation:

rise over run

slope is -9/3 and the y intercept is 10

Answer:

slope: -1

y-intercept: 3

Step-by-step explanation:

Slope-intercept form:

y = mx + b, in which:

y = (x , y)

m = slope

x = (x , y)

b = y-intercept

The given equation is:

\(y = -x + 3\)

Note that when there is a negative sign with no number directly next to the x, it means that the slope (m) is -1. 3 is also in the space of the y-intercept (variable b).

Your answer is:

slope: -1

y-intercept: 3

~

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Answer:

17) b. commutative property

18) NOT SURE d. associative property

19) NOT SURE d. associative property

20) a. commutative property

When 100 ft of rope is out, the rate at which the boat is approaching the dock is 0 ft/min which means the boat is not moving towards the dock at that moment.

Let x represent the horizontal distance between the boat and the dock (in feet).

Let y represent the vertical distance between the boat and the pulley (in feet).

Since the rope is attached to the front of the boat, which is 10 feet below the level of the pulley, we have y = 10 ft.

The rate of change of the length of the rope, which is related to the distance between the boat and the dock, is dx/dt = 12 ft/min.

We want to find the rate at which the boat is approaching the dock, which is the rate of change of x with respect to time (dx/dt) when 100 ft of rope is out.

Now, let's set up the similar triangles between the boat, the pulley, and the dock.

x / y = (x + 100) / 100

Now, we can differentiate both sides of this equation with respect to time t:

d(x/y)/dt = d((x + 100) / 100)/dt

To solve for dx/dt, we need to differentiate x/y and (x + 100)/100 with respect to time.

Using the quotient rule, we have:

(dx/dt × y - x × dy/dt) / (y²) = (1/100) × (dx/dt)

Substituting y = 10 and dy/dt = 0 (since the pulley is fixed), we get:

(dx/dt × 10 - x × 0) / (10²) = (1/100) × (dx/dt)

10 × dx/dt = (1/100)×(dx/dt)

10 × dx/dt - (1/100) × dx/dt = 0

(1000/100 - 1/100) × dx/dt = 0

(999/100) × dx/dt = 0

dx/dt = 0

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Therefore, the correct choice is:

A. The function has a relative minimum value of f(x, y) = 10 at (x, y) = (0, 0).

To find the relative maximum and minimum values of the function f(x, y) = 3x^2 + 7y^2 + 10, we need to take partial derivatives with respect to x and y and set them equal to zero.

∂f/∂x = 6x = 0

∂f/∂y = 14y = 0

From the first equation, we find that x = 0, and from the second equation, we find that y = 0.

To determine whether these critical points correspond to relative maximum or minimum values, we can use the second partial derivative test.

Taking the second partial derivatives:

∂²f/∂x² = 6

∂²f/∂y² = 14

Since both second partial derivatives are positive, we conclude that the function has a relative minimum value at the critical point (0, 0).

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Answer:

22

Step-by-step explanation:

Use a proportion.

30/15 = 44/x

30x = 15 × 44

2x = 44

x = 22

Answer:

22

Step-by-step explanation:

30 miles/ 15 sessions of tract practice

Per practice Jack is running a total of 2 miles.

So 44 miles/2 miles

22 sessions of track practice

According to the question Approximately 60% of Americans are in the working class and 80% of the people in the U.S. are lower middle class. The correct answer is D. \(\(60\%\)\) and \(\(10\%\)\).

The working class typically comprises individuals involved in manual labor, skilled trades, or service-oriented jobs. They often earn wages and may have lower income levels compared to other classes.

The percentage of Americans in the working class can vary based on factors such as economic conditions, industry trends, and societal changes. The lower middle class generally includes individuals who have achieved some level of education beyond high school and hold white-collar or technical jobs.

They often have moderate incomes and may have attained some level of financial stability. The percentage of people in the U.S. who fall into the lower middle class can also fluctuate based on economic factors and social dynamics.

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Given that X has k-stage Erlang distribution with parameter 1. Therefore, the probability density function of X can be given as follows: f(x)={λkxk−1e−λx(k−1)!for x≥0otherwiseY=2XX2 = 2kXX has the chi-square distribution with 2k degrees of freedom.

Therefore, we need to prove the moment generating function of Y equals the moment generating function of a chi-square distribution with 2k degrees of freedom. Moment generating function of X can be given as follows: MX(t) = (1−t/λ)−k Therefore, moment generating function of Y can be given as follows: MY(t) = E(etY)= E[et(2kXX2)] ... Equation (1)Since X has k-stage Erlang distribution with parameter 1, let’s represent it as the sum of k independent exponentially distributed random variables with mean 1/λ as follows: X=∑i=1kExpiwhere Exp is an exponentially distributed random variable with mean 1/λ.

Therefore, Equation (1) can be written as follows:MY(t) = E [et(2kX(Expi)22)] = E [et∑i=1k(2kExpi)22] = ∏i=1k E [et(2kExpi)22] ... Equation (2)The moment generating function of an exponentially distributed random variable Exp with parameter λ can be given as follows:ME(t) = E(etExp) = ∫0∞etxe−λxdx = λλ−tThe moment generating function of Xpi can be calculated by replacing λ with kλ in the moment generating function of Exp.

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Answer: 30% of 105 (= 31.5)

Step-by-step explanation:

30% of 105 is 31.5 and

32% of 98 is 31.36

Answer:

Step-by-step explanation:

There are two bases of 3x4 and 4 sides of 2x4

Total surface area is: