3x-5x+6=5x-3

What is x-?

The polynomials span P₂ implies any polynomial of degree 2 written as linear combination of these polynomials.

To determine whether the given polynomials span P₂,

Check whether any polynomial of degree 2 can be written as a linear combination of these polynomials.

Let us consider a general polynomial of degree 2.

p(x) = ax² + bx + c

We need to find coefficients k₁, k₂, k₃, and k₄ such that.

p(x) = k₁(1-x+2x²) + k₂(3+x) + k₃(5-x+4x²) + k₄(-2-2x+2x²)

Expanding the right side and collecting like terms, we get,

p(x) = (2k₁+4k₃+2k₄)x² + (-k₁-k₂+k₃-2k₄)x + (k₁+3k₂+5k₃-2k₄+1)

This equation must hold for any value of x.

Equate the coefficients of the powers of x on both sides,

2k₁ + 4k₃ + 2k₄ = a

-k₁ - k₂ + k₃ - 2k₄ = b

k₁ + 3k₂ + 5k₃ - 2k₄ + 1 = c

Solve this system of linear equations for k₁, k₂, k₃, and k₄.

Write this in matrix form as,

[tex]\left[\begin{array}{cccc}2&0&4&2\\-1&-1&1&-2\\1&3&5&-2\end{array}\right][/tex][tex]\left[\begin{array}{ccc}k_{1} \\k_{2}\\k_{3}\end{array}\right][/tex]  [tex]= \left[\begin{array}{ccc}a \\b\\c-1\end{array}\right][/tex]

Solve this system using Gaussian elimination or other methods.

However, a simpler way to check whether the polynomials span P₂ is to check whether the matrix of coefficients is invertible.

If the matrix is invertible, then there is a unique solution for any value of a, b, and c.

If the matrix is not invertible,

Then there are some values of a, b, and c for which there is no solution.

And the polynomials do not span P₂.

To check whether the matrix is invertible, compute its determinant,

[tex]\left|\begin{array}{cccc}2&0&4&2\\-1&-1&1&-2\\1&3&5&-2\end{array}\right|[/tex]

= 12

Since the determinant is non-zero,

The matrix is invertible, and the polynomials span P₂.

Therefore, matrix is invertible implies polynomials span P₂, any polynomial of degree 2 written as linear combination of these polynomials.

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The above question is incomplete, the complete question is:

Determine whether the following polynomials span P₂ (polynomial of degree 2):

p₁=1−x+2x²,

p₂=3+x,

p₃=5−x+4x²,

p₄=−2−2x+2x²

The equivalence classes are disjoint, and their union covers all of A. Also, each element in A belongs to exactly one equivalence class.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

a) To show that R is an equivalence relation on A, we need to verify three properties: reflexivity, symmetry, and transitivity.

Reflexivity: For any x in A, we have f(x) = f(x) by definition of a function. Therefore, (x,x) is in R for any x in A, which means R is reflexive.

Symmetry: For any (x,y) in R, we have f(x) = f(y). This implies that f(y) = f(x), and hence (y,x) is in R. Therefore, R is symmetric.

Transitivity: For any (x,y) and (y,z) in R, we have f(x) = f(y) and f(y) = f(z). This implies that f(x) = f(z), and hence (x,z) is in R.

Therefore, R is transitive.

b) The equivalence classes of R are the sets of elements in A that have the same function value under f.

In other words, the equivalence class of an element x in A is the set of all elements y in A such that f(x) = f(y). We can write this as:

[x] = {y in A | f(x) = f(y)}

For example, if A = {1,2,3,4,5} and f(x) = x², then the equivalence classes of R are:

[1] = {1, -1}

[2] = {2, -2}

[3] = {3, -3}

[4] = {4}

[5] = {5, -5}

Hence, the equivalence classes are disjoint (i.e., they have no common elements), and their union covers all of A. Also, each element in A belongs to exactly one equivalence class.

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Step-by-step explanation:

Every year you get 3 % of 1800 pounds added to the account

  what is 3% of 1800 ?         3 % is .03 in decimal form

     1800 * .03 = 54  pounds interest per year

in TWO years you will have   54 * 2   added to 1800 = 1908 pounds

Answer:

A). increase by 54  per year.

B). After two years the account has 1908

Step-by-step explanation:

A). 800 * 3% * 1 = 54

B). 1800 + 1800 * 3% * 2  = 1908

Hope this helps

Answer: 361

Step-by-step explanation:

To calculate the p-value, we need to use a one-tailed t-test since we're interested in the probability of getting a sample mean greater than 12.10 oz.

First, we need to calculate the t-statistic:

t = (sample mean - hypothesized mean) / (standard deviation / sqrt(sample size))

t = (12.15 - 12.10) / (0.05 / sqrt(25))

t = 3.1623

Next, we need to find the degrees of freedom, which is the sample size minus one:

df = 25 - 1 = 24

Using a t-distribution table with 24 degrees of freedom and a significance level of 0.01, we find that the critical value is 2.492.

Since the calculated t-value (3.1623) is greater than the critical value (2.492), we can reject the null hypothesis and conclude that there is evidence that the mean volume is greater than 12.10 oz.

To find the p-value, we need to calculate the probability of getting a t-value greater than 3.1623 with 24 degrees of freedom:

p-value = P(t > 3.1623) = 0.0028 (calculated using a t-distribution table or software)

Therefore, the p-value is 0.0028, which is less than the level of significance (0.01), indicating strong evidence against the null hypothesis.

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No, it is not true that an elementary n × n matrix has either n or n+1 nonzero entries. because, the number of nonzero entries in an elementary matrix is not fixed and can vary depending on the row operation used to obtain the matrix.

An elementary matrix is a square matrix obtained by performing a single elementary row operation (i.e., adding a multiple of one row to another or multiplying a row by a nonzero scalar) on the identity matrix. The number of nonzero entries in an elementary matrix depends on the specific row operation performed.

For example, the elementary matrix obtained by multiplying the second row of the 3×3 identity matrix by 2 is:

[1 0 0]

[0 2 0]

[0 0 1]

This matrix has 4 nonzero entries, not 3 or 4.

Similarly, the elementary matrix obtained by adding 3 times the third row to the first row of the 3×3 identity matrix is:

[1 0 3]

[0 1 0]

[0 0 1]

This matrix also has 4 nonzero entries.

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Answer: C

Step-by-step explanation:

area of rectangle = (5--2) × (2--2) = 7 × 4 = 28

area of triangle = ((12-5) × (6--6)) ÷ 2 = (7 × 12) ÷ 2 = 84 ÷ 2 = 42

total area = 28 + 42 = 70

Answer:

(i) To simplify (b²-a²) ÷ (2a²+ab-3b²), we can factor the numerator and denominator using the difference of squares formula, which states that a² - b² = (a + b)(a - b).

(b²-a²) = (b + a)(b - a)

(2a²+ab-3b²) = (2a-b)(a+3b)

Thus, we can rewrite the expression as:

(b + a)(b - a) / (2a-b)(a+3b)

(ii) To simplify (3x-3y) ÷ (ax-ay-x+y), we can factor out the common factor of 3 from the numerator and the common factor of (a-1) from the denominator:

3(x-y) / (a-1)(x-y)

We can then cancel the common factor of (x-y) to get the simplified form:

3 / a-1

(iii) To simplify (y²-6y-7) ÷ (2y²-17y+21), we can factor both the numerator and the denominator:

(y-7)(y+1) / (2y-3)(y-7)

We can then cancel out the common factor of (y-7) to get the simplified form:

(y+1) / (2y-3)

(iv) To simplify (a²-ab-ac+bc) ÷ (a²+ab-ac-bc), we can factor out the -1 from the denominator:

(a²-ab-ac+bc) ÷ -1(a²-ab+ac-bc)

We can then factor out the common factor of (a-b) from both the numerator and the denominator:

(a-b)(a-c) ÷ -1(a-b)(a+c)

Cancelling out the common factor of (a-b) gives us the simplified expression:

(c-a) / (a+c)

Answer:

66 coins.

Step-by-step explanation:

There are 4 quarters in a year, so 2 years is 8 quarters.

Since Janice adds 5 coins every 3 months, in one year (or 4 quarters), she will add:

5 coins/3 months x 4 quarters = 20 coins

So in 2 years (8 quarters), she will add:

20 coins/year x 2 years = 40 coins

Therefore, after 2 years, the total number of coins in Janice's collection will be:

26 + 40 = 66 coins.

The statement 'In order to calculate MACRS depreciation, the business needs to know the asset's original cost, the depreciation period, and which depreciation method (i.e. 200% declining balance) is used.' is false. because, the asset was used for both personal and business purposes, the business must calculate the percentage of business use in order to determine the correct depreciation deduction.

While the asset's original cost, depreciation period, and depreciation method are all necessary inputs for calculating MACRS depreciation, there are other pieces of information that are also required.

For example, the business must determine the applicable MACRS recovery period and the property's placed-in-service date, as well as any applicable conventions for calculating the depreciation.

Additionally, if the asset was used for both personal and business purposes, the business must calculate the percentage of business use in order to determine the correct depreciation deduction.

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The equation that Mai could have written, using the Fermi process is A. ( 4x 10 ⁵ ) / ( 4x 10 ⁻⁴ ) = 1 x 10 ⁹.

A technique initiated by Fermi is used for predicting a rough figure or estimation of something, typically with minuscule or no details about the particulars of the thing being predicted.

The volume of the pack of gums would be:

= 1 / 5 x 1 / 10 x 1 / 50

= 0. 0004

= 4x 10 ⁻⁴

The volume of the gymnasium would be:

= 100 x 80 x 50

= 400, 000

= 4x 10 ⁵

So this can be written as:

( 4x 10 ⁵ ) / ( 4x 10 ⁻⁴ ) = 1 x 10⁹ .

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The appropriate statistical test for this scenario would be a paired t-test for means, since the same group of individuals were tested twice under two different conditions (using word processor A and B).

The null hypothesis would be that there is no significant difference in typing speed between the two word processors, while the alternative hypothesis would be that there is a significant difference. To conduct the test, the differences between the typing speeds for each individual would be calculated (speed when using word processor A minus speed when using word processor B), and the mean and standard deviation of these differences would be calculated.

Then, a t-statistic would be calculated using the formula (mean difference / standard error of the mean difference), and the p-value would be determined based on the t-distribution with degrees of freedom equal to the sample size minus 1. If the p-value is less than the chosen level of significance (usually 0.05), then we reject the null hypothesis and conclude that there is a significant difference in typing speed between the two-word processors.

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Answer: 4/7

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Answer: 4/7 Probability

Step-by-step explanation:

Take the total number of even numbers (4), then the total number of all numbers (7). Slap them together as a fraction, then you're chilling.

a) The points[tex](x, y) = (-1, 2)[/tex] are the equilibrium solutions

b) The solutions are decreasing since [tex](y-2)[/tex] is negative and [tex](x+1)[/tex] is positive.

c) The solution curves concave up if y'' is positive, and concave down if y'' is negative.

a)  Either [tex]y = 2 or x = -1[/tex]is required for this equation to be true. Therefore, the points[tex](x, y) = (-1, 2)[/tex] are the equilibrium solutions.

We must set [tex]y = 0[/tex] and solve for y in order to find the equilibrium solutions. So:

[tex](y-2)(x+1) = 0[/tex]

b) We need to look at the sign of y' in various areas of the xy-plane to figure out where the solutions are rising or decreasing. The solutions are growing if y' is positive; they are shrinking if y' is negative.

Since [tex](y-2)[/tex] and [tex](x+1)[/tex]are both negative, y' is positive and the solutions are increasing if[tex]x -1[/tex]and [tex]y 2.[/tex] When [tex]x > -1[/tex]and [tex]y 2, (y-2)[/tex] becomes negative and (x+1) becomes positive, indicating that y' is negative and the solutions are getting smaller. If [tex]y > 2[/tex], then y' is positive and the solutions are getting bigger because [tex](y-2)[/tex] and [tex](x+1)[/tex] are both positive. for x is greater than [tex]-1[/tex] and y is greater than [tex]2[/tex], the solutions are decreasing since [tex](y-2)[/tex] is negative and [tex](x+1)[/tex] is positive. This is the case for [tex]y 2.[/tex]

The expression for y' shows that when [tex](y-2)[/tex] and [tex](x+1)[/tex]have the same sign, and when they have the opposite sign, respectively, it will be positive.

c) We need to look at the sign of y'' in various areas of the xy-plane to figure out where the solution curves are concave up-concave down. By taking the derivative of y', we may find y'':

[tex]y'' = (y-2) - 2(x+1)[/tex]

The solution curves concave up if y'' is positive, and concave down if y'' is negative.

We may deduce that y'' is positive when[tex]y > 2 + 2(x+1)[/tex] and negative when [tex]y 2 + 2(x+1)[/tex] from the expression for y''. As a result, when the solution curves are above the line[tex]y = 2 + 2(x+1)[/tex], they are concave up, and when they are below it, they are concave down.

a) [tex]y > 2 + 2(x+1)[/tex]

b) [tex]y 2 + 2(x+1)[/tex]

c)[tex]y = 2 + 2(x+1)[/tex]

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Complete Question:

Let y' =(y-2)(x+1). a) Determine all equilibrium solutions. b) Determine the region in the xy - plane where the solutions are increasing, and where the solutions are decreasing. c) Determine the regions in the xy - plane where the solution curves are concave up, and determine those regions where they are concave down. Solve the following differential equations.

a) dy + 2xy2 = 0 doc ? =

b) x - y = 2x?y, y(i)=1 y *1) b dy dx

A)  The average velocity of the particle is 4.5 feet per second

B)  The particle reaches its average velocity at time c is 5.25 seconds.

The average velocity of a particle on the interval [1,3] is given by the formula:

The change in displacement is given by

The change in time is 3 - 1 = 2 seconds.

Therefore, the average velocity of the particle is:

To find the time c when the particle reaches its average velocity, we need to solve the equation:

where s(t) = -t² + 5t.

Substituting s(t) into the equation, we get:

Simplifying the equation, we get:

Solving for c, we get:

Therefore, the particle reaches its average velocity at time c = 5.25 seconds.

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The frequency for the number of computer science students enrolled in Stat 155 is 127, and the relative frequency is approximately 0.552 (to 3 decimal places).

In Stat 155, there are 230 students enrolled, and 127 of them are majoring in computer science. The frequency for the number of computer science students enrolled in Stat 155 is 127. To find the relative frequency, divide the frequency by the total number of students:

Relative frequency = (Number of computer science students) / (Total number of students)
Relative frequency = 127 / 230 ≈ 0.552

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We can use the equation F = 7s + 35 to find the total number of fence sections standing after any number of Saturdays s.

Let's start by breaking down the information given in the problem:

Jaden has already put up 35 fence sections.

He plans to add 7 sections every Saturday until he is finished.

We can use this information to create an equation that relates the total number of fence sections F to the number of Saturdays s. Since Jaden starts with 35 fence sections and adds 7 more each Saturday, we can write:

F = 7s + 35

In this equation, s represents the number of Saturdays that have passed, and F represents the total number of fence sections standing after that many Saturdays.

For example, after 3 Saturdays, Jaden would have added 7 sections each time, for a total of 21 additional sections. Adding this to the 35 sections he started with gives:

F = 7(3) + 35 = 21 + 35 = 56

Therefore, after 3 Saturdays, there would be 56 fence sections standing.

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Diversity in People can be seen in terms of:

In terms of Ethnicity,  people from various ethnic traditions bring a rich tapestry of sophistications, traditions, and outlooks to society. Access to Healthcare: Ensuring all has access to value healthcare improves the health and comfort of individuals, offspring, and communities.

In terms of Gender, embracing neutral diversity helps promote equal time for all genders in various fields and fights feminine-based bias. : Providing access to instruction for all members of society, although socio-economic rank, etc.

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See text below

Now we make a table and mention two examples of diversity in people and two examples of common good in society

The future value of the ordinary annuity is $35,134.71 rounded to the nearest cent.

To find the future value of an annuity, we can use the formula:

FV = PMT x (((1 + r)ⁿ - 1) / r)

Where:

PMT = the amount of the periodic payment (in this case, $450 per month)

r = the interest rate per period (5% / 12 months = 0.004167 per month)

n = the total number of periods (18 years x 12 months per year = 216 months)

Plugging in the numbers, we get:

FV = $450 x (((1 + 0.004167)²¹⁶ - 1) / 0.004167)

FV = $450 x (78.077126)

FV = $35,134.71

Therefore, the future value of the ordinary annuity is $35,134.71 rounded to the nearest cent.

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The term that best describes the investment is $1,250 invested at 4% compounded interest. The Option A is correct.

We will use compound interest: A = P(1 + r/n)^(nt) formula to get the annual total investment. In this case, P = $1,250, r = 0.04, n = 1 and t = 1.

Plugging the values, we get:

A = 1250(1 + 0.04/1)^(1*1)

A = 1250(1.04)

A = $1,300

Therefore, term that best describes the investment is $1,250 invested at 4% compounded interest.

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It is not true that if A is an n × n matrix with detA = 2, then detA³ = 6.

det(A³) = det(A × A × A) = det(A) × det(A) × det(A) = (det(A))³ = 2³ = 8

detA³ = 8, not 6.

The determinant of a matrix is a scalar value that encodes various properties of the matrix.

One of the properties is the volume scaling factor that is induced by the matrix transformation.

The determinant has the property that det(kA) = kⁿ × det(A) for any scalar k and n × n matrix A, where n denotes the dimension of the matrix.

Therefore, we have:

det(A³) = det(A × A × A) = det(A) × det(A) × det(A) = (det(A))³ = 2³ = 8

detA³ = 8, not 6.

The determinant of a matrix raised to a power is not simply obtained by raising the determinant to the same power.

The determinant of a matrix raised to a power can be obtained by raising the determinant of the matrix to the power and then multiplying by the scaling factor induced by the matrix transformation.

Specifically, for a matrix A with detA = 2 and an integer k, we have:

[tex]det(A^k) = (det(A))^k \times scaling factor induced by A^k[/tex]

The scaling factor induced by [tex]A^k[/tex] can be computed by considering the effect of [tex]A^k[/tex] on the unit hypercube.

This calculation requires the use of linear algebra and is beyond the scope of this answer.

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The answer  is that there are 165 ways to distribute eight identical bottles of water to three friends.

We can use the formula for distributing identical objects to distinct groups, which is (n+r-1) choose (r-1), where n is the number of objects and r is the number of groups. In this case, n=8 and r=3.

So the formula becomes (8+3-1) choose (3-1), which simplifies to 10 choose 2. Using the combination formula, 10 choose 2 equals 45. However, this only accounts for cases where all three friends receive at least one bottle of water.

To account for cases where some friends may receive no water, we need to add the number of ways where two friends receive water and one friend receives no water, and the number of ways where one friend receives water and two friends receive no water.

There are three ways to choose which friend receives no water, and then we need to distribute the remaining eight bottles of water among the remaining two friends. Using the formula from earlier, this gives us (8+2-1) choose (2-1) = 9.

So for the case where two friends receive water and one friend receives no water, there are 3 * 9 = 27 ways. Similarly, for the case where one friend receives water and two friends receive no water, there are 3 * 9 = 27 ways.

Adding these cases to the initial case where all three friends receive at least one bottle of water, we get a total of 45 + 27 + 27 = 99 ways. However, we still need to account for cases where all eight bottles of water go to a single friend, which is just 3 ways.

So the final answer is 99 + 3 = 102 ways to distribute eight identical bottles of water to three friends, where some friends may receive no water.

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Answer: its B) test statistic

Step-by-step explanation:

The probabilities of Sue catching the bus TO the shops, catching the bus FROM the shops, walking to the shops, and walking from the shops are 0.4, 0.7, 0.6, and 0.3, respectively, given that she either catches the bus or walks when visiting the shops.

Let's denote the event of Sue catching the bus TO the shops as A and the event of her catching the bus FROM the shops as B. Then, we can use the following probabilities:

P(A) = 0.4

P(B) = 0.7

Since Sue either catches the bus or walks, these two events are mutually exclusive and exhaustive. Therefore, the probability of her walking to the shops is:

P(not A) = 1 - P(A) = 1 - 0.4 = 0.6

Similarly, the probability of her walking from the shops is:

P(not B) = 1 - P(B) = 1 - 0.7 = 0.3

We can also use the law of total probability to find the probability of Sue catching the bus:

P(bus) = P(A) + P(B) = 0.4 + 0.7 = 1.1

This value is greater than 1, which is not possible since probabilities cannot be greater than 1. This means that there is an error in the given probabilities. However, we can still use the above calculations for the given probabilities to determine the probabilities of walking and catching the bus.

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The complete question is :

What are the probabilities of Sue catching the bus TO the shops, catching the bus FROM the shops, walking to the shops, and walking from the shops, if the probability of Sue catching the bus TO the shops is 0.4 and the probability of her catching the bus FROM the shops is 0.7, and it is known that she either catches the bus or walks when visiting the shops?

The amounts for each type of paintbrush is given as follows:

The amounts are obtained by a system of equations, for which the variables are given as follows:

A total of 10 paintbrushes were purchased, hence:

x + y = 10.

The number of small paintbrushes in the set was 4 times the number of large paintbrushes in the set, hence:

x = 4y.

Replacing into the first equation, the value of y is obtained as follows:

4y + y = 10

5y = 10

y = 2.

Hence the value of x is of:

x = 4 x 2

x = 8.

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Country B has a higher labor productivity than country A. This means that, on average, each worker in country B is producing more final goods per hour worked compared to each worker in country A.
Based on the given information, we can calculate the labor productivity of both countries.

Country A:
- Labor force = 800
- Hours worked per day = 8
- Total labor hours per day = 800 x 8 = 6,400
- Total final goods produced per day = 128,000
- Labor productivity = 128,000 / 6,400 = 20

Country B:
- Labor force = 1,800
- Hours worked per day = 6
- Total labor hours per day = 1,800 x 6 = 10,800
- Total final goods produced per day = 270,000
- Labor productivity = 270,000 / 10,800 = 25

Therefore, country B has a higher labor productivity than country A. This means that, on average, each worker in country B is producing more final goods per hour worked compared to each worker in country A.

Based on the provided information, Country A has a population of 1,000 with 800 people working 8 hours a day, producing 128,000 final goods. Country B has a population of 2,000, with 1,800 people working 6 hours a day, resulting in 270,000 final goods.

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To determine the minimum sample size needed for the researcher to be confident that his estimate is within "mm" hg of the true mean systolic blood pressure, we will use the following formula:

n = (Z * σ / E)²

where:
n = minimum sample size
Z = Z-score corresponding to the desired confidence level
σ = population standard deviation (in this case, "mm" hg)
E = margin of error (in this case, "mm" hg)

1. Determine the Z-score corresponding to the desired confidence level. Common confidence levels include 90%, 95%, and 99%, which correspond to Z-scores of 1.645, 1.960, and 2.576, respectively. Choose the appropriate Z-score based on the desired confidence level.

2. Substitute the given values of σ and E (both in "mm" hg) and the chosen Z-score into the formula: n = (Z * σ / E)²

3. Carry out the calculations, rounding the result up to the nearest whole number. This will ensure that the sample size is the minimum whole number that satisfies the requirements.

4. The result is the minimum sample size needed for the researcher to be confident that his estimate is within "mm" hg of the true mean systolic blood pressure.

Note: Since this question haven't provided specific values for standard deviation, margin of error, and desired confidence level . follow the steps with the specific values you have to find the minimum sample size.

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The probability that a point within the white circle lies on the red circle is P = 0.44

The probability will be given by the quotient between the area of the red square and the area of the circle.

On the diagram we can only see two squares, so Im assuming that it should say "square" instead of circle.

The area of the large square is:

A = 3*3 = 9

The area of the red square is:

A' = 2*2 = 4

Then the probability is:

P = 4/9 = 0.44

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If one dimension was quadrupled, second dimension was halved and third dimension did not change, the volume of a cube would be changed by 27 cubic centimeters to 54 cubic centimeters.

First we identify the original dimensions of the cube. It is given that volume of the cube is 27 cubic centimeters, each side of the cube will be 3 centimeters (3x3x3=27).

According to the question:

One dimension was quadrupled: 3 x 4 = 12cm

Second dimension was halved: 3 / 2 = 1.5cm

Third dimension did not change: 3cm

So, new dimensions are 12cm x 1.5cm x 3 cm

To find the new volume, we will use the formula:

V = product of dimensions

12 cm x 1.5 cm x 3 cm = 54 cubic centimeters

Therefore, the new volume of the cube would be 54 cubic centimeters.

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a. The probability that the network crashes on both Monday and Tuesday is 0.1024; b. The probability that the network crashes for the first time on Thursday is 0.1389 ; c. The probability that the network crashes every day is 0.0001; d. The probability that the network crashes on at least one day is 0.3222.

We can approach this problem using the binomial distribution. Let X be the number of days in a week that the network crashes. Then X follows a binomial distribution with parameters n = 5 (number of days in a workweek) and p = 0.16 (probability of a crash on any given day).

a. The probability that the network crashes on both Monday and Tuesday is:

P(X = 2) = (5 choose 2) * (0.16)² * (1-0.16)³

= 0.1024

b. The probability that the network crashes for the first time on Thursday is the probability that it does not crash on Monday, Tuesday, or Wednesday, but does crash on Thursday and/or Friday. So:

P(X = 1) * P(no crashes on Monday, Tuesday, Wednesday) = (5 choose 1) * (0.16) * (1-0.16)⁴ * (0.84)³

= 0.3808 * 0.3652

= 0.1389

c. The probability that the network crashes every day is:

P(X = 5) = (5 choose 5) * (0.16)⁵ * (1-0.16)⁰

= 0.0001

d. The probability that the network crashes on at least one day is the complement of the probability that it does not crash at all during the week:

P(X >= 1) = 1 - P(X = 0)

= 1 - (5 choose 0) * (0.16)⁰ * (1-0.16)⁵

= 1 - 0.6778

= 0.3222

Therefore, a. The probability that the network crashes on both Monday and Tuesday is 0.1024; b. The probability that the network crashes for the first time on Thursday is 0.1389 ; c. The probability that the network crashes every day is 0.0001; d. The probability that the network crashes on at least one day is 0.3222.

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