Figure 2 shows some typical Lorenz curves. They all pass through the points (0,0) and (1, 1) and are concave upward. In the extreme case L(x) = x, society is perfectly egalitarian: the poorest a% of the population receives a % of the total income and so everybody receives the same income. The area between a Lorenz curve y - L(x) and the line y = x measures how much the income distribution differs from absolute equality. The Gini index (sometimes called the Gini coefficient or the coefficient of inequality) is the area between the Lorenz curve and the line y = x(shaded in Figure 3) divided by the area under y = x. (0.8.0.51, 1. (a) Show that the Gini index G is twice the area between the Lorenz curve and the line y = x, that is, 10.4.0.12) G=2 = 2 [[x - L(x)] dx 0 0.2 0.4 0.6 0.8 1 FIGURE 1 Lorenz curve for the US in 2010 (b) What is the value of G for a perfectly egalitarian society (everybody has the same income)? What is the value of G for a perfectly totalitarian society (a single person receives all the income?)

Answer:

f(x) = -3(2^x)

g(x) = -3 + 2x

h(x) = 2(3^x)

j(x) = -3 – 2x

So,

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

f(2) = -3(4)

f(2) = -12

g(-2) = -3 + 2(-2)

g(-2) = -3 -4

g(-2) = -7

h(2) = 2(3^2)

h(2) = 2(9)

h(2) = 18

j(-2) = -3 – 2(-2)

j(-2) = -3 –4

j(-2) = -7

The probability that the flip results in a head is given as Pr[H] = 23. Therefore, the probability that the flip results in a tail is Pr[T] = 1 - Pr[H] = 1 - 23 = 13.

Let A be the event that a white marble is selected. We need to find the conditional probability Pr[H|A], i.e., the probability that the flip resulted in a head given that a white marble was selected.

Using Bayes' theorem, we have:

Pr[H|A] = (Pr[A|H]*Pr[H]) / Pr[A]

Pr[A|H] is the probability of selecting a white marble given that the flip resulted in a head. This is given by (9/25), since there are 9 white marbles out of 25 in the first urn.

Pr[A] is the total probability of selecting a white marble, which can be found using the law of total probability:

Pr[A] = Pr[A|H]*Pr[H] + Pr[A|T]*Pr[T]

= (9/25)*0.23 + (1/11)*0.13

= 0.0888 + 0.0118

= 0.1006

Pr[A|T] is the probability of selecting a white marble given that the flip resulted in a tail. This is given by (1/11), since there is only 1 white marble out of 11 in the second urn.

Therefore,

Pr[H|A] = (9/25 * 0.23) / 0.1006 = 0.6508

Hence, the probability that the flip resulted in a head given that a white marble was selected is 0.6508 (or approximately 0.65).

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Answer:

10 units

Step-by-step explanation:

Here, legs = base and perpendiculars.

So, Clearly given Base = 6 units Perpendicular = 8 cm

Square's Side = Hypotenuse.

By Pythagoras theorem,

H² = B²+P²

H ² = 6²+8²

H² = 36+64 = (10)²

H = 10 units.

Square's Side length = 10 units

Jenny works at Sammy's Restaurant and is paid according to the rates in the following

table.

Jenny's weekly wage agreement

Basic wage $600.00

PLUS

$0.90 for each customer served.

In a week when Jenny serves n customers, her weekly wage, W, in dollars, is given by

the formula

W = 600+ 0.90n.

(i) Determine Jenny's weekly wage if she served 230 customers.

(ii) In a good week, Jenny's wage is $1 000.00 or more. What is the LEAST number

of customers that Jenny must serve in order to have a good week?

(iii) At the same restaurant, Shawna is paid a weekly wage of $270.00 plus $1.50 for

each customer she serves.

If W, is Shawna's weekly wage, in dollars, write a formula for calculating Shawna's

weekly wage when she serves m customers.

(iv) In a certain week, Jenny and Shawna received the same wage for serving the same

number of customers.

How many customers did they EACH serve?

3 / 3

(i) If Jenny serves 230 customers, her weekly wage is

W = 600 + 0.90n = 600 + 0.90(230) = $807.00

Therefore, Jenny's weekly wage if she serves 230 customers is $807.00.

(ii) We want to find the least number of customers, n, that Jenny must serve in order to earn $1,000 or more. That is,

600 + 0.90n ≥ 1,000

0.90n ≥ 400

n ≥ 444.44

Since n must be a whole number, Jenny must serve at least 445 customers in order to earn $1,000 or more in a week.

(iii) Shawna's weekly wage, W, in dollars, when she serves m customers is given by the formula:

W = 270 + 1.50m

Therefore, Shawna's weekly wage when she serves m customers is $270.00 plus $1.50 for each customer she serves.

(iv) Let's assume that Jenny and Shawna received the same wage, W, for serving the same number of customers, x. Then we have:

Jenny's wage = 600 + 0.90x

Shawna's wage = 270 + 1.50x

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

600 + 0.90x = 270 + 1.50x

330 = 0.60x

x = 550

Therefore, Jenny and Shawna each served 550 customers.

According to the provided scenario, if you weigh 160 pounds, then 3 drinks in four hours would make you definitely illegal.

If a person weighs 160 pounds and drinks alcohol at a moderate rate, then after 3 drinks in four hours, their BAC (blood alcohol concentration) would be around 0.08, which is considered legally impaired and definitely illegal. However, it is important to note that this estimate is based on various factors such as the person's gender, age, and metabolism, and can vary from person to person.

Therefore, it is always advisable to drink responsibly and not drive after consuming alcohol.

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The smallest possible values for "a" and "b" that satisfy these conditions are when a = 1 and b = 3. This is because 1 is the smallest factor of 15, and 3 is the smallest multiple of 3.

What is integers?

The Latin term "Integer," which implies entire or intact, is where the word "integer" first appeared. Zero, positive numbers, and negative numbers make up the particular set of numbers known as integers.

Let's call the two unknown numbers as "a" and "b"

Since "a" is a factor of 15, the possible values for "a" are 1, 3, 5, and 15.

Since "b" is a multiple of 3, the possible values for "b" are 3, 6, 9, 12, 15, and so on.

The smallest possible values for "a" and "b" that satisfy these conditions are when a = 1 and b = 3. This is because 1 is the smallest factor of 15, and 3 is the smallest multiple of 3.

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Let's break this problem down step by step.

First, we can work out the weight of one white ball by subtracting the weight of 6 white and 3 orange balls (20 pounds) from the weight of 2 white and 5 orange balls (8 pounds):

Weight of 1 white ball = 8 - 20 = -12

Next, we can work out the weight of one orange ball by subtracting the weight of 2 white and 5 orange balls (8 pounds) from the weight of 6 white and 3 orange balls (20 pounds):

Weight of 1 orange ball = 20 - 8 = 12

Now that we know how much one white and one orange ball weigh, we can work out the weight of 4 white and 4 orange balls together:

Weight of 4 white and 4 orange balls = (4 x -12) + (4 x 12) = 0

Therefore, the weight of 4 white and 4 orange balls together is 0 pounds.

By solving two given simultaneous equations using algebra, we find that the combined weight of 4 white balls and 4 orange balls is 16 pounds.

The subject matter of this question pertains to simultaneous equations. The equations in question are, 2w + 5o = 8 and 6w + 3o = 20, where w stands for the weight of a white ball and o stands for the weight of an orange ball. We can solve these equations to find the individual weights of these balls. After finding these values, we substitute these into a new equation, 4w + 4o, to find the total weight of 4 white and 4 orange balls together. By solving these equations, we find that the weight of 4 white balls and 4 orange balls together is 16 pounds.

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Answer: At 250 miles, plan B is the most cost effective.

Step-by-step explanation:

Answers are under the questions below.

I. In complete sentences, translate each function into a verbal model describing the total cost of the rental in terms of the number of miles that the car is driven.

Plan A will charge .12 per mile plus a one time fee of $75.

Equation is y = .12x + 75 with the $75 being the y intercept or one time fee.

Plan B will charge .35 per mile with no one time fee.

Equation is y = .35x with the y intercept being (0) zero.

II. For each function, determine how the rate of change will affect the total cost of a car rental.

Initially the Plan A will be more expensive because of the one time fee, even though the rate is .12 per mile, which is much less than Plan B.

Plan B charges .35 per mile, but will eventually catch up in cost.

At approximately 326 miles the cost of each plan will equal at $114.13.

After 326 miles, Plan B will cost more than Plan A.

III. For a car rental that will include a maximum of 250 miles for the duration of the rental, which plan is the most cost effective?

Plan A, substitute 250 miles for x

Cost = .12 (250)  + 75

Cost = $105

Plan B, substitute 250 miles for x

Cost = .35 (250)

Cost = $87.50

At 250 miles, plan B is the most cost effective, saving $17.50.

graph is attached.

The 90% confidence interval for the true difference between the mean amounts of weight lost by dieters who supplement with calcium and those who do not is: resulting in a confidence interval of (4.337, 6.263) pounds.

To construct a 90% confidence interval to estimate the true difference between the mean amounts of weight lost by dieters who supplement with calcium and those who do not, we can use a two-sample t-test with unequal variances.

The null hypothesis is that the true difference between the mean amounts of weight lost by the two groups is zero, and the alternative hypothesis is that the true difference is not zero.

We can use the following formula to calculate the confidence interval:

CI = (x_1 - x_2) ± tα/2 * √((s_1)²/n_1 + (s_2)²/n_2)

where

x_1 = 15.6 (mean amount of weight lost by group a)

x_2 = 10.3 (mean amount of weight lost by group b)

s_1 = 1.2 (standard deviation of group a)

s_2 = 1.9 (standard deviation of group b)

n_1 = n_2 = 26 (sample size of both groups)

tα/2 = t0.05/2,24 = 1.711 (degrees of freedom = n_1 + n_2 - 2 = 50)

Plugging in the values, we get:

CI = (15.6 - 10.3) ± 1.711 * √(1.2²/26 + 1.9²/26)

CI = 5.3 ± 0.963

CI = (4.337, 6.263)

Therefore, with 90% confidence, we can estimate that the true difference between the mean amounts of weight lost by dieters who supplement with calcium and those who do not is between 4.337 and 6.263 pounds.

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The value of x in the equation 1/4(4 + x) = 4/3 is x = 4/3.

Multiply both sides of the equation by 4 to eliminate the fraction on the left-hand side:

1/4(4 + x) = 4/3

4 * 1/4(4 + x) = 4 * 4/3

Simplifying:

4 + x = 16/3

Subtract 4 from both sides of the equation:

4 + x - 4 = 16/3 - 4

Simplifying:

x = 16/3 - 12/3

x = 4/3

A fraction is a mathematical concept used to represent a part of a whole or a ratio between two quantities. It is typically written in the form of a numerator (top number) over a denominator (bottom number), separated by a horizontal line. For example, the fraction 1/2 represents one out of two equal parts, or half of a whole. Similarly, the fraction 3/4 represents three out of four equal parts, or three-quarters of a whole.

Fractions are an essential part of mathematics and are used in a wide range of applications, including measurements, cooking, and financial calculations. They can be added, subtracted, multiplied, and divided just like whole numbers, but they require a bit more care in their manipulation due to their unique structure.

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Answer:

9

Step-by-step explanation:

The base of the prism is a right triangle.

The hypotenuse measures 5. One leg measures 4.

The other leg must measure 3 since it is the Pythagorean triple 3, 4, 5.

Area of the base = 3 × 4 / 2 = 6

V = Bh

54 = 6 × h

h = 9

(a) Parameter being estimated in the given information is the difference between two proportions (p_1 - p_2).

(b) A 95% confidence interval is given by  (0.075, 0.225)

(a) The parameter being estimated is the difference between two population proportions, which is denoted by (p_1 - p_2).

(b) The margin of error for a 95% confidence interval is 5%, which means that the critical value of z is 1.96 (obtained from a standard normal distribution table). Using the formula for the margin of error, we can write:

1.96 * √(p_1_hat*(1-p_1_hat)/n_1 + p_2_hat*(1-p_2_hat)/n_2) = 0.05

where p_1_hat and p_2_hat are the sample proportions from the two samples, and n1 and n2 are the sample sizes.

Solving for p_1_hat - p_2_hat, we get:

p1_hat - p2_hat = ±0.075

Since we are interested in a 95% confidence interval, we can subtract and add this value from P1 - P2 to obtain the interval:

P_1 - P_2 ± 0.075

Substituting the given value of P_1 - P_2 = 0.15, we get:

95% Confidence Interval: (0.075, 0.225)

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As per the points A, B, and C are colinear points and the point C lies between points A and B on the line.

Let's begin by finding the distance between points A and B. Using the distance formula equation, we can substitute the values of the x and y coordinates of A and B:

AB = √((0 - 5)² + (5 - (-5))²)

= √(25 + 100)

= √125

= 5√5

Therefore, the distance between points A and B is 5√5.

Similarly, we can find the distance between points B and C:

BC = √((2 - 0)² + (1 - 5)²)

= √4 + 16

= √20

= 2√5

Finally, we can find the distance between points A and C:

AC = √((2 - 5)² + (1 - (-5))²)

= √9 + 36

= √45

= 3√5

Alternatively, we can use the equation of the line passing through any two of these points and check if the third point lies on that line.

Let's use the points A and B to find the equation of the line passing through them:

y - (-5) = ((5 - (-5)) / (0 - 5))(x - 5)

y + 5 = (10 / (-5))(x - 5)

y + 5 = -2(x - 5)

y + 5 = -2x + 10

y = -2x + 5

Now, let's check if point C lies on this line by substituting its coordinates into the equation:

1 = -2(2) + 5

1 = 1

Since the equation is true, we can conclude that points A, B, and C are collinear. Moreover, since point C lies between points A and B on the line, we can say that C lies on segment AB.

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Complete Question:

Given points A, B, and C. Find AB, BC, and AC. Are A, B, and C collinear? If so, which point lies between the other two? A(5, −5), B(0,5), C(2, 1)

The subject with the maximum average is Mathematics, while the subject with the minimum average is Social studies; moreover, there is a difference of 6 between Hindi and English.

Based on the graph and the values given about each subject, it can be concluded that the subject with the maximum average is Mathematics which has an average of 91, while the subject with the minimum average is social studies with an average of 55.

English average: 62

Hindi average: 56

62 - 56 = 6

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Answer:

-9

Step-by-step explanation:

x = -5

3x + 6

Since x = -5..

Do this

3(-5) + 6

Perform

-15 + 6

Answer: -9

Therefore, when x is equal to -5, the value of 3x + 6 is -9.

An equation is a statement that expresses the equality of two mathematical expressions using mathematical symbols such as variables, numbers, and mathematical operations. The equality is represented by an equal sign "=" between the two expressions. Equations are used to represent mathematical relationships and solve problems in various fields such as physics, chemistry, engineering, and economics.

Given by the question.

3x + 6 = 3(-5) + 6

= -15 + 6

= -9

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2/7 + 1/4 = 15/28 ≅ 0.5357143

Add: 2/7 + 1/4 = 2 · 4/7 · 4 + 1 · 7/4 · 7 = 8/28 + 7/28 = 8 + 7/28 = 15/28 

It is suitable to adjust both fractions to a common (equal, identical) denominator for adding, subtracting, and comparing fractions. The common denominator you can calculate as the least common multiple of both denominators - LCM(7, 4) = 28. It is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 7 × 4 = 28. In the following intermediate step, it cannot further simplify the fraction result by canceling.In other words - two sevenths plus one quarter is fifteen twenty-eighths.

All parts are define in the below points.

A random sample is a subset of a population in which each individual or element in the population has an equal chance of being selected. It is a sampling method used in statistics and research to minimize bias and increase the generalizability of the findings to the larger population.

a. Hypotheses: Null Hypothesis: The mean DNA concentration of the ocean floor specimens is not significantly different from the healthy concentration (µ = 0.31g/m2). Alternative Hypothesis: The mean DNA concentration of the ocean floor specimens is significantly different from the healthy concentration (µ ≠ 0.31g/m2). Using a two-tailed t-test with alpha = 0.05, we find a p-value of 0.0003, which is less than the significance level. Therefore, we reject the null hypothesis and conclude that the mean DNA concentration of the ocean floor specimens is significantly different from the healthy concentration.

b. We would perform a one-tailed t-test with the alternative hypothesis that the mean DNA concentration is less than 0.31g/m2 if the goal was to determine whether the mean DNA concentration of the floor specimens was lower than what is regarded as a healthy concentration. It would have a p-value of 0.00015.

c. Using the 95% confidence interval approach, we construct a one-sided confidence interval for the mean DNA concentration. If the lower bound of the confidence interval is less than 0.31g/m2, we can conclude that the mean DNA concentration is less than the healthy concentration. The 95% confidence interval for the mean is (0.2457g/m2, 0.3105g/m2), which does not include the healthy concentration of 0.31g/m2. Therefore, we can conclude that the mean DNA concentration of the ocean floor specimens is less than the healthy concentration.

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a). We discover a p-value of 0.0003 using a two-tailed t-test with alpha = 0.05, which is below the significance level.

b). Its p-value would be 0.00015.

c). The safe concentration of [tex]0.31g/m^2[/tex] is not included in the 95% confidence interval for the mean, which is [tex](0.2457g/m^2,\ 0.3105g/m^2)[/tex].

A random sample is a portion of a community in which every person or component has an equal chance of being chosen. In statistics and research, it is a sampling technique used to reduce bias and improve the generalizability of the results to a broader population.

A). An hypothesis is a The null hypothesis states that there is no discernible difference between the mean DNA concentration of the ocean bottom samples and the healthy concentration [tex](\mu=0.31g/m^2)[/tex]. Alternative Hypothesis: The mean DNA concentration of the ocean floor samples differs considerably from the healthy concentration [tex](\mu\neq 0.31g/m^2)[/tex] in a statistically significant way. We discover a p-value of 0.0003 using a two-tailed t-test with alpha = 0.05, which is below the significance level. We therefore reject the null hypothesis and come to the conclusion that the mean DNA concentration of the samples from the ocean bottom differs significantly from that of healthy individuals.

B). If the objective was to determine whether the mean DNA concentration of the floor specimens was lower than what is considered as a healthy concentration, we would conduct a one-tailed t-test with the alternative hypothesis that the mean DNA concentration is less than [tex]0.31g/m^2[/tex]. Its p-value would be 0.00015.

C). We create a one-sided confidence interval for the mean DNA concentration using the 95% confidence interval method. The mean DNA concentrationis less than the healthy concentration if the lower limit of the confidence interval is less than [tex]0.31g/m^2[/tex]. The safe concentration of [tex]0.31g/m^2[/tex] is not included in the 95% confidence interval for the mean, which is [tex](0.2457g/m^2,\ 0.3105g/m^2)[/tex]. As a result, we can say that the average DNA concentration of the samples from the ocean bottom is lower than the healthy concentration.

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The value of sec A as required to be determined in the task content is; -√13 / 2.

As evident from the task content; it follows that the terminal ray of angle A, drawn in standard position, passes through the point (-4, -6).

Therefore, the length that the line from the origin to A has length;

L = √((-4)² + (-6)²)

L = √52.

On this note, it follows that the value of sec A which is represented by; hypothenuse/ adjacent is;

sec (A) = -√52 / 4

sec (A) = -√13 / 2.

Ultimately, the value of sec (A) as required is; -√13 / 2.

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The probability of someone returning exactly 1 of the 3 packages can be calculated as:P(1 out of 3 packages is returned) = C(3, 1) × P(0 or 1 diskette is defective)¹ × (1 - P(0 or 1 diskette is defective))²P(1 out of 3 packages is returned) = C(3, 1) × (0.9043820371)¹ × (0.0956179629)²P(1 out of 3 packages is returned) = 0.2448700124Therefore, the required probability of someone returning exactly 1 of the 3 packages is 0.2448700124.

The given data from the question is that the company produces diskettes which have the probability of being defective as 0.01. The packages that are sold have a size of 10 and the guarantee says that there can be at most one defective diskette in the package. Now, the question is to find the probability of someone returning exactly 1 of the 3 packages that they have bought.  So, the given data can be summarized as:Given:Probability of the diskette being defective, p = 0.01Guarantee: At most one diskette in the package of size 10 is defective.Now, let's solve the problem using probability theory

Probability of 1 diskette being defective in a package of size 10 can be calculated as:P(defective) = p = 0.01P(non-defective) = 1 - p = 0.99Using the given guarantee, probability of at most one defective diskette in a package of size 10:P(0 or 1 diskette is defective) = P(0 defective) + P(1 defective)P(0 or 1 diskette is defective) = C(10, 0) × (0.99)¹⁰ + C(10, 1) × (0.99)⁹ × (0.01)P(0 or 1 diskette is defective) = 0.9043820371Using the above probability

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In set theory, two sets are considered equal if they have the same elements. In this case, A is a set containing the elements 1, 2, and 3, while B is an empty set (also known as the null set),

A contains three distinct elements, and B contains none, we can conclude that A and B are not equal, i.e., A is not equal to B.

A ≠ B

Set theory is a branch of mathematics that studies collections of objects, called sets, and the relationships between them. A set is defined as a well-defined collection of distinct objects, which can be anything from numbers and letters to more abstract concepts like functions and geometrical shapes. The set theory provides a foundation for other areas of mathematics, including algebra, topology, and logic.

One of the fundamental concepts of set theory is the notion of membership, which states that an object either belongs to a set or does not. Sets can also be combined through operations such as union, intersection, and complementation, and the relationships between sets can be represented using Venn diagrams.

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3

Step-by-step explanation:

you can see the ratio of flour to sugar is 3:1 so for every 3 cups of flour is 1 cup of sugar so if you have 5 sugar cups you must have 3 times that of flour so 5×3=15 so 15 cups

Answer:

Step22-by-step explanation:

In total, the knockout stage of the UEFA champions league includes players from 79 different nationalities.

The UEFA champions league is a very popular soccer tournament played in Europe. This tournament includes 16 different teams in the knockout stage. Some of the most popular teams are Manchester City, Liverpool, Napoli, and Real Madrid.

Moreover, even if there are only 16 teams, the players in them are from many different nationalities. Indeed, this year there are players from 79 different countries.

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a) The total number of marbles is 6 + 12 = 18.P(a red marble) = 6/18 = 1/3

b) There are 12 odd-numbered marbles in total. P(an odd-numbered marble) = 12/18 = 2/3

The probability that the marble drawn from the jar is red can be found using the formula for probability. The probability formula can be written as the ratio of the number of favorable outcomes to the total number of possible outcomes. P(a red marble) = number of red marbles / total number of marbles. In this case, there are 6 red marbles and 12 blue marbles in the jar. Therefore, the total number of marbles is 6 + 12 = 18.P(a red marble) = 6/18 = 1/3(b) The probability that the marble drawn from the jar is odd-numbered can be found using the formula for probability. The probability formula can be written as the ratio of the number of favorable outcomes to the total number of possible outcomes. P(an odd-numbered marble) = number of odd-numbered marbles / total number of marbles. In this case, there are 6 red marbles numbered 1 to 6 and 12 blue marbles numbered 1 to 12 in the jar. Therefore, the total number of marbles is 6 + 12 = 18.To find the number of odd-numbered marbles, we need to count the number of red and blue marbles numbered 1, 3, 5, 7, 9, 11. There are 6 odd-numbered red marbles and 6 odd-numbered blue marbles. Therefore, there are 12 odd-numbered marbles in total. P(an odd-numbered marble) = 12/18 = 2/3

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Roy purchased whole pizzas for P 190.00 each. To determine the number of whole pizzas he bought, we can divide the total cost of the pizzas by the cost of each pizza. Therefore, the calculation P 950.00 / P 190.00 results in 5, indicating that Roy bought five whole pizzas.

Roy spent a total of P 1070 for pizza with 3 sets of additional toppings. Since each set of additional toppings costs P 40.00, then the total cost of the toppings is 3 x P 40.00 = P 120.00. Subtracting this from the total amount spent gives us P 950.00, which is the cost of the pizzas alone.

Since each whole pizza costs P 190.00, we can divide the cost of the pizzas by the cost of each pizza to find the number of whole pizzas Roy bought. Therefore, P 950.00 / P 190.00 = 5.

Thus, Roy bought 5 whole pizzas.

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Answer:

-0.5x[tex]x^{2}[/tex]+3x=0

Step-by-step explanation:

Answer:

Step-by-step explanation:

Given:

Side a and Side b are 6 and 5.

The angle C is 131.

This is an obtuse scalene triangle as identified.

Area = ab * sin (C)/2  = 11.32064

The player should be willing to pay up to $1.33 to play this game and not lose money in the long run.

The expected value is the sum of the products of each possible outcome and its probability. Let's calculate the expected value of the game:

E(X) = (1/6) * $4 + (5/6) * (-$2)

E(X) = $0.67

This means that on average, the player can expect to win $0.67 per game. Since it costs $2 to play, the player should not be willing to pay more than $2 - $0.67 = $1.33 to play the game and not lose money in the long run.

Probability theory is based on axioms, which are basic assumptions about the nature of probability. It is used to quantify uncertainty and to make predictions based on the available information. Probability is expressed as a number between 0 and 1, with 0 meaning an event is impossible, and 1 meaning an event is certain.

The concept of probability is used in a variety of fields, including statistics, economics, engineering, and physics. In statistics, probability is used to model random variables, estimate parameters, and test hypotheses. In economics, probability is used to model financial risks and decision-making under uncertainty. In engineering and physics, probability is used to model complex systems and predict the behavior of particles.

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Answer:

Let's use the formula for the volume of a right circular cone to solve this problem:

V = (1/3)πr^2h

We are given that the radius is increasing at a rate of 1.4 in/s and the height is decreasing at a rate of 2.6 in/s. We want to find the rate at which the volume is changing when the radius is 144 in. and the height is 138 in. In other words, we want to find dV/dt when r = 144 and h = 138.

Using the chain rule of differentiation, we can express the rate of change of the volume as follows:

dV/dt = (dV/dr) (dr/dt) + (dV/dh) (dh/dt)

To find dV/dr and dV/dh, we differentiate the formula for the volume with respect to r and h, respectively:

dV/dr = (2/3)πrh

dV/dh = (1/3)πr^2

Substituting the given values and their rates of change, we have:

dV/dt = (2/3)π(144)(138)(1.4) + (1/3)π(144)^2(-2.6)

dV/dt = 55,742.4 - 1,994,598.4

dV/dt = -1,938,856 in^3/s

Therefore, when the radius is 144 in. and the height is 138 in., the volume of the cone is decreasing at a rate of approximately 1,938,856 cubic inches per second.

Step-by-step explanation:

Proved that the sum of measure of three angles of triangle is 180 using the Polygon Angle Sum Theorem

To prove that the sum of the measures of three angles of a triangle is 180 degrees, we can use the Polygon Angle Sum Theorem, which states that the sum of the measures of the interior angles of a polygon with n sides is equal to (n-2) times 180 degrees.

A triangle is a polygon with three sides, so we can apply the Polygon Angle Sum Theorem to a triangle to find the sum of its interior angles. Using n=3, we have:

Sum of measures of interior angles of triangle = (n-2) × 180 degrees

= (3-2) × 180 degrees [since we are dealing with a triangle]

= 1 × 180 degrees

= 180 degrees

Therefore, the sum of the measures of the interior angles of a triangle is 180 degrees. This means that the sum of the measures of the three angles in a triangle is always 180 degrees, regardless of the size or shape of the triangle.

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