30 POINTS!!!!! What is the domain for each function below?
Picture included below!
-For the 1st equation, the options are the greater than/equal to symbol and the less than/equal to symbol, and then 3, -3, and 0.
-For the 2nd equation, the options are the greater than/equal to symbol and the less than/equal to symbol, and 0 and 3.

Determining the 3PL with the most distribution space requires specific data on the size and capacity of each company's facilities.

The 3PL identified as having the most distribution (warehousing) space in square feet cannot be determined without specific information or data.

There are many asset-based 3PLs in the logistics industry, and their distribution space can vary significantly based on factors such as company size, industry focus, geographic coverage, and investments in facilities.

Without specific data on the distribution space of each asset-based 3PL, it is not possible to determine which one has the most square footage.

Asset-based 3PLs are companies that own and operate their own assets, such as warehouses, trucks, and equipment, to provide logistics and supply chain services.

These companies often make significant investments in their facilities to ensure efficient storage and distribution of goods for their clients.

Some large 3PL providers may have extensive warehousing networks and substantial distribution space, while smaller or specialized providers may have more focused or limited warehouse capacities.

Therefore, determining the 3PL with the most distribution space requires specific data on the size and capacity of each company's facilities.

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There is an 88% chance that he will complete his next free throw.

Now, When Steph Curry has an 88% free throw rating, that means on average he makes 88 free throws out of 100 attempts.

Now, let's look at the probability of him making 16 free throws in a row.

Since each free throw is independent of the others, the probability of making 16 in a row is simply 0.88 to the power of 16

since he has an 88% chance of making each one).

That comes out to about, 0.284, or 28.4% chance of making 16 in a row.

Therefore, the probability that he completes his next free throw after making 15 in a row is still 88%,

since each free throw is independent of the others.

So, there is an 88% chance that he will complete his next free throw.

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After considering the given data we come to the conclusion that the vertex for the given quadratic equation is  (-1,-4).

Here, the vertex form of a quadratic function is represented by f (x) = a(x - h)² + k,
Here
(h, k) = vertex of the parabola .
The given quadratic function p(x) = (x - 1)(x + 3) could be expanded to p(x) = x² + 2x - 3. Now comparing this with the vertex form of a quadratic function, we can understand that the vertex is (-1, -4) .

Hence, the vertex of p(x) = (x - 1)(x + 3) is (-1,-4).
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The complete question is
A quadratic function is defined by p (x)= (x - 1) ( x +  3) .What is the vertex of p (x) ?

The length of the line segment that connects the points (4, -1) and (9, 7) is approximately 9.43 units.

The distance formula used in finding the distance between two points is expressed as;

[tex]d = \sqrt{(x_2 - x_1)^2+( y_2 - y_1)^2}[/tex]

Given that; the coordinates are (4, -1) and (9, 7), so we have:

x₁ = 4

y₁ = -1

x₂ = 9

y₂ = 7

Substituting these values into the distance formula, we get:

[tex]d = \sqrt{(x_2 - x_1)^2+( y_2 - y_1)^2}\\\\d = \sqrt{(9 - 4)^2+( 7 - (-1))^2}\\\\d = \sqrt{(5)^2+( 7 + 1)^2}\\\\d = \sqrt{(5)^2+( 8)^2}\\\\d = \sqrt{ 25 + 64}\\\\d = \sqrt{ 89}\\\\d = 9.43[/tex]

Therefore, the length of the line segmnet is 9.43 units.

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Answer:

missing : refrence

Step-by-step explanation:

The general solution of the given system is x(t) = c1t(1 1) + c2t(1 -1).

To find the general solution of x′ = (−1 1 −1 −1 ) x, we first need to find the eigenvalues and eigenvectors of the matrix A = (−1 1 −1 −1).

The characteristic polynomial of A is given by det(A - λI) = 0, where I is the 2x2 identity matrix and λ is the eigenvalue:

|−1-λ 1 |

|-1 -λ| = (-1-λ)(-1-λ) - (-1)(1) = λ^2 + 2λ = λ(λ+2) = 0.

So the eigenvalues of A are λ1 = 0 and λ2 = -2.

To find the eigenvectors corresponding to each eigenvalue, we need to solve the equations:

(A - λ1I)x1 = 0 and (A - λ2I)x2 = 0.

For λ1 = 0, we have:

(A - λ1I)x1 =

| -1 1 |

| -1 -1 | x1 = 0.

Solving this system of equations, we get x1 = t(1 1), where t is any scalar.

For λ2 = -2, we have:

(A - λ2I)x2 =

| 1 1 |

| -1 -3 | x2 = 0.

Solving this system of equations, we get x2 = t(1 -1), where t is any scalar.

Thus, the general solution of x′ = (−1 1 −1 −1 ) x can be written as a linear combination of the eigenvectors:

x(t) = c1t(1 1) + c2t(1 -1),

where c1 and c2 are constants that depend on the initial conditions.

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As the population (t) grows, the probability of no birthday collision reduces. This is due to the fact that as the population grows, the likelihood of two or more people having the same birthday rises.

The probability of no birthday collision among t people can be computed using the formula:

P(no collision) = 1 x (364/365) x (363/365) x ... x [(365-t+1)/365]

For t = 10, we have:

P(no collision) = 1 x (364/365) x (363/365) x ... x (356/365)
P(no collision) = 0.883
Therefore, the probability of no birthday collision among 10 people is 0.883 or approximately 88.3%.

For t = 25, we have:

P(no collision) = 1 x (364/365) x (363/365) x ... x (341/365)
P(no collision) = 0.568
Therefore, the probability of no birthday collision among 25 people is 0.568 or approximately 56.8%.

For t = 40, we have:

P(no collision) = 1 x (364/365) x (363/365) x ... x (326/365)
P(no collision) = 0.108
Therefore, the probability of no birthday collision among 40 people is 0.108 or approximately 10.8%.

In general, the probability of no birthday collision decreases as the number of people (t) increases. This is because the likelihood of two or more people sharing the same birthday increases as the number of people increases.

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The missing coordinate of point P is sqrt(5/9). The complete coordinates of P in quadrant II are (-2/3, sqrt(5/9)).

To find the missing coordinate of p, we need to use the fact that p lies on the unit circle in the given quadrant. The coordinates of a point on the unit circle are (cosθ, sinθ), where θ is the angle that the point makes with the positive x-axis.
In this case, we know that p lies in quadrant ii, which means that its x-coordinate is negative and its y-coordinate is positive. We also know that the length of the vector OP, where O is the origin and P is the point on the unit circle, is 1.
Using the Pythagorean theorem, we can write:
(OP)^2 = x^2 + y^2 = 1
Substituting the given coordinates of p, we get:
(-2)^2 + 3^2 = 1
4 + 9 = 1
This is clearly not true, so there must be an error in the given coordinates of p.
Therefore, we cannot find the missing coordinate of p using the given information.
Thus, the missing coordinate of point P is sqrt(5/9). The complete coordinates of P in quadrant II are (-2/3, sqrt(5/9)).

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The false statement is option d. summary(patient History) can be used if you precede it by entering attach(patientHistory$summary)

a. patient History$age will print all the elements of the age column.

This statement is true.

If you want to print all the elements of the age column in a data frame called patient History, you can use the dollar sign ($) operator to access the column by name.

So patient History$age will give you a vector containing all the age values.

b. gender <("M", "F", NA) is a valid column vector.

This statement is false.

The syntax of this statement is not correct. If you want to create a valid column vector with the values "M", "F", and NA, you can use the c() function like this: gender <- c("M", "F", NA). The c() function is used to combine values into a vector.

c. Using the attach() function can reduce the number of characters needed to create an R script. This statement is true, but it's important to be aware of the potential risks of using attach().

The attach() function can make it easier to access columns in a data frame without having to specify the data frame name every time.

However, it can also create confusion and errors if there are multiple objects with the same name in different environments. It's generally recommended to avoid using attach() and instead use the $ operator or the with() function.

d. summary(patient History) can be used if you precede it by entering attach(patientHistory).

This statement is false. The correct syntax to use the summary() function on a data frame is summary(patient History), without the attach() function.

The summary() function provides a summary of the statistics for each column in the data frame.

So, the false statement is option d. I hope this helps! Let me know if you have any further questions.

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Answer: 4056

Step-by-step explanation:

take cube root of 17576= 26

26*26*6=4056

Based on the given scatterplot, the trend appears to be a negative association between age and GPA. As age increases, GPA tends to decrease.

In a scatterplot, the trend represents the general pattern or direction of the relationship between two variables. In this case, the variables are age and GPA. The scatterplot shows that as age increases, there is a general tendency for GPA to decrease. This suggests a negative association between the two variables.

There could be several reasons for this negative association. It could be that older students have more responsibilities and less time to devote to their studies, leading to lower GPAs. Alternatively, it could be that older students are more likely to have completed more difficult courses earlier in their college careers, leading to lower GPAs in subsequent courses.

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By using the multiplication concept, we found that 1/5 of 30 is equal to 6. The following problem can be solved by multiplying 1/5 × 30. It is one of the fundamental arithmetic operations.

The multiplication 1/5 × 30 is used to solve the problem of finding the result when 1/5 of 30 is taken. Multiplication is a fundamental arithmetic operation taught to students in the early grades. Multiplication can be used to solve a variety of mathematical problems, including those that involve finding the total value of multiple items or the number of items in a set. In this case, the multiplication 1/5 × 30 is used to solve the problem of finding the result when 1/5 of 30 is taken.

To find the result of 1/5 of 30, we must multiply 30 by 1/5. To multiply a fraction by a whole number, we can multiply the numerator of the fraction by the whole number and then divide the result by the denominator of the fraction. So,

= 1/5 × 30

= (1 × 30)/5

= 30/5

= 6

Therefore, the result of 1/5 of 30 is 6. This means that if we divide 30 into five equal parts, each part will have a value of 6. The multiplication 1/5 × 30 can solve the problem of finding the result when 1/5 of 30 is taken. By using the multiplication formula, we found that 1/5 of 30 is equal to 6.

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Characteristic polynomial of the matrix [tex]p(x) = (x+1)(x-2)^2[/tex]

For the matrix [8 -4 0 -4], the characteristic polynomial is found by taking the determinant of the matrix [8-x -4 0 -4; 0 8-x -4 0; 0 0 8-x -4; 0 0 0 8-x] and simplifying it. This results in p(x) = [tex](x-8)^4[/tex].

For the matrix [3 0 4 -3 -4 -1 0 -1 0], the characteristic polynomial is found by taking the determinant of the matrix [3-x 0 4; -3 -4-x -1; 0 -1 -x 0;] and simplifying it. This results in [tex]p(x) = (x+1)(x-2)^2[/tex].

The determinant of the matrix (A - lam*I), where I is the identity matrix of the same size as A, is found by computing the characteristic polynomial of a square matrix A, represented by P(lam), which is a polynomial function of a scalar variable lambda. We refer to the eigenvalues of the matrix A as the roots of the characteristic polynomial. Important details about the matrix, including its diagonalizability, rank, trace, and determinant, are revealed by the characteristic polynomial. It frequently appears in applications like systems of linear equations, differential equations, and linear transformations.

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Answer:

a and -b

Third answer choice

Step-by-step explanation:

If (x - a)(x - b) = 0

then one or both of the terms must be zero

Therefore one solution can be found when (x- a) = 0
x - a = 0 ==> x = a

The other solution is when (x+ b) = 0
x + b = 0 ==> x = - b

So the solution set is
x = a and x = -b

Third answer choice

Answer: The radius of convergence of the series Σ(-1)ⁿ xⁿ 3ⁿ ln(n) with n=2 is 3.

To find the radius of convergence of the series Σ(-1)ⁿ xⁿ 3ⁿ ln(n) from n=2 to infinity, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges absolutely, and the radius of convergence r is the reciprocal of the limit. If the limit is greater than 1, then the series diverges, and if the limit is equal to 1, the test is inconclusive.

So, applying the ratio test to our series, we have:

|(-1)(ⁿ+¹+¹) x(ⁿ+¹) 3(ⁿ+¹) ln(n+1)| / |(-1)ⁿ xⁿ 3ⁿ ln(n)|

= |x|/3 * ln(ⁿ+¹)/ln(n)

As n approaches infinity, the limit of this expression is:

lim n->inf |x|/3 * ln(n+1)/ln(n) = |x|/3 * 1 = |x|/3

So the series converges absolutely if |x|/3 < 1, or equivalently, if |x| < 3. Therefore, the radius of convergence is r = 3.

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We first list all the partitions of integers from 1 to 7, then use these lists to compute the values of the partition function p(n) for n from 1 to 7. Therefore, the values of the partition function for integers from 1 to 7 are 1, 2, 3, 5, 7, 11, and 15, respectively.

A partition of a positive integer n is a way of writing n as a sum of positive integers, where the order of the summands does not matter. For example, the partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. To compute the partition function p(n), we count the number of partitions of n.

Here are the partitions of integers from 1 to 7:

1: {1}

2: {2}, {1,1}

3: {3}, {2,1}, {1,1,1}

4: {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1}

5: {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, {1,1,1,1,1}

6: {6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1,1}, {1,1,1,1,1,1}

7: {7}, {6,1}, {5,2}, {5,1,1}, {4,3}, {4,2,1}, {4,1,1,1}, {3,3,1}, {3,2,2}, {3,2,1,1}, {3,1,1,1,1}, {2,2,2,1}, {2,2,1,1,1}, {2,1,1,1,1,1}, {1,1,1,1,1,1,1}

Using this list, we can compute the values of the partition function p(n) for n from 1 to 7:

p(1) = 1

p(2) = 2

p(3) = 3

p(4) = 5

p(5) = 7

p(6) = 11

p(7) = 15

Therefore, the values of the partition function for integers from 1 to 7 are 1, 2, 3, 5, 7, 11, and 15, respectively.

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If W = U ⊕ V, then U ∩ V = {0} which can be proved by proving {0} is an element of U ∩ V and there are no other elements in U ∩ V besides {0} for the vector space.

To show that if W = U ⊕ V, then U ∩ V = {0}, we need to prove two things:

1. {0} is an element of U ∩ V.
2. There are no other elements in U ∩ V besides {0}.

Step 1: Show that {0} is an element of U ∩ V.

Since U and V are subspaces of the vector space W, they both must contain the zero vector (0) as per the definition of a subspace. Therefore, the zero vector is in both U and V, which implies that 0 is an element of U ∩ V.

Step 2: Show that there are no other elements in U ∩ V besides {0}.

Suppose there is a nonzero vector x that belongs to U ∩ V. This means x is in both U and V. Since W = U ⊕ V, any vector in W can be uniquely written as the sum of a vector from U and a vector from V. Thus, x can be written as:

x = u + v

where u is a vector from U and v is a vector from V. However, x is also in both U and V, so we can rewrite the equation as:

x = x + 0

Since the sum of vectors from U and V is unique, we must have u = x and v = 0. But this contradicts our initial assumption that x is a nonzero vector, as x ∈ V and we assumed x ≠ 0. Therefore, there can be no other elements in U ∩ V besides {0}.

In conclusion, if W = U ⊕ V, then U ∩ V = {0}.

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i) The percentage of enquires where no sale is made is 42%.

ii) If in a month 2,800 inquiries are made, the company would expect to make sales of 1,624.

The percentage refers to the quotient of a number or value multiplied by 100.

The quotient is the result of a division operation that compares a portion of a quantity with the whole.

The percentage of people who make telephone enquires and buy the company's products = 58%

i) The percentage of the people who make telephone inquiries but do not buy the company's products = 42% (100% - 58%)

ii) The number of inquiries made in a month = 2,800

The expected number of sales for the month = 1,624 (2,800 x 58%)

Thus, based on the percentage of expected sales, when 2,800 inquiries are made, the company should make 1,624 sales.

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The correct equation is 0.75w + 0.5w + 2 = 17.

To determine the error in the given equation and find the correct number of laps needed before the sprint to the finish, we can follow these steps:

Step 1: Examine the given equation, w(1.5 + 1)^2 = 17, and identify the error.

The equation provided is incorrect because it does not account for the width and height of the rectangle, as well as the 2-mile sprint to the finish line.

Step 2: Formulate the correct equation based on the given information.

The correct equation should be 0.75w + 0.5w + 2 = 17, where w represents the number of laps. This equation includes the distance covered by riding laps around the block (0.75w + 0.5w) and the additional 2-mile sprint to the finish line.

Step 3: Solve the equation to find the correct number of laps.

By simplifying the equation, we get 1.25w + 2 = 17. Subtracting 2 from both sides gives us 1.25w = 15. Dividing both sides by 1.25 yields w = 12.

Therefore, the correct equation is 0.75w + 0.5w + 2 = 17, and the cyclists will need to ride 12 laps before the sprint to the finish. The given equation and the incorrect number of laps mentioned in the other options do not accurately represent the race distance and the laps required.

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The z-score corresponding to a sample mean of 84 for a sample size of 9 scores, with a population mean of 80 and a population standard deviation of 12, is 1.

To find the z-score corresponding to a sample mean of m = 84 with a population mean (μ) of 80 and a population standard deviation (σ) of 12, the z-score can be calculated using the formula z = (x - μ) / (σ / √n).

In this case, the population mean (μ) is 80 and the population standard deviation (σ) is 12. The sample mean (m) is given as 84, and the sample size (n) is 9.

To calculate the z-score, we use the formula:

z = (x - μ) / (σ / √n)

Substituting the given values, we have:

z = (84 - 80) / (12 / √9)

Simplifying the expression, we get:

z = 4 / (12 / 3)

z = 4 / 4

z = 1

Therefore, the z-score corresponding to a sample mean of 84 for a sample size of 9 scores, with a population mean of 80 and a population standard deviation of 12, is 1. This indicates that the sample mean is one standard deviation above the population mean. The z-score allows us to compare the sample mean to the population distribution and assess how unusual or typical the sample mean is relative to the population.

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Option B suggests that the answer options could be interpreted differently by different users. This could lead to inconsistencies in how respondents rate the variety of material available. Different interpretations of the rating scale or varying perceptions of what constitutes a high or low variety could impact the survey results.

Option C states that the survey is biased because it is being taken only by the service's users. This introduces a potential sampling bias since the survey is limited to the service's user base. The opinions and experiences of non-users are not included, which may not provide a comprehensive understanding of the variety of material available. The results may be skewed towards the preferences and perspectives of the service's existing users.

Option A and Option D are not directly related to potential influences on the survey results. Option A addresses who can take the survey, but it does not pertain to the potential biases or variations in responses. Option D discusses the mode of survey administration, but it does not specifically address factors that could affect the survey results themselves.

Therefore, options B and C are the choices that could affect the results of the survey.

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Answer:

Step-by-step explanation:

A squiggly line can refer to a variety of different shapes or patterns. To create an equation using cosine that resembles a squiggly line, you can use a combination of sine and cosine functions with different frequencies and amplitudes. Here's an example equation that produces a squiggly pattern:

y = A * cos(B * x) + C * sin(D * x)

In this equation, A, B, C, and D are constants that you can adjust to modify the shape and characteristics of the squiggly line. By experimenting with different values for these constants, you can create various squiggly patterns. Keep in mind that the specific equation may vary depending on the exact shape and features you have in mind for the squiggly line.

Displacement is defined as the magnitude and direction of an object's change in position from the starting point.What is displacement?Displacement refers to the overall change in the position of an object over a specified period of time. It takes both magnitude and direction into account.

Displacement, as opposed to distance traveled, is a vector amount that considers not only the total distance traveled but also the direction in which the object moved.

Displacement is the length of the straight line connecting the beginning and ending positions of an object, as well as the direction of this line.

There are a few key things to keep in mind about displacement:Displacement is calculated using the formula: Displacement (Δd) = Final Position - Initial Position (d₂ - d₁)

Displacement is a vector amount since it includes both magnitude and direction.

If an object moves around in a circle and finishes where it began, its displacement will be zero but the distance it travels will not.

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df(0) = [0 0 0; 0 0 0]

d(g o f)(0) = [0 0; 0 0].

We have f: R^3 → R^2 and g: R^2 → R^2.

Using the chain rule, we have:

d(g o f)(0) = dg(f(0)) ◦ df(0)

First, let's find df(0):

df(0) = [∂f₁/∂x₁(0) ∂f₁/∂x₂(0) ∂f₁/∂x₃(0); ∂f₂/∂x₁(0) ∂f₂/∂x₂(0) ∂f₂/∂x₃(0)]

We know that f(0) = (0, 0, 0) and f(0) = (1, 2), so:

f₁(0) = 1, f₂(0) = 2

∂f₁/∂x₁(0) = ∂f₁/∂x₂(0) = ∂f₁/∂x₃(0) = 0

∂f₂/∂x₁(0) = ∂f₂/∂x₂(0) = ∂f₂/∂x₃(0) = 0

Next, let's find dg(f(0)):

dg(x, y) = [∂g₁/∂x ∂g₁/∂y; ∂g₂/∂x ∂g₂/∂y]

dg(1, 2) = [2 1; 6 3]

Finally, we can find d(g o f)(0):

d(g o f)(0) = dg(f(0)) ◦ df(0) = [2 1; 6 3] ◦ [0 0 0; 0 0 0] = [0 0; 0 0]

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The derivative of the composition g o f at (0) given by applying the chain rule is [2 4; 3 1].

The problem requires finding the derivative of the composition g o f at (0).

Using the chain rule, we can express this derivative as the product of the Jacobian matrix of g with respect to its inputs and the Jacobian matrix of f with respect to its inputs, evaluated at (0).

The Jacobian matrix of g is given by:

[ 2y 1 2x ]

[ 3y 3x ]

If T : Rn → R

m is a linear transformation, then T(0) = 0.

Evaluating this at f(0) = (1, 2) gives:

[ 4 2 ]

[ 6 3 ]

The Jacobian matrix of f is given by:

[ 1 0 0 ]

[ 0 1 0 ]

Evaluating this at 0 gives:

[ 1 0 0 ]

[ 0 1 0 ]

Multiplying these two matrices, we get:

[ 2 4 ]

[ 3 1 ]

Therefore, d(g o /)(0) = [2 4; 3 1].

In summary, we used the chain rule to find the derivative of the composition g o f at (0), which is given by the product of the Jacobian matrix of g and the Jacobian matrix of f, both evaluated at the same point.

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The value of probability is P(A^c∩B^c) = 0.2.

Using the formula P(A) = P(A ∩ B) + P(A ∩ B^c) and P(A^c) = 1 - P(A), we can compute P(A) and P(B) as follows:

P(A) = P(A ∩ B) + P(A ∩ B^c) = 0.3 + 0.15 = 0.45

P(A^c) = 1 - P(A) = 1 - 0.45 = 0.55

Similarly, we can compute P(B) using P(B ∩ A) + P(B ∩ A^c) = P(B ∩ A) + P(A^c ∩ B) = 0.35P, which gives P(B) = 0.35P.

Using the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B), we can compute P(A ∪ B) as follows:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.45 + 0.35P - 0.3 = 0.15 + 0.35P

Since P(A ∪ B) + P(A^c ∪ B^c) = 1, we have

P(A^c ∪ B^c) = 1 - P(A ∪ B) = 1 - (0.15 + 0.35P) = 0.85 - 0.35P

Finally, using the formula P(A^c ∩ B^c) = 1 - P(A ∪ B) = 1 - (0.15 + 0.35P) = 0.85 - 0.35P. Therefore, P(A^c ∩ B^c) = 0.85 - 0.35P.

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A. P(6, then 1) = 1/90

B. P(even, then 5) = 1/18

C. P(8, then odd) = 1/18

D. P(3, then prime) = 2/45

E. P(prime, composite) = 4/15

F. P(even, then 3, then 5) = 1/144

Given:

Total number of cards: 10

A. P(6, then 1):

P(6, then 1) = 1/10 x 1/9

= 1/90

B. P(even, then 5):

Number of favorable outcomes: 5 x 1 = 5

P(even, then 5) = 5/10 x 1/9

= 1/18

C. P(8, then odd):

Number of favorable outcomes: 1 x 5 = 5

P(8, then odd) = 1/10 x 5/9

= 1/18

D. P(3, then prime):

Number of favorable outcomes: 1 x 4 = 4

P(3, then prime) = 1/10 x 4/9

= 2/45

E. P(prime, composite):

Number of favorable outcomes: 4 x 6 = 24

P(prime, composite) = 4/10 x 6/9

= 4/15

F. P(even, then 3, then 5):

Number of favorable outcomes: 5 x 1 x 1 = 5

P(even, then 3, then 5) = 5/10 x 1/9 x 1/8

= 1/144

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In mathematics, an integral is a mathematical object that represents the area between a function and the x-axis on a graph, or the accumulation of a quantity over time.

(a) Integrate by parts, letting dv = 5 − x dx.

Using integration by parts, we can write:

∫x(5-x) dx = x ∫(5-x) dx - ∫[d/dx(x) ∫(5-x) dx] dx

= x [5x - (1/2)x^2] - ∫(0 - (5-x)dx)

= x [5x - (1/2)x^2] - (5x - (1/2)x^2) + C

= - (1/2)x^2 + 10x + C, where C is the constant of integration.

(b) Integrate by substitution, letting u = 5 − x.

Using u-substitution, we can write:

∫x(5-x) dx = ∫(5-u)u du

= ∫(5u - u^2) du

= (5/2)u^2 - (1/3)u^3 + C

= (5/2)(5-x)^2 - (1/3)(5-x)^3 + C, where C is the constant of integration.

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The simplified ratio is:    

1 : 2 : 6 or 1 : 2/1 : 6/1.

The question requires us to determine the ratio of the earnings of the three children in the Skosana family. Given that the eldest, middle and youngest children earn R270, R180 and R90 respectively, we can express these amounts as a ratio as follows:

R270 : R180 : R90

Let's simplify this ratio by dividing each amount by the highest common factor of 90.

R270 ÷ 90 = 3

R180 ÷ 90 = 2

R90 ÷ 90 = 1

Therefore, the ratio of the earnings of the three children in the Skosana family is:3 : 2 : 1.

An explanation of how we came up with the answer is as shown below:

We can express the eldest child's earning as a ratio of the total earnings as follows:

R270 : (R270 + R180 + R90)

Simplifying this ratio, we get:

3R : (3R + 2R + R) = 3R : 6R

= 1 : 2

Similarly, we can express the middle child's earning as a ratio of the total earnings as follows:

R180 : (R270 + R180 + R90)

Simplifying this ratio, we get:

2R : (3R + 2R + R) = 2R : 6R

= 1 : 3

Finally, we can express the youngest child's earning as a ratio of the total earnings as follows:

R90 : (R270 + R180 + R90)

Simplifying this ratio, we get:

R : (3R + 2R + R) = R : 6R

= 1 : 6

Therefore, the ratio of the earnings of the three children in the Skosana family is:

1 : 2 : 6.

However, we can further simplify this ratio by dividing each amount by the highest common factor of

6.1 ÷ 1 = 12 ÷ 2

= 36 ÷ 6

= 1

Therefore, the simplified ratio is:

1 : 2 : 6 or 1 : 2/1 : 6/1.

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The general solution is x(t) = c1[cos(t), sin(t)] + c2[cos(t), -sin(t)].

The general solution of a differential equation x'(t) = Ax(t), with matrix A = [0 -1; 1 0], can be found by determining the eigenvalues and eigenvectors of the matrix A.

For this matrix, the eigenvalues are λ1 = i and λ2 = -i. The corresponding eigenvectors are x₁= [1, i] and x₂ = [1, -i].

The general solution of the differential equation is given by the linear combination of the eigenvector solutions:

x(t) = c₁x₁(t) + c₂x₂(t), where c₁ and c₂ are constants.

The solutions x₁(t) and x₂(t) can be expressed as:

x₁(t) = [cos(t), sin(t)] x₂(t) = [cos(t), -sin(t)]

Thus, the general solution is x(t) = c₁[cos(t), sin(t)] + c₂[cos(t), -sin(t)].

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