a pair of fair dice is tossed. define the following events: a: 5you will roll a 7 1i.e., the sum of the numbers of dots on the upper faces of the two dice is equal to 7).6 b: 5at least one of the two dice is showing a 4.6 a. identify the sample points in the events a, b, a b, a b, and ac . b. find p1a2, p1b2, p1a b2, p1a b2, and p1ac 2 and by summing the probabilities of the appropriate sample points. c. use the additive rule to find p1a b2 . compare your answer with that for the same event in part b. d. are a and b mutually exclusive? why?

The given piecewise function has a range of:

Range = {46, 48, 50, 52, 54, 56}.

The range of a function is the set of all possible output values, or the set of all y-coordinates that correspond to the x-coordinates in the domain of the function.

It is, in other words, the entire set of values that the function is capable of returning as its result.

From the given piecewise function, we can see that the y-coordinates of the function are limited to the values 46, 48, 50, 52, 54, and 56.

These values correspond to the y-intercepts of each of the line segments in the graph.

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This is because when hawks are extremely rare, their payoff is higher than that of doves regardless of the value of c. Therefore, there is no point at which their payoffs will be equal.

The mathematical field of game theory helps shed light on how it emerges. Game theory is “the study of mathematical models of strategic interaction among rational decision-makers” (according to Wikipedia).

Game theory applies to “games” as varied as economics, politics, chess, and tic-tac-toe. In each case, there are some rules, some “players” or “agents”, and a set of strategies available to them.

Each player has a concept of “utility” – a “currency” they seek to individually maximize through the strategies they play.

The currency of evolution is the concept of fitness.

That is, the chance of being represented in the next generation. Genes and traits which increase the odds of survival to reproductive age are more likely to be passed on to future generations. Therefore, they confer a greater fitness to the individual which "hosts” them.

The evolutionary game theory takes the concepts from game theory and applies them in an evolutionary context.

In the hawk-dove game, at what value of c do doves and hawks have equal payoffs when hawks are extremely rare? The answer is that there is no such value. This is because when hawks are extremely rare, their payoff is higher than that of doves regardless of the value of c. Therefore, there is no point at which their payoffs will be equal.

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The Toy Train Collectors' Club has 140 members who are 42 years of age or younger.

To find the number of members who are 42 years of age or younger, we need to look at the bars in the histogram that represent these age groups. Specifically, we need to add up the heights of the bars that represent members aged 0-42 years old.

The histogram shows that there are 30 members in the age group 0-10,

50 members in the age group 11-20,

40 members in the age group 21-30,

20 members in the age group 31-40, and

15 members in the age group 41-50.

To find the number of members who are 42 years of age or younger, we need to add up the heights of the bars that represent members aged 0-42 years old, which is 30+50+40+20 = 140.

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The two quantities needed to determine the total amount of money Monte paid his employees this month are:

A. the hourly salary of each employee

C. the number of hours each employee worked this month.

With four employees at the Monte Bakery, who earn a different hourly wages, to work out the total salary for the month, Monte needs to multiply the hourly wage of each employee by the total number of hours each worked.

Thus, in this mathematical operation, we first use multiplication and then add each individual employee's total salary to sum the total salary for the month.

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Answer:

b

Step-by-step explanation:

Answer:

[tex]r = \sqrt{ {3}^{2} + {( - 7)}^{2} } = \sqrt{9 + 49} = \sqrt{58} [/tex]

[tex] {x}^{2} + {y}^{2} = 58[/tex]

The research company can then conduct an appropriate statistical test, such as a t-test or an ANOVA, to analyze the data and determine whether to accept or reject the null hypothesis. If the null hypothesis is rejected, it would imply that there is a significant difference in the amount of vitamin D between the two types of multi-vitamins.

To examine if there is a difference in the amount of vitamin D in a brand name multi-vitamin and generic brand multi-vitamin, the appropriate hypotheses for the research company would be as follows:

Null Hypothesis (H0): There is no significant difference in the amount of vitamin D in the brand name multi-vitamin and the generic brand multi-vitamin. In other words, the mean amount of vitamin D in both types of multi-vitamins is equal.

Alternative Hypothesis (H1): There is a significant difference in the amount of vitamin D in the brand name multi-vitamin and the generic brand multi-vitamin. This means that the mean amount of vitamin D in one type of multi-vitamin is not equal to the other.

The research company can then conduct an appropriate statistical test, such as a t-test or an ANOVA, to analyze the data and determine whether to accept or reject the null hypothesis. If the null hypothesis is rejected, it would imply that there is a significant difference in the amount of vitamin D between the two types of multi-vitamins.

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The possible number of pounds of candy that Linda will buy can be shown as p > 5 .

Let's use algebra to solve the problem. We know that Linda will spend more than $30 on candy, so we can write:

6p > 30

Dividing both sides by 6, we get :

p > 5

This means that Linda must buy more than 5 pounds of candy in order to spend more than $30. And this can be shown by the expression, p > 5.

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The required, graph of both lines has been shown where the common solution is (0, 2).

I apologize for the error in my previous solution. Here's the corrected solution:

To solve the system of linear equations x + 2y = 4 and -2x + 5y = 10 graphically, we need to plot the graphs of the two equations on the same set of axes and find the point where they intersect.

The graph of both lines has been shown where the common solution is (0, 2).

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The solution of the system of equations. -2x-y=6 over -2x+8y=-39 is the ordered pair (-0.5, -5).

In order to to graph the solution to the given system of equations on a coordinate plane, we would use an online graphing calculator to plot the given system of equations and then take note of the point of intersection;

-2x-y=6     ......equation 1.

-2x+8y=-39  ......equation 2.

In this exercise, we would use an online graphing calculator to plot the given system of equations as shown in the graph attached below.

Based on the graph shown in the image attached below, we can logically deduce that the solution to this system of equations is the point of intersection of the lines on the graph representing each of them, which is given by the ordered pair (-0.5, -5).

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Answer:

24 slices

Step-by-step explanation:

There are 3 whole pizzas. He cuts each of them into eighths. That means eight slices. 3 times 8 is 24. There are a total of 24 slices. This can also look like this: 24/8   or 24 over 8.  Which also equals to 3.

The Central Limit Theorem can also be used to investigate unusual events. An unusual event is one that occurs with a probability of less than 1%

The Central Limit Theorem can be used to investigate unusual events by calculating the probability of a sample mean being a certain number of standard deviations away from the population mean.

If we assume that the population is normally distributed, then we can use the normal distribution to calculate the probability of observing a sample mean that is a certain number of standard deviations away from the population mean.

An unusual event is typically defined as an event that occurs with a low probability, usually less than 5% or 1%. So, if we observe a sample mean that is more than 2 standard deviations away from the population mean, we can say that this is an unusual event that occurs with a probability of less than 5%. Similarly, if we observe a sample mean that is more than 3 standard deviations away from the population mean, we can say that this is an unusual event that occurs with a probability of less than 1%.

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The correct option is (C) Triangle HJK is congruent to triangle NLM

A triangle is a closed, two-dimensional geometric figure with three sides and three angles. It is the simplest polygon and is formed by connecting three non-collinear points with line segments.

Congruent is a term used in geometry to describe two or more geometric figures that have the same shape and size. If two geometric figures, such as triangles or line segments, are congruent, it means that they are identical in shape and size, and they can be superimposed onto each other.

According to the given information:

In Triangle HKJ

Angle H is 58 degrees.

Angle J is 62 degrees.

Angle K is 60 degrees

In Triangle LMN (Arranging in the same order as the angles above)

Angle N is 58 degrees.

Angle L is 62 degrees.

Angle M is 60 degrees.

Therefore, we can say that:

Triangle HJK is congruent to Triangle NLM.

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The way that data sets can be used to compare two data

When attempting to compare two sets of data, there are numerous ways one can utilize visual representations to help recognize similarities and variances between the two. Here are some common examples:

Measures of center: Data displays such as box plots or histograms can be employed to quickly display measures of central tendency – like the mean, median, and mode – then contrast them amongst the two datasets.

Measures of variability: To compare two collections of data in terms of their measures of variability – including range, interquartile range, and standard deviation

Trends: Scatterplots or line graphs can be used to discern trends among the data and observe distinguishing characteristics between the two sets.

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To cover the entire area of pyramid, Serena would need approximately 63.36 square inches of paper.

The area of the square base is

4.8 in x 4.8 in = 23.04 sq in

The area of each triangular face is

(1/2) x base x height = (1/2) x 4.8 in x 4.2 in = 10.08 sq in

Since there are four triangular faces, the total area of the triangular faces is

4 x 10.08 sq in = 40.32 sq in

Therefore, the total surface area of the pyramid is

23.04 sq in + 40.32 sq in = 63.36 sq in

So, it would take 63.36 square inches of paper to cover the entire pyramid.

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The value of y for the inverse variation when x = -3 is derived to be equal to -8, and the equation that relating x and y is: y = 24/x.

Inverse variation is a mathematical relationship between two variables, in which an increase in one variable leads to a proportional decrease in the other variable. Mathematically, inverse variation can be expressed as y = k/x, where y and x are the two variables, k is a constant of proportionality, and the product of y and x is always equal to k.

when x = -4 and y = -6, then k is derived as:

-6 = k/-4

k = 24 {cross multiplication}

equation relating x and y is:

y = 24/x

when x = -3, y is derived as:

y = 24/-3

y = -8

Therefore, the value of y for the inverse variation when x = -3 is derived to be equal to -8, and the equation that relating x and y is: y = 24/x.

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Answer: $3.75

Step-by-step explanation:

Since apples cost $1.25 per pound and she needs to buy 3 pounds, the total cost would be $1.25*3 = $3.75.

The domain of the function g(4) is equal to 16, which makes the option (1) correct.

A function is composite when the co- domain of the first mapping is the domain of the second mapping

We shall evaluate the domain of the function g(4) as follows:

Given g(x) = 5f(x) +1, from the table of values for f(x), f(4) = 3 so;

g(4) = 5(f(4)) + 1

g(4) = 5(3) + 1

g(4) = 15 + 1

g(4) = 16

Therefore, the domain of the function g(4) is derived to be equal to 16.

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Heidi's rate of elimination is also decreasing over time, but at a slower rate than Clara's which decreases exponentially over time

Given data ,

Let the exponential equation be represented as A

Now , the value of A is

f(t) = 300(0.946)^t

For Clara, we can calculate the rate of elimination by finding the difference in the amount of medicine present between two consecutive time points, and dividing by the time elapsed:

From t=1 to t=2: f(2) - f(1) = 223.73 - 236.5 = -12.77 mg

Rate of elimination = -12.77 mg / (2-1) hours = -12.77 mg/hour

From t=2 to t=3: f(3) - f(2) = 211.65 - 223.73 = -12.08 mg

Rate of elimination = -12.08 mg / (3-2) hours = -12.08 mg/hour

From t=3 to t=4: f(4) - f(3) = 200.22 - 211.65 = -11.43 mg

Rate of elimination = -11.43 mg / (4-3) hours = -11.43 mg/hour

From t=4 to t=5: f(5) - f(4) = 189.41 - 200.22 = -10.81 mg

Rate of elimination = -10.81 mg / (5-4) hours = -10.81 mg/hour

Hence , this expression tells us that the rate of elimination for Heidi is proportional to the amount of medicine in her body at any given time, and decreases exponentially over time as the amount of medicine decreases

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The graph of this function would be a straight line passing through the point (0, c) with a slope of m.

What is the function?

A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.

If the function increases at a constant rate of m and has a y-intercept of (0, c), then its equation can be written as:

f(x) = mx + c

The graph of this function would be a straight line passing through the point (0, c) with a slope of m.

Here's an example of the graph of f(x) = 2x + 3:

Attachment

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Knowing that t.05 = 1.812 when df = 10 helps us to determine the critical value of t for hypothesis testing in this scenario.

If we know that t.05 = 1.812 when df = 10, then it is correct to say that the critical value of t at the 5% significance level for a two-tailed test with 10 degrees of freedom is 1.812.

The t-distribution is a family of probability distributions that depend on the sample size and the degrees of freedom (df). The degrees of freedom represent the number of independent observations available to estimate the population parameters. In this case, when df = 10, the t-distribution has thicker tails compared to the standard normal distribution.

The value t.05 represents the critical value of t at the 5% significance level (or alpha level) in a two-tailed test. This means that if the calculated t-value falls outside the range of -t.05 to +t.05, then the null hypothesis can be rejected.

Therefore, knowing that t.05 = 1.812 when df = 10 helps us to determine the critical value of t for hypothesis testing in this scenario.

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This means that for a t-distribution with 10 df, 95% of the data lies within ±1.812 standard deviations from the mean.

If we know that t.05= 1.812 when df = 10,

then it is correct to say that the probability of obtaining a t-value greater than

1.812 or less than -1.812 is 0.05,

assuming a two-tailed test and a significance level of 0.05.

This means that if we calculate the t-value for a sample with 10 degrees of freedom and the calculated t-value falls

outside the range of -1.812 to 1.812,

we can reject the null hypothesis at the 0.05 significance level.

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The mean of a binomial probability distribution with p = 0.32 and n = 150 is 48

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success, denoted by p. In the case of a binomial distribution with n trials and probability of success p, the mean, or expected value, is equal to the product of the number of trials and the probability of success, which is np.

In this case, the problem provides the values of p and n, which are p = 0.32 and n = 150, respectively. Therefore, the mean can be calculated by multiplying these two values

μ = np = 150 x 0.32 = 48

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The population N(t) at any time t≥0 is given by the function N(t) = 100t^2 + 800t + 1332.

a) To find the population after 14 days, we need to integrate the given rate function from 0 to 14:

N(14) - N(0) = ∫(0 to 14) N'(t) dt

N(14) - 1500 = ∫(0 to 14) [200(t+2) - 12] dt

N(14) - 1500 = [100[tex]t^2[/tex] + 800t - 168] evaluated from t=0 to t=14

N(14) = 1500 + [100(14[tex])^2[/tex] + 800(14) - 168] - [100(0[tex])^2[/tex] + 800(0) - 168]

N(14) = 1500 + 17,512

N(14) = 19,012

So the population after 14 days is 19,012 cells.

To find the population after 34 days, we can use the same method:

N(34) - N(0) = ∫(0 to 34) N'(t) dt

N(34) - 1500 = ∫(0 to 34) [200(t+2) - 12] dt

N(34) - 1500 = [100[tex]t^2[/tex] + 800t - 168] evaluated from t=0 to t=34

N(34) = 1500 + [100(34[tex])^2[/tex]+ 800(34) - 168] - [100(0[tex])^2[/tex] + 800(0) - 168]

N(34) = 1500 + 116,768

N(34) = 118,268

So the population after 34 days is 118,268 cells.

b) To find the population N(t) at any time t≥0, we can integrate the given rate function from 0 to t:

N(t) - N(0) = ∫(0 to t) N'(x) dx

N(t) - 1500 = ∫(0 to t) [200(x+2) - 12] dx

N(t) = 1500 + ∫(0 to t) [200(x+2) - 12] dx

N(t) = 1500 + [100[tex]x^2[/tex] + 800x - 168] evaluated from x=0 to x=t

N(t) = 1500 + 100[tex]t^2[/tex] + 800t - 168

N(t) = 100t^2 + 800t + 1332

So the population N(t) at any time t≥0 is given by the function N(t) = 100[tex]t^2[/tex]+ 800t + 1332.

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a) The population after 14 days is 22,788 cells, and the population after 34 days is 126,056 cells.

b) The population at any time t≥0 is given by the equation N(t) = [tex]100t^2[/tex] + 388t + 1500, where t is measured in days and N(t) is the number of cells in the culture at time t.

a) To find the population after 14 days, we can use the given rate equation to calculate the growth rate at t=14 as follows:

N'(14) = 200(14+2) - 12 = 2808 cells/day

Then we can use the following formula to calculate the population after 14 days:

N(14) = N(0) + ∫N'(t) dt, from t=0 to t=14

N(14) = 1500 + ∫(200(t+2) - 12) dt, from t=0 to t=14

N(14) = 1500 + [tex][100t^2 + 400t - 12t][/tex]from t=0 to t=14

N(14) = 1500 + [tex][100(14)^2 + 400(14) - 12(14)] - [100(0)^2 + 400(0) - 12(0)][/tex]

N(14) = 1500 + 21288

N(14) = 22788 cells

Similarly, to find the population after 34 days, we can use the given rate equation to calculate the growth rate at t=34 as follows:

N'(34) = 200(34+2) - 12 = 6772 cells/day

Then we can use the same formula as above to calculate the population after 34 days:

N(34) = N(0) + ∫N'(t) dt, from t=0 to t=34

N(34) = 1500 + ∫(200(t+2) - 12) dt, from t=0 to t=34

N(34) = [tex]1500 + [100t^2 + 400t - 12t] from t=0 to t=34[/tex]

N(34) =[tex]1500 + [100(34)^2 + 400(34) - 12(34)] - [100(0)^2 + 400(0) - 12(0)][/tex]

N(34) = 1500 + 124556

N(34) = 126056 cells

b) To find the population N(t) at any time t≥0, we can use the same formula as above:

N(t) = N(0) + ∫N'(t) dt, from t=0 to t=t

N(t) = 1500 + ∫(200(t+2) - 12) dt, from t=0 to t=t

N(t) = [tex]1500 + [100t^2 + 400t - 12t][/tex] from t=0 to t=t

N(t) = [tex]1500 + 100t^2 + 388t[/tex]

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Answer:

4 -4

Step-by-step explanation:

The function f(x) that has a local minimum at (0,0) and a local maximum at (1,3) is f(x) = 2x³ - 3x².

From the information, we can set up a system of equations as follows:

f'(x) = 3ax² + 2bx + c = 0 (at x=0)

f'(x) = 3ax² + 2bx + c = 0 (at x=1)

Plugging in x=0, we get:

f'(0) = c = 0

Plugging in x=1, we get:

f'(1) = 3a + 2b + c = 0

3a + 2b = -c = 0

3a + 2b = 0 (since c=0)

Solving the system of equations, we get:

a = 2

b = -3

c = 0

d = 0

The function is f(x) = 2x³ - 3x²

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Answer:

Step-by-step explanation:

To determine which function has the greatest y-intercept, we need to look at the constant term, which represents the y-intercept, of each function. The function with the largest constant term will have the greatest y-intercept.

For example, consider the following three functions:

1. f(x) = 2x + 5

2. g(x) = 3x - 7

3. h(x) = -4x + 10

The constant term for each function is 5, -7, and 10 respectively. Therefore, h(x) has the greatest y-intercept of 10, since its constant term is larger than those of f(x) and g(x).

option 1 is correct , trigonometric ratio to solve for the missing side is sine.

we have an angle of 54 degrees and a hypotenuse of 12 units. We can use trigonometric ratios to solve for the missing side, which is the perpendicular side.

In a right triangle, the three primary trigonometric ratios are:

Sine (sin): the ratio of the length of the side opposite the angle to the hypotenuse.

Cosine (cos): the ratio of the length of the side adjacent to the angle to the hypotenuse.

Tangent (tan): the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Given that we know the hypotenuse (12 units) and the angle (54 degrees), we can use the sine ratio to solve for the length of the perpendicular side. The sine ratio is defined as:

sin(x) = opposite/hypotenuse

Plugging in the given values:

sin(54 degrees) = opposite/12

To solve for the length of the perpendicular side (opposite), we can rearrange the equation:

opposite = sin(54 degrees) * 12

Using a calculator, we can find the value of sin(54 degrees) and multiply it by 12 to find the length of the perpendicular side.

therefore, option 1 is correct , trigonometric ratio to solve for the missing side is sine.

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The solution to the initial value problem is [tex]y = (3 - e^x)/(e^x - 1)[/tex]. The domain of this solution is all values of x except x = ln(3).

A method for resolving specific kinds of first-order ordinary differential equations is called variable separation. It entails moving the equation's terms so that those involving the independent variable x are on the other side and those involving the dependent variable y are on the one side. In order to arrive at a general solution, we can then integrate both sides with respect to their respective variables. The name of the method comes from the fact that the variables on either side of the equation are separated.

The given differentiation is given as:

[tex]dy/dx = 9e^{x - y}[/tex]

Separating the variables we have:

[tex]dy/(9e^{x - y}) = dx[/tex]

Now, integrating on both sides we have:

[tex]\int dy/(9e^{x - y}) = \int dx[/tex]

Using partial fraction decomposition we have:

[tex]\int [1/(3-y) - 1/(3e^{x-y})] dy = x + C[/tex]

Integrating each term we have:

[tex]ln|3-y| - ln|3e^{x-y}| = x + C\\ln|3-y| - ln (\|y-3e^x\| = x + C\\ln|(3-y)/(y-3e^x)| = x + C\\(3-y)/(y-3e^x) = ke^x[/tex]

, where k is a constant of integration

We can solve for y:

[tex]y = (3ke^x + 3)/(ke^x + 1)[/tex]

Now, for the initial condition y = 4 and x = 0 we have:

[tex]4 = (3k + 3)/(k + 1)\\4k + 4 = 3k + 3\\k = -1[/tex]

Hence, the solution to the initial value problem is [tex]y = (3 - e^x)/(e^x - 1)[/tex].

Now, the domain of this solution is all values of x except x = ln(3)

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