DISCRETE MATH
What is the maximum number of edges a graph with n vertices can have if all n vertices also have loops at the vertex?

Answer: Let's use the formula for the density of a mixture to solve this problem:

ρ_mix = (m_A + m_B) / (V_A + V_B)

where ρ_mix is the density of the mixture, m_A and m_B are the masses of liquids A and B, and V_A and V_B are the volumes of liquids A and B.

Since we know the densities of liquids A and B, we can use them to calculate the masses of the liquids:

m_A = ρ_A * V_A

m_B = ρ_B * V_B

where ρ_A and ρ_B are the densities of liquids A and B.

We are given that the volume of liquid B is 50 cm³, so we can write:

V_B = 50 cm³

To solve for x, we need to find the value of V_A. Let's use the fact that x cm³ of liquid A are mixed with 50 cm³ of liquid B to write:

V_A + V_B = x + 50 cm³

Substituting the expressions for V_B and m_B into the formula for the density of the mixture, we get:

ρ_mix = (m_A + ρ_B * V_B) / (V_A + V_B)

Substituting the expressions for m_A and m_B, we get:

ρ_mix = (ρ_A * V_A + ρ_B * V_B) / (V_A + V_B)

We are given that the density of the resulting mixture is 2.7 g/cm³, so we can write:

ρ_mix = 2.7 g/cm³

Substituting all these values into the formula for the density of the mixture, we get:

2.7 g/cm³ = (ρ_A * V_A + 2.4 g/cm³ * 50 cm³) / (x + 50 cm³)

Simplifying this expression, we get:

2.7 g/cm³ = (ρ_A * V_A + 120 g) / (x + 50 cm³)

Multiplying both sides by (x + 50 cm³), we get:

2.7 g/cm³ * (x + 50 cm³) = ρ_A * V_A + 120 g

Expanding the left side, we get:

2.7 g/cm³ * x + 2.7 g/cm³ * 50 cm³ = ρ_A * V_A + 120 g

Simplifying this expression, we get:

ρ_A * V_A = 2.7 g/cm³ * x - 2.7 g/cm³ * 50 cm³ + 120 g

ρ_A * V_A = 2.7 g/cm³ * (x - 50 cm³) + 120 g

Now we can substitute the density of liquid A into this expression and solve for x:

ρ_A = 3.5 g/cm³

ρ_A * V_A = 2.7 g/cm³ * (x - 50 cm³) + 120 g

3.5 g/cm³ * V_A = 2.7 g/cm³ * (x - 50 cm³) + 120 g

3.5 g/cm³ * V_A = 2.7 g/cm³ * x - 2.7 g/cm³ * 50 cm³ + 120 g

3.5 g/cm³ * V_A = 2.7 g/cm³ * x - 54 g + 120 g

3.5 g/cm³ * V_A = 2.7 g/cm³ * x + 66 g

V_A = (2.7 g/cm³ *x + 66 g) / 3.5 g/cm³

V_A = (0.7714 x + 77.14) cm³/g

Now we can substitute this expression for V_A into the earlier equation we obtained for the density of the mixture, and solve for x:

2.7 g/cm³ = (ρ_A * V_A + 2.4 g/cm³ * 50 cm³) / (x + 50 cm³)

2.7 g/cm³ = (3.5 g/cm³ * (0.7714 x + 77.14) cm³/g + 2.4 g/cm³ * 50 cm³) / (x + 50 cm³)

2.7 g/cm³ = (2.6999 x + 227.6) / (x + 50 cm³)

Multiplying both sides by (x + 50 cm³), we get:

2.7 g/cm³ * (x + 50 cm³) = 2.6999 x + 227.6

Expanding the left side, we get:

2.7 g/cm³ * x + 2.7 g/cm³ * 50 cm³ = 2.6999 x + 227.6

Simplifying this expression, we get:

0.0001 x = 1.4

x = 14,000 cm³

Therefore, 14,000 cm³ of liquid A should be mixed with 50 cm³ of liquid B to obtain a mixture with density 2.7 g/cm³.

The x-intercepts of the function f(x) = |x| - 5 are x = 5 and x = -5, and the y-intercept is y = -5. The x-intercepts of the function p(x) = |x - 3| - 1 are x = 4 and x = 2, and the y-intercept is y = 2.

(a) To determine the x-intercepts of the function f(x) = |x| - 5, we set f(x) = 0 and solve for x.

0 = |x| - 5

|x| = 5

This equation has two solutions: x = 5 and x = -5. Therefore, the x-intercepts are x = 5 and x = -5.

To determine the y-intercept, we substitute x = 0 into the function:

f(0) = |0| - 5 = -5

Therefore, the y-intercept is y = -5.

(b) To determine the x-intercepts of the function p(x) = |x - 3| - 1, we set p(x) = 0 and solve for x.

0 = |x - 3| - 1

| x - 3| = 1

This equation has two solutions: x - 3 = 1 and x - 3 = -1. Solving these equations, we find x = 4 and x = 2. Therefore, the x-intercepts are x = 4 and x = 2.

To determine the y-intercept, we substitute x = 0 into the function:

p(0) = |0 - 3| - 1 = |-3| - 1 = 3 - 1 = 2

Therefore, the y-intercept is y = 2.

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The number of simple events in this experiment is 16.

The correct answer to the given question is option c.

The probability of an event can be calculated by dividing the number of favorable outcomes by the number of possible outcomes. A simple event is one in which only one of the outcomes can occur. For example, if a coin is tossed, a simple event would be the outcome of the coin being heads or tails.

The total number of possible outcomes in the experiment of tossing 4 unbiased coins simultaneously is 2⁴, since there are two possible outcomes for each coin. Thus, the total number of possible outcomes is 16.

Each coin has two possible outcomes: heads or tails. If all four coins are flipped, there are two possible outcomes for the first coin, two possible outcomes for the second coin, two possible outcomes for the third coin, and two possible outcomes for the fourth coin. Therefore, the total number of possible outcomes is 2 × 2 × 2 × 2 = 16.

Therefore, the number of simple events in this experiment is 16, which is option (c).

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Answer:

A.

Twice the sum of a number and three means that the number and three are in parentheses.

The equation representing y, the total amount that Arelli charges her clients after the Consultation fee and x hours of personal training is y = 12 + 18x.

Let y be the total amount Arelli charges her clients after the consultation fee and x hours of personal training.The total cost would be the sum of the consultation fee and the hourly rate charge.

Using the information given, the equation that represents the total amount that Arelli charges her clients after the consultation fee and x hours of personal training would be:y = 12 + 18xThe consultation fee is $12 which is added to $18 per hour of personal training.

In order to get the total amount charged for x hours of training, multiply the hourly rate by the number of hours:18x

Thus, the total amount Arelli charges for x hours of personal training is:y = 12 + 18x

Therefore, the equation representing y, the total amount that Arelli charges her clients after the consultation fee and x hours of personal training is y = 12 + 18x.

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Answer:

11+n

Step-by-step explanation:

eleven increased by a number

11+n

hope it helps

The area of the Triangle given in the question is 4.7

Area of Triangle = 1/2(base × height )

Using the vertex , we can obtain the height of the triangle thus:

height = 1.873

Inserting the parameters into the Area formula :

height = 1.873

base = 5

Area of Triangle = 1/2 × 5 × 1.873

Area of Triangle= 4.68

Therefore, the area of the triangle is 4.7

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The bank that pays the highest interest for this 4-year IRA account is Chase Bank.

It should be noted that interest rates simply means the cost of borrowing or the gain from savings.

In this case, Destiny wants to compare interest rates quoted by four local banks for a deposit.

Chase Bank: = 2 1/2% = 2.5%

Jefferson City Trust:l = 1.7%,

Bank of America = 2.12%

First State Bank = 1 3/4% = 1.75%

Therefore, the highest is Chase Bank as it gives 2.5%.

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Thus, the obtained area of sector for the given circle is found as 43.96 in².

Two radii that meet at the centre to form a sector define a circle. The sector is the portion of the circle created by these two radii. Knowing a circle's central angle assessment and radius measurement are both crucial for solving circle-related difficulties.

The curved portion that runs along the circle's perimeter and joins the ends of a two radii that make up a sector is known as the sector arc.

given data:

Formula for the area of sector:

area of sector = Ф /360 * (πr²)

area of sector = 140/360 * (3.14 *6²)

area of sector = 7/18 * 3.14 *36

area of sector = 43.96 in²

Thus, the obtained area of sector for the given circle is found as 43.96 in².

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Answer:

the mixed number form is 10 7/8

Answer:

2x^2+14x-240=0

Step-by-step explanation:

Remember Pythagoras's theorem?

a^2+b^2=c^2

So over here,

x^2+ (x+7)^2= 17^2 [ we need to now simplify (x+7)^2 --> (x+7) (x+7)  ]

x^2+ x^2+7x+7x+49= 289

2x^2+ 14x+49= 289

2x^2+14x+49=0 [ we need to make the right side zero coz thats

the rule for ]  

2x^2+14x-240=0

(idk, but i think the "show that  x^2.....) is wrong)

Answer:

No

Step-by-step explanation:

A trapezoid is never a square. A trapezoid only has one pair of parallel lines and a square has two.

Answer:

y=14-8x

x= -2 + 3y/9x

Step-by-step explanation:

8x+y=14

isolate y so you get:

y=14-8x

-9x+3y=18

isolate x so you get:

-9x=18-3y

divide on both sides to get x by itself

-9x/-9x  18-3y/-9x

x= -2 + 3y/9x

Hope this helps :)

True. Marginal cost refers to the additional cost incurred by producing one more unit of output.

Since the production of one more unit requires the use of additional variable factors of production (such as labor or raw materials), marginal cost always reflects the cost of those variable factors. Fixed costs, on the other hand, do not change with changes in output and are not included in marginal cost calculations. Therefore, marginal cost only reflects the change in cost associated with variable factors of production.

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The fundamental set of solutions of the given system of differential equations x' =(1 2 0, -3 -1 3, 3 2 -2) is to be identified from the given options.

The correct answer is option (a) x1 = e^-2t(2 -3 3), X2 = e^-t(1 -1 1), X3 = e^t(1 0 1).

To verify this, we can calculate the Wronskian of the three solutions and show that it is non-zero, which confirms that they form a fundamental set of solutions. Another way to check is to substitute the solutions into the differential equation and verify that they satisfy it. In this case, both methods give us the same result - the solutions satisfy the differential equation and are linearly independent, hence form a fundamental set of solutions. Therefore, the correct answer is (a).

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Answer:

62.5%

Step-by-step explanation:

There are a total of 8 options.

We are trying to get 5 of the 8.

5/8 = 0.625 = 62.5%

The probability of having 90 or more wins in the next three hours of playing is 0.0506 or 5.06% (approx).

We are required to find the probability of having 90 or more wins in the next three hours of playing a mobile phone game similar to Pokémon, given that the number of successive win follows a Poisson distribution with a mean of 27 wins per hour.

The given mean of Poisson distribution is λ = 27.

The Poisson distribution formula is:P(X = x) = (e^-λ λ^x) / x!

We need to calculate the probability of having 90 or more wins in 3 hours.

We can combine these three hours and treat them as one large interval, for which λ will be λ1 + λ2 + λ3= (27 wins/hour) * 3 hours= 81 wins.

P(X ≥ 90) = 1 - P(X < 90)

To calculate P(X < 90), we can use the Poisson distribution formula with λ = 81.P(X < 90) = Σ (e^-81 * 81^k) / k!, k = 0, 1, 2, ....89= 0.9494

Using the above formula and values, we get P(X ≥ 90) = 1 - P(X < 90)= 1 - 0.9494= 0.0506

Therefore, the probability of having 90 or more wins in the next three hours of playing is 0.0506 or 5.06% (approx).

Hence, the probability of having 90 or more wins in the next three hours of playing is 0.0506 or 5.06% (approx).

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Answer:

RP=21.

Step-by-step explanation:

Since 51/17=3 and 111/37=3, we can divide 63 by 3 to get the answer, 21.

Juana would have to pay approximately $29.40 in interest for the $10,686.00 loan over a 20-day period, assuming an annual interest rate of 5.79%.

The interest Juana would have to pay in 20 days can be calculated using the formula:

Interest = Principal × Interest Rate × Time

In this case, the principal amount is $10,686.00 and the interest rate is 5.79% per annum. To calculate the interest for 20 days, we need to convert the time to a fraction of a year. Since there are 365 days in a year, the time in years would be 20/365.

Using the formula and substituting the values:

Interest = $10,686.00 × 0.0579 × (20/365)

Calculating this expression, we find that the interest amount Juana would have to pay in 20 days is approximately $29.40.

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Answer:

Step-by-step explanation:

Not liking rock mostly doesn't effect not liking just rap, since half like both rock and rap.

Though, there are 2 more people that like just rock compared to just rap, so you are more likely to like just rock.

Additionally, if you don't like rock, there is a 31.4 % that you don't like rap as well, since 17 people like neither.

Answer:

a, b, and c

Step-by-step explanation:

The points, if present on the circle, will make the equation true.

Evaluating the options :

(-1, 0)

(5, 0)

(0, √5)

(3, √5)

The options a, b, and c lie on the circle.

Answer:

Area of circle = π(12^2) = 144π

Area of red triangle = (1/2)(24)(12) = 144

P(point in circle is in triangle)

\( \frac{144}{144\pi} = \frac{1}{\pi} = .32\)

The probability that point falls in red shaded triangle is 0.32

probability is the ratio of the number of favorable to the total possible outcomes of an experiment.

Total possible outcomes= Area of circle

Area = π*12² = 144π

Favorable outcomes = Area of Triangle

Area = 1/2 * 24 * 12 = 144

P(shaded triangle) = 144/144π = 0.318

Therefore, probability that point falls in red shaded triangle is 0.32

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True. Nonsampling errors refer to errors that occur during the data collection process that are not related to the sampling method used. These errors can occur in any type of data collection method, whether it be through surveys, experiments, or observational studies.

Nonsampling errors can arise due to a variety of reasons, such as poor survey design, biased sampling methods, inadequate training of surveyors, errors in data entry or processing, or even deliberate fraud.

Nonsampling errors can have a significant impact on the quality of the data collected, as they can introduce biases and inaccuracies into the results. These errors can result in incorrect conclusions and decisions based on the data, which can have serious consequences. Therefore, it is important to take steps to minimize nonsampling errors during the data collection process, such as carefully designing surveys, ensuring proper training of surveyors, and conducting thorough quality checks on the data collected.

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The ratio of their areas is (3:8)² which simplifies to 9:64.

Area of smaller circle is 256/9 π.

The ratio of their perimeters is 5:3 since they are regular polygons with proportional side lengths.

The ratio of the areas of two similar polygons is equal to the square of the ratio of their corresponding sides. Since the scale factor of the polygons is 3:8, the ratio of their corresponding sides is 3:8. Therefore, the ratio of their areas is (3:8)^2, which simplifies to 9:64.

The area of a circle is proportional to the square of its radius. Let r be the radius of the smaller circle, then the radius of the larger circle is 3/2 times r. The area of the larger circle is given as 64π, so (3/2)^2 times the area of the smaller circle must also equal 64π. Solving for the area of the smaller circle, we get:

(9/4)πr^2 = 64π

r^2 = (64/9) * (4/π)

r^2 = 256/9π

Area of smaller circle = πr^2 = π * (256/9π) = 256/9 π.

The ratio of the areas of two regular polygons is equal to the square of the ratio of their side lengths. Let s1 and s2 be the side lengths of the first and second pentagons, respectively. Then we have:

Area of first pentagon / Area of second pentagon = (s1^2 / s2^2)

We are given the areas of the two pentagons, so we can plug them in and simplify:

150√3 / 54√3 = (s1² / s2²)

25 / 9 = (s1^2 / s2^2)

s1 / s2 = √(25/9) = 5/3

Therefore, the ratio of their perimeters is 5:3 since they are regular polygons with proportional side lengths.

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Answer:

\(2m=45\)

Step-by-step explanation:

He earned twice as much washing cars, which is \(2m\). This is given to be equal to 45.

Answer:

5s

Step-by-step explanation:

Algebraic rule

s is a variable and it is an unspoken rule to multiply them if they are next to one another without a mathematical sign

The coordinates of A'', B'' and C'' are (-1, 4), (-4, 3) and (-1, 2) respectively.

Transformation of geometrical figures or points is the manipulation of a given figure to some other way.

Different types of transformations are Rotation, Reflection, Glide reflection, Translation and Dilation.

Given triangle has coordinates,

A(2, -2), B(5, -3) and C(2, -4).

First the triangle is translated by the rule (x, y) → (x - 1, y + 6)

A(2, -2) becomes A'(2 - 1, -2 + 6) = A'(1, 4).

B(5, -3) becomes B'(5 - 1, -3 + 6) = B'(4, 3)

C(2, -4) becomes C'(2 - 1, -4 + 6) = C'(1, 2)

Then the translated triangle is reflected across the Y axis.

When reflected a point (x, y) across the Y axis, y coordinate remains same and x coordinate flips.

A'(1, 4) becomes A''(-1, 4)

B'(4, 3) becomes B''(-4, 3)

C'(1, 2) becomes C''(-1, 2)

Hence the vertices of the triangle after undergoes translation and reflection becomes A''(-1, 4), B''(-4, 3) and C''(-1, 2).

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The arc length of the curve defined by y = (e^x + e^(-x))/2 on the interval [-ln(2), ln(2)) can be found by integrating with respect to x and is approximately 4.219 units.

To find the arc length, we use the formula for arc length:

L = ∫(a to b) √(1 + (dy/dx)^2) dx,

where dy/dx is the derivative of y with respect to x.

In this case, the given equation y = (e^x + e^(-x))/2 can be rewritten as:

y = (1/2)(e^x + e^(-x)).

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

dy/dx = (1/2)(e^x - e^(-x)).

Substituting this derivative into the arc length formula, we have:

L = ∫(-ln(2) to ln(2)) √(1 + ((1/2)(e^x - e^(-x)))^2) dx.

Integrating this expression is a complex task and may not have a closed-form solution. However, we can use numerical methods, such as numerical integration techniques or software tools, to approximate the integral.

Using numerical integration, the arc length of the curve is found to be approximately 4.219 units.

Therefore, the arc length of the curve defined by y = (e^x + e^(-x))/2 on the interval [-ln(2), ln(2)) is approximately 4.219 units.

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