Arya has 5 cupcakes and wants to
share them with 8 friends equally.
How many cupcakes will each friend
receive?

(a) The density function of X can be computed by considering the cumulative distribution function (CDF) of X. Since X is uniformly distributed over the interval (0, 1), the CDF of X is given by F_X(x) = x for 0 ≤ x ≤ 1. To find the density function f_X(x), we differentiate the CDF with respect to x, resulting in f_X(x) = d/dx(F_X(x)) = 1 for 0 ≤ x ≤ 1. Therefore, X is uniformly distributed with density 1 over the interval (0, 1).

Similarly, the density function of Y can be obtained by considering the CDF of Y. Since Y is also uniformly distributed over the interval (0, 1), the CDF of Y is given by F_Y(y) = y for 0 ≤ y ≤ 1. Differentiating the CDF with respect to y, we find that the density function f_Y(y) = d/dy(F_Y(y)) = 1 for 0 ≤ y ≤ 1. Hence, Y is uniformly distributed with density 1 over the interval (0, 1).

(b) To compute the expected value of the area of the rectangle with corners (0, 0) and (X, Y), we can consider the product of X and Y, denoted by Z = XY. The expected value of Z can be calculated as E[Z] = E[XY]. Since X and Y are independent random variables, the expected value of their product is equal to the product of their individual expected values. Therefore, E[Z] = E[X]E[Y].

From part (a), we know that X and Y are uniformly distributed over the interval (0, 1) with density 1. Hence, the expected value of X is given by E[X] = ∫(0 to 1) x · 1 dx = [x²/2] evaluated from 0 to 1 = 1/2. Similarly, the expected value of Y is E[Y] = 1/2. Therefore, E[Z] = E[X]E[Y] = (1/2) · (1/2) = 1/4.

Thus, the expected value of the area of the rectangle with corners (0, 0) and (X, Y) is 1/4.

(c) The covariance between X and Y can be computed using the formula Cov(X, Y) = E[XY] - E[X]E[Y]. Since we have already calculated E[XY] as 1/4 in part (b), and E[X] = E[Y] = 1/2 from part (a), we can substitute these values into the formula to obtain Cov(X, Y) = 1/4 - (1/2) · (1/2) = 1/4 - 1/4 = 0.

Therefore, the covariance between X and Y is 0, indicating that X and Y are uncorrelated.

In conclusion, the density of X is 1 over the interval (0, 1), the density of Y is also 1 over the interval (0, 1), the expected value of the area of the rectangle with corners (0, 0) and (X, Y) is 1/4, and the covariance between X and Y is 0.

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The probability of second heart attack does the equation predict for someone who has taken the anger treatment course and whose anxiety level is 75 is 1.54%.Given that data on 20 patients is used to fit the logistic regression with the two predictor variables as follows: PredictorCoefSE CoefZPConstant-6.363473.21390-1.980.048AngerTrt-1.024111.17108-0.870.382Anxiety0.1190450.05497902.170.030.

Then, the logistic regression equation is:logit (p) = -6.363473 - 1.024111 x AngerTrt + 0.119045 x AnxietyWhere p is the probability of a second heart attack within 1 year, AngerTrt is whether the patient completed an anger-control treatment course (yes = 1), and Anxiety is the patient's score on a trait anxiety scale (a higher score means more anxious).

Now, let's find the probability of second heart attack for someone who has taken the anger treatment course and whose anxiety level is 75. For this, substitute AngerTrt = 1 and Anxiety = 75 in the equation.logit (p) = -6.363473 - 1.024111 x 1 + 0.119045 x 75= -6.363473 - 1.024111 + 8.930125= 1.542541p = exp(logit (p)) / (1 + exp(logit (p)))= exp(1.542541) / (1 + exp(1.542541))= 0.015423 = 1.54%Therefore, the probability of second heart attack does the equation predict for someone who has taken the anger treatment course and whose anxiety level is 75 is 1.54%.Option (C) is correct.

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By using the method of reduction of order as in differential equation to find a second linearly independent solution of the: Equation x2y" + xy' – 9y = 0 (x > 0); yı(x) = x3 has general solution is y(x) = c1x^3 + c2x^(-2),

Equation  4y" – 4y' + y = 0; yı(x) = ex/2 general solution is y(x) = c1exp(x/2) + c2*exp(-x/2),

Equation x2y" – x(x + 2)y' + (x + 2)y = 0 (x > 0); yı(x) = x has general solution  y2(x) = (C3x^(3/2) + C4)e^(-x).

Equation (x + 1)y" - (x + 2)y' + y = 0 (x > -1); yı(x) = ex has the general solution y(x) = c1ex + [c2 - ln(|2x + 1|)/2]ex.

Using the method of reduction of order, assume a second solution of the form y2(x) = u(x)y1(x). Then we have:

y'1(x)u(x) + y1(x)u'(x) = 0

u'(x) = -y'1(x)u(x)/y1(x)

Integrating both sides:

ln|u(x)| = -ln|y1(x)| + C

u(x) = K/x^3

Plugging this into the differential equation:

x^2y'' + xy' - 9y = 0

x^2[u''(x)y1(x) + 2u'(x)y1'(x) + u(x)y1''(x)] + x[u'(x)y1(x) + u(x)y1'(x)] - 9u(x)y1(x) = 0

Simplifying and dividing by x^2y1(x):

u''(x) - 6/x^2 u(x) = 0

Equation r(r-1) - 6 = 0, which has roots r = 3 and r = -2. Therefore, the general solution is y(x) = c1x^3 + c2x^(-2).

Using the method of reduction of order, assume a second solution of the form y2(x) = u(x)y1(x). Then we have:

y'1(x)u(x) + y1(x)u'(x) = 0

u'(x) = -y'1(x)u(x)/y1(x)

Integrating both sides:

ln|u(x)| = -2ln|y1(x)| + C

u(x) = Kexp(-x/2)

Plugging this into the differential equation:

4y'' - 4y' + y = 0

4[u''(x)y1(x) + 2u'(x)y1'(x) + u(x)y1''(x)] - 4[u'(x)y1(x) + u(x)y1'(x)] + u(x)y1(x) = 0

Simplifying and dividing by 4y1(x):

u''(x) - u(x)/4 = 0

equation r^2 - 1/4 = 0, has roots r = 1/2 and r = -1/2. Therefore, the general solution is y(x) = c1exp(x/2) + c2*exp(-x/2).

Let y2(x) = v(x)y1(x), where v(x) is a function to be determined.

Then, y'2(x) = v'(x)y1(x) + v(x)y'1(x) and y"2(x) = v"(x)y1(x) + 2v'(x)y'1(x) + v(x)y"1(x).

Substituting y1(x) and y2(x) into the given differential equation, we get:

x^2(v"(x)y1(x) + 2v'(x)y'1(x) + v(x)y"1(x)) - x(x+2)(v'(x)y1(x) + v(x)y'1(x)) + (x+2)v(x)y1(x) = 0

Simplifying and dividing by x^2y1(x), we obtain:

v"(x) + (2/x - (x+2)/x^2)v'(x) + ((x+2)/x^2 - 1/x^2)v(x) = 0

Let u(x) = v'(x). Then, the above equation can be written as a first-order linear differential equation:

u'(x) + (2/x - (x+2)/x^2)u(x) + ((x+2)/x^2 - 1/x^2)v(x) = 0

Using an integrating factor of exp(∫[(2/x - (x+2)/x^2)dx]), we get:

u(x)/x^2 = C1 + C2∫exp(-2lnx)exp((x+2)/x)dx

u(x)/x^2 = C1 + C2/x^2e^(x+2)

v(x) = C3x^(1/2)e^(-x) + C4x^(-3/2)e^(-x)

Therefore, the second linearly independent solution is:

y2(x) = (C3x^(3/2) + C4)e^(-x)

41. (x + 1)y" - (x + 2)y' + y = 0, one solution y1(x) = ex.

We assume that the second solution has the form y2(x) = v(x)ex.

We can then find y2'(x) and y2''(x) as follows:

y2'(x) = v'(x)ex + v(x)ex

y2''(x) = v''(x)ex + 2v'(x)ex + v(x)ex

We can substitute y1(x) and y2(x) into the differential equation and simplify using the above expressions for y2'(x) and y2''(x):

(x + 1)[v''(x)ex + 2v'(x)ex + v(x)ex] - (x + 2)[v'(x)ex + v(x)ex] + v(x)ex = 0

Simplifying and dividing by ex, we get:

xv''(x) + (2x + 1)v'(x) = 0

This is a first-order linear differential equation, which we can solve using separation of variables:

v'(x) = -1/(2x + 1) dv/dx

Integrating both sides

v(x) = C1 - ln(|2x + 1|)/2

where C1 is a constant of integration.

Therefore, the second linearly independent solution is:

y2(x) = v(x)ex = [C1 - ln(|2x + 1|)/2]ex

So, the general solution is:

y(x) = c1ex + [c2 - ln(|2x + 1|)/2]ex

where c1 and c2 are constants of integration.

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____The given question is incomplete, the complete question is given below:

In each of Problems 38 through 42, a differential equation and one solution yı are given. Use the method of reduction of or- der as in Problem 37 to find a second linearly independent solution y2. 38. x2y" + xy' – 9y = 0 (x > 0); yı(x) = x3 39. 4y" – 4y' + y = 0; yı(x) = ex/2 40. x2y" – x(x + 2)y' + (x + 2)y = 0 (x > 0); yı(x) = x 41. (x + 1)y" - (x + 2)y' + y = 0 (x > -1); yı(x) = ex

The correct matching of the algebraic equation to the verbal description of an equivalent equation is given below:

An algebraic equation, sometimes known as a polynomial equation, in mathematics, is a formula of form P=0, where P is a polynomial with coefficients in some field, typically the field of the rational numbers.

The equivalent description has been matched to the given algebraic equation above,

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The number line for the set of jump distances to make a new record.

Option B is the correct answer.

It is the representation of numbers in real order.

The difference between the consecutive numbers in a number line is always positive.

We have,

The school record in the long jump = 518 cm

Now,

To make a new record the set of jump distances should be greater than 518 cm.

To represent the set of jump distances on a number line we can not have a black dot on 513 on the number line.

The dot should be an open dot.

Thus,

Option B is the number line for the set of jump distances to make a new record.

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The solution for the question is  give n as

A: the obvious transformation is 16 points upward

B:  k=16

C:   g(x) = f(x)+k = f(x)+16

This is further explained below.

Generally, There are four distinct techniques to alter the form and/or location of a point, line, or geometric figure, and they are all included under the umbrella term "transformation." We refer to the object's pre-transformation shape as the "Pre-Image," while its post-transformation form, or "Image," describes its new form and location.

In conclusion,

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Answer:

  396 ft³

Step-by-step explanation:

You want the volume of the triangular prism with right triangle base having legs 8 ft and 9 ft, and prism height 11 ft.

The area of the triangular base is ...

  A = 1/2bh

  A = 1/2(8 ft)(9 ft) = 36 ft²

The height of the whole prism is shown as 11 feet.

The volume of the prism is given by ...

  V = Bh . . . . . where B is the base area, and h is the height

  V = (36 ft²)(11 ft) = 396 ft³

The volume of the prism is 396 ft³.

__

Additional comment

You know that the area of the base triangle is half the area of its enclosing rectangle. This means the volume of the prism is half the volume of the enclosing cuboid. It could be figured as V = 1/2(LWH).

<95141404393>

Yes, m∠ABC and m∠XYZ are congruent

No, lines FG and DE are not congruent

Yes, OR and OS are congruent

Yes, 2∠ and ∠4 are congruent.

Congruent angles are angles that have the same measure, in degrees and are often represented by the symbol "≅". While congruent line segments are two or more line segments that have the same length.

m∠XYZ = 180° - 140° {supplementary angles}

m∠XYZ = 40°

so;

m∠ABC and m∠XYZ are congruent

line FG = and line DE =

so;

FG and DE are not congruent

OR = 2in and OS = 2in

so;

OR and OS are congruent

∠2 and ∠4 are vertical angles and are equal so;

2∠ and ∠4 are congruent

Therefore, (m∠ABC and m∠XYZ), (2∠ and ∠4), and (OR and OS), are two pairs of angles and line segments respectively which are congruent, while the lines segments FG and DE are not congruent.

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Answer:

5  7/24

Step-by-step explanation:

Answer:

5.29

Step-by-step explanation:

Answer:

  11 -√20

Step-by-step explanation:

If you want ...

  √20 + x = 11

You can find x by subtracting √20 from both sides of the equation:

  x = 11 -√20

The number you're looking for is 11 -√20.

Answer:

SAS

Step-by-step explanation:

In triangle ABC and triangle EFD

1. BC = CD (S) Given

2. < BCD = < EFD (A) being vertically opposite. angles.

3. AC = CE (S) Given

Hence

By SAS postulate Triangle ABC and triangle EFD are congruent.

HOPE IT HELPS :)❤

Answer:

The answer would be C.

Step-by-step explanation:

0.2 x 5 = 1

1 x 5 =5

5 x 5 = 25

So your next number in the sequence should be 125.

Answer:

I think D the function has a maximum value of -61

Step-by-step explanation:

The probability of the commuting time being between 50 and 60 minutes is determined for a train with a uniformly distributed commuting time between 40 and 90 minutes.

In a uniform distribution, the probability density function (PDF) is constant within the range of the distribution. In this case, the commuting time is uniformly distributed between 40 and 90 minutes. The PDF for a uniform distribution is given by:

f(x) = 1 / (b - a)

where 'a' is the lower bound (40 minutes) and 'b' is the upper bound (90 minutes) of the distribution.

To find the probability that the commuting time falls between 50 and 60 minutes, we need to calculate the area under the PDF curve between these two values. Since the PDF is constant within the range, the probability is equal to the width of the range divided by the total width of the distribution.

The width of the range between 50 and 60 minutes is 60 - 50 = 10 minutes. The total width of the distribution is 90 - 40 = 50 minutes.

Therefore, the probability that the commuting time will be between 50 and 60 minutes is:

P(50 ≤ x ≤ 60) = (width of range) / (total width of distribution) = 10 / 50 = 1/5 = 0.2, or 20%.

Thus, there is a 20% probability that the commuting time on this particular train will be between 50 and 60 minutes.

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The supremum of the set A does not exist (it is negative infinity), and the infimum of the set A is 1.

To define the supremum and infimum of the set A, we first need to determine the properties of the set.

The set A is defined as A = {(x-3)/(x-2) ∈ R : x < 0}.

To find the supremum (also known as the least upper bound) of A, we need to find the smallest value that is greater than or equal to all the elements of A. In other words, we are looking for the least upper bound of the set A.

Let's analyze the elements of A:

For x < 0, the expression (x-3)/(x-2) can take on different values depending on the value of x. We need to find the maximum value that this expression can reach for all x < 0.

As x approaches 0 from the left side, (x-3)/(x-2) approaches negative infinity. Therefore, there is no finite supremum for the set A.

Next, let's find the infimum (also known as the greatest lower bound) of A. We need to find the largest value that is less than or equal to all the elements of A. In other words, we are looking for the greatest lower bound of the set A.

Again, analyzing the elements of A:

As x approaches negative infinity, (x-3)/(x-2) approaches 1. Therefore, the infimum of the set A is 1.

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The  3 examples of quadratic equations written in standard form is:

(1) a\(x^2\) + bx + c = 0

(2) a\((x - h)^2\) + k = 0

(3) a (x - p)(x - q) = 0

a\(x^2\) + bx + c = 0 is the quadratic equation's standard form, where 'a' is the leading coefficient and a non-zero real number. Due to the word "quad" meaning "square," this equation, whose degree is 2, is referred to as "quadratic." A quadratic equation can be expressed in other ways in addition to the usual form.

Vertex Form: a\((x - h)^2\) + k = 0

Intercept Form: a (x - p)(x - q) = 0

\(ax^2 + bx + c = 0\), where a 0 and a, b, and c are real numbers, is the conventional form of a quadratic equation with a variable x.

a is the \(x^2\) coefficient, b is the x coefficient, c is the constant.

In this case, b and c might either be zeros or non-zero values.

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The 95% confidence interval for the proportion of voters planning to vote for the incumbent candidate is approximately 0.536 to 0.664.

a one-sample z interval for a proportion, we need the sample proportion, sample size, and the desired level of confidence. Let's use the given information to calculate the confidence interval.

Sample proportion (p(cap)) = 60% = 0.60 Sample size (n) = 150

Let's assume a desired level of confidence of 95%. This means we want to construct a 95% confidence interval.

To construct the interval, we follow these steps:

The standard error (SE) of the sample proportion: SE = √(p(cap) × (1 - p(cap)) / n)

SE = √(0.60 × 0.40 / 150)

The critical value (z) corresponding to the desired level of confidence. For a 95% confidence interval, the z value is approximately 1.96. You can find the specific value using a standard normal distribution table or a statistical software.

The margin of error (ME)

ME = z × SE

ME = 1.96 × SE

The lower and upper bounds of the confidence interval:

Lower bound = p(cap) - ME

Upper bound = p(cap) + ME

Lower bound = 0.60 - ME

Upper bound = 0.60 + ME

Now, substitute the calculated values into the formulas

SE = √(0.60 × 0.40 / 150)

ME = 1.96 × SE

Lower bound = 0.60 - ME

Upper bound = 0.60 + ME

Calculate the values to find the confidence interval

SE ≈ 0.0326

ME ≈ 0.064

Lower bound ≈ 0.60 - 0.064 ≈ 0.536

Upper bound ≈ 0.60 + 0.064 ≈ 0.664

Therefore, the 95% confidence interval for the proportion of voters planning to vote for the incumbent candidate is approximately 0.536 to 0.664.

The sample size is 150, and 60% of the selected voters (0.6 × 150 = 90 voters) plan to vote for the incumbent candidate. Since both the number of successes (90) and the number of failures (60) in the sample are greater than 10, the condition of normality is met.

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The number of individuals with the virus, N, follows a binomial distribution with n = 100 (the sample size) and p = 0.05 (the probability of an individual having the virus).

(a) The probability distribution of N follows a binomial distribution since we have a fixed sample size (n = 100) and each individual either has the virus (success) or does not have the virus (failure). The probability of an individual having the virus is p = 0.05, and the probability of an individual not having the virus is q = 1 - p = 0.95. Therefore, the probability distribution of N can be described by the binomial distribution B(n, p), where n = 100 and p = 0.05.

(b) The expected number E(N) of individuals with the virus in the sample can be calculated as E(N) = n * p = 100 * 0.05 = 5 individuals. This means, on average, we expect to find 5 individuals with the virus in the sample of 100 individuals.

(c) The probability of getting exactly 5 individuals with the virus in our sample of 100 individuals can be calculated using the binomial probability formula: P(N = 5) = C(n, 5) * p^5 * q^(n-5), where C(n, 5) represents the number of ways to choose 5 individuals from the sample of 100.

(d) The standard deviation of N, denoted as σ(N), can be calculated as the square root of the variance, where the variance is given by Var(N) = n * p * q. Therefore, σ(N) = sqrt(n * p * q) = sqrt(100 * 0.05 * 0.95).

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The rate of change of y with respect to x is given by dy/dx = xy - (3/2)x²y.

To find the rate of change of y with respect to x, we need to differentiate the given equation. The rate of change can be determined by taking the derivative of both sides of the equation with respect to x.

First, let's differentiate each term separately using the rules of differentiation.

Differentiating x²y with respect to x gives us 2xy using the product rule.

To differentiate 5, we know that a constant has a derivative of 0.

Differentiating 2ln(y) with respect to x requires the chain rule. The derivative of ln(y) with respect to y is 1/y, and then we multiply by dy/dx. So, the derivative of 2ln(y) is 2/y * dy/dx.

Differentiating x³ gives us 3x² using the power rule.

Now, we can rewrite the equation with its derivatives:

2xy - 2/y * dy/dx = 3x²

To solve for dy/dx, we can isolate it on one side of the equation. Rearranging the equation, we get:

2xy = 2/y * dy/dx + 3x²

To isolate dy/dx, we move the term 2/y * dy/dx to the other side:

2xy - 2/y * dy/dx = 3x²

2xy = 2/y * dy/dx + 3x²

2/y * dy/dx = 2xy - 3x²

Now, we can solve for dy/dx by multiplying both sides by y/2:

dy/dx = (2xy - 3x²) * (y/2)

Simplifying further, we have:

dy/dx = xy - (3/2)x²y

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Answer:

5.75 , 5.76, 5.73

Step-by-step explanation:

Hope this helps!

Answer:

x= -5 or x=4

Step-by-step explanation:

Lets first turn this into quadratic formula form

(-1 ±\(\sqrt{1^2-4*1*-20}\)) / 2

(-1 ± 9 )/2

-10 / 2   =x or 8/2 =x

x= -5 or x=4

It was hard to type it in the comments hope this helped

The probability that in 63 tosses of a fair die, fewer than 6 fours will be obtained is approximately 0.068. The probability that in 48 tosses of a fair die, fewer than 9 fours will be obtained is approximately 0.026.

1. Let X be the number of fours in 63 tosses of the fair die. X is a binomial random variable with n = 63 and p = 1/6 (since there is 1 favorable outcome and 6 possible outcomes on each toss). We need to find P(X < 6), which can be calculated using the normal approximation to the binomial distribution. The mean and standard deviation of X are:

μ = np

μ = 63/6

μ = 10.5

\(\sigma = \sqrt{np(1-p)}\)

\(\sigma = \sqrt{63/6 \times 5/6}\)

σ ≈ 2.485

To use the normal approximation, we standardize X:

Z = (X - μ)/σ = (5.5 - 10.5)/2.485 ≈ -2.01

We want to find P(X < 6), which is equivalent to P(Z < (6 - 10.5)/2.485) = P(Z < -1.80). From a standard normal distribution table or calculator, we find that P(Z < -1.80) ≈ 0.03593. Therefore, P(X < 6) ≈ 0.03593, which when rounded to the nearest thousandth is approximately 0.068.

2. Let X be the number of fours in 48 tosses of the fair die. X is a binomial random variable with n = 48 and p = 1/6. We need to find P(X < 9), which can be approximated using the normal distribution.

μ = np

μ = 48/6

μ = 8

\(\sigma = \sqrt{np(1-p)}\)

\(\sigma = \sqrt{48/6 \times 5/6}\)

σ ≈ 2.044

To use the normal approximation, we standardize X:

Z = (X - μ)/σ = (8.5 - 8)/2.044 ≈ 0.25

We want to find P(X < 9), which is equivalent to P(Z < (9 - 8)/2.044) = P(Z < 0.49). From a standard normal distribution table or calculator, we find that P(Z < 0.49) ≈ 0.68681. Therefore, P(X < 9) ≈ 0.68681, which when rounded to the nearest thousandth is approximately 0.026.

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Answer:

1) 136

2) 136

3) 44

Step-by-step explanation:

trapezoid = 360 degrees

44 = the 3rd angle because they are equal

add the two angles(44 and 44) 44 + 44 = 88

that makes up half the shape

next 360 - 88 to find the last two

272 is the remaining measurements, split in half to find 1 and 2

272/2 = 136

In a regression analysis, the regression equation given is y = 12 - 6x. The correct option for the coefficient of correlation is b. -0.7.

The terms SSE (sum of squared errors) and SST (total sum of squares) are provided, with values 510 and 1000, respectively. To determine the coefficient of correlation (r), we need to first calculate the coefficient of determination (R²), which is given by the formula:
R² = (SST - SSE) / SST
Substituting the given values, we get:

R² = (1000 - 510) / 1000 = 490 / 1000 = 0.49
Now, we need to find the correlation coefficient (r), which is the square root of the coefficient of determination (R²). However, we need to determine the sign (positive or negative) based on the regression equation. Since the slope of the equation (in this case, -6) is negative, the correlation coefficient will also be negative. Therefore, we have:
r = -√0.49 = -0.7

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Answer:

16+(kx3)

Step-by-step explanation:

You add 16 to kx3

The fourth term of the geometric sequence is 6.4.

We have to given that,

The first term of a geometric sequence is 20 and the second term is 8.

Let's denote the common ratio of the geometric sequence by r.

We know that the first term is 20,

so a₁ = 20,

And the second term is 8,

so a₂ = 20r = 8.

Solving for r, we get:

r = a₂/a₁ = = 8/20 = 2/5

Now, we want to find the fourth term of the sequence, which is a₄.

We can use the formula for the nth term of a geometric sequence, which is:

a (n) = a₁ rⁿ⁻¹

Plugging in n=4, a₁=20, and r=2/5, we get:

a₄ = 20 (2/5)³

a₄ = 6.4

Therefore, the fourth term of the geometric sequence is 6.4.

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Answer:

Complex numbers

Step-by-step explanation:

Given

5 + i and 5 - i

Required

What are they called?

In numbering system,

Complex numbers are of the form:

\(a + bi\)

Comparing the given numbers to the form of complex number, we'll see that there's a perfect match between the two

i.e.

\(a + bi = 5 + i\)

\(a + bi = 5 - i\)

Hence, they are referred to as complex numbers

The least squares solution of the inconsistent linear system is x1 = 0.754, x2 = 0.027, x3 = 1.169, y1 = 0, y2 = 0, y3 = 0.

To find the least squares solution of the inconsistent linear system, we first need to rewrite the system in matrix form, Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants.

A =

|1 1 1 0 0 0|

|0 0 1 0 0 0|

|1 0 1 0 0 0|

|2 0 5 0 0 0|

|-7 8 0 0 0 0|

|1 2 -1 0 0 0|

x =

|x1|

|x2|

|x3|

|y1|

|y2|

|y3|

b =

|3|

|1|

|2|

|8|

|0|

|1|

Since the system is inconsistent, we cannot find a solution that satisfies all the equations. Therefore, we need to find the least squares solution, which is the solution that minimizes the sum of the squares of the residuals.

To find the least squares solution, we need to find the vector x that minimizes ||Ax - b||^2, where ||.|| is the Euclidean norm.

The least squares solution is given by x = (A^T A)^-1 A^T b.

First, we calculate A^T A and A^T b:

A^T A =

|55 -10 11 0 0 0|

|-10 70 -3 0 0 0|

|11 -3 19 0 0 0|

|0 0 0 0 0 0|

|0 0 0 0 0 0|

|0 0 0 0 0 0|

A^T b =

|5|

|7|

|5|

|8|

|-56|

|4|

Next, we calculate (A^T A)^-1:

(A^T A)^-1 =

|0.0377 -0.0066 0.0063 0 0 0|

|-0.0066 0.0149 0.0020 0 0 0|

|0.0063 0.0020 0.0569 0 0 0|

|0 0 0 0 0 0|

|0 0 0 0 0 0|

|0 0 0 0 0 0|

Finally, we calculate x = (A^T A)^-1 A^T b:

x =

|0.754|

|0.027|

|1.169|

|0|

|0|

|0|

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Answer:

The lateral surface of an object is all of the sides of the object , except/excluding its base and top.

I think I am correct I might sure I read it somewhere

by this explanation I guess you know your answer

have a nice day

#Captainpower

Given:

The scale factor is 1:12.

Dimension of model = 32 cm

To find:

The actual dimension in m.

Solution:

Let x be the actual dimension.

The scale factor is 1:12 and the dimension of model is 32 cm.

\(\dfrac{1}{12}=\dfrac{32}{x}\)

On cross multiplication, we get

\(x=32\times 12\)

\(x=384\ cm\)

\(x=3.84\ m\)             \([\because 1\ m=100\ cm]\)

Therefore, the actual dimension is 3.84 m.