A simple random sample of size n= 49 is obtained from a population that is skewed right with µ = 83 and σ = 7. (a) Describe the sampling distribution of x. (b) What is P (x > 84.9) ? (c) What is P (x ≤ 76.7) ?
(d) What is P (78.1 < x < 85.2) ?

The solution of the linear equation:

7x – 13 = ─2x + 5

is x = 2

Here we want to find the value of x that is a solution of:

7x - 13 = -2x + 5

To solve it, we need to isolate x in one of the sides of the equation.

7x - 13 = -2x + 5

7x + 2x = 5 + 13

9x = 18

x = 18/9

x = 2

The value x = 2 makes the given linear equation true.

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Given the function h(x) = 2x - 3 and g(x) = -2x + 2, g(h(x)) = -4x + 8

A function is a mathematical relationship that shows the relationship between two variables.

Given the functions h(x) = 2x - 3 and g(x) = -2x + 2

We desire to find (goh)(x), we proceed as follows.

Since we have the functions h(x) = 2x - 3 and g(x) = -2x + 2

We notice that  (goh)(x) = g(h(x))

So, substituting the values of the variables into the equation, we have that

(goh)(x) = g(h(x))

= -2x + 2

Substituting h(x) = 2x - 3 into the equation, we have that

g(h(x)) = -2x + 2

= -2(2x - 3) + 2

= -4x + 6 + 2

= -4x + 8

So, g(h(x)) = -4x + 8

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a.  The probability distribution of the test statistic is approximately a standard normal distribution.

b. The p-value for the test of  factors that affect the prevalence of tuberculosis among intravenous drug users is 0.0202.  

c. We can conclude that there is a statistically significant difference between the two groups in terms of their proportions of positive skin test results.

d. The 95% confidence interval does not contain zero, so there is a statistically significant difference between the two groups in terms of their proportions of positive skin test results.

To estimate the common value of p assuming that the population proportions of positive skin test results are equal, we can compute the pooled proportion:

p-hat = (x1 + x2) / (n1 + n2)

= (24.7 + 17.4) / (97 + 161)

= 0.195

where x1 and x2 are the number of individuals with positive skin test results in the two groups, and n1 and n2 are the sample sizes.

a. To test the null hypothesis that the proportions of intravenous drug users who have positive tuberculin skin test results are identical for those who share needles and those who do not, we can use a two-sample z-test for proportions. The test statistic is:

z = (p1 - p2) / sqrt(phat * (1 - phat) * (1/n1 + 1/n2))

where p1 and p2 are the sample proportions, phat is the pooled proportion, and n1 and n2 are the sample sizes.

Plugging in the values, we get:

z = (0.247 - 0.174) / sqrt(0.195 * (1 - 0.195) * (1/97 + 1/161))

= 2.05

The probability distribution of the test statistic is approximately a standard normal distribution, since the sample sizes are large enough (both n1 and n2 are greater than 30).

b. The p-value for the test is the probability of observing a z-value of 2.05 or more extreme under the null hypothesis. From a standard normal distribution table or calculator, we find that the p-value is approximately 0.0202 (or 0.0404 for a two-tailed test).

Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that the proportions of intravenous drug users who have positive tuberculin skin test results are not identical for those who share needles and those who do not.

c. To construct a 95% confidence interval for the true difference in proportions, we can use the formula:

(p1 - p2) ± z* sqrt(phat * (1 - phat) * (1/n1 + 1/n2))

where z is the critical value for a 95% confidence interval from a standard normal distribution (z = 1.96).

Plugging in the values, we get:

(0.247 - 0.174) ± 1.96 * sqrt(0.195 * (1 - 0.195) * (1/97 + 1/161))

= 0.073 ± 0.090

Therefore, we can be 95% confident that the true difference in proportions of intravenous drug users who have positive tuberculin skin test results between those who share needles and those who do not is between 0.073 and -0.073 (which can be written as an absolute value of 0.073).

d.  We can infer that there is a statistically significant difference between the two groups in terms of the proportions of positive skin test results because the interval does not contain zero.

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Among the given numbers, -0.52 is closer to -1/2, 0.35 is closer to 0, and 0.25 is closer to both 0 and 1/2.

To determine which number is closer to -1/2, 0, and 1/2, we need to find the absolute value of the difference between each number and the three given values, and then compare the results.

- For 0.35: The absolute value of the difference between 0.35 and -1/2 is approximately 0.85, the absolute value of the difference between 0.35 and 0 is 0.35, and the absolute value of the difference between 0.35 and 1/2 is approximately 0.15. Therefore, 0.35 is closer to 0 than to -1/2 or 1/2.

- For -3/5: The absolute value of the difference between -3/5 and -1/2 is approximately 0.05, the absolute value of the difference between -3/5 and 0 is approximately 0.6, and the absolute value of the difference between -3/5 and 1/2 is approximately 0.95. Therefore, -3/5 is closer to 0 than to -1/2 or 1/2.

- For -0.52: The absolute value of the difference between -0.52 and -1/2 is approximately 0.02, the absolute value of the difference between -0.52 and 0 is approximately 0.52, and the absolute value of the difference between -0.52 and 1/2 is approximately 1.02. Therefore, -0.52 is closer to -1/2 than to 0 or 1/2.

- For 0.25: The absolute value of the difference between 0.25 and -1/2 is approximately 0.75, the absolute value of the difference between 0.25 and 0 is 0.25, and the absolute value of the difference between 0.25 and 1/2 is approximately 0.25. Therefore, 0.25 is closer to 0 and 1/2 than to -1/2.

- For 3/5: The absolute value of the difference between 3/5 and -1/2 is approximately 1.15, the absolute value of the difference between 3/5 and 0 is approximately 0.6, and the absolute value of the difference between 3/5 and 1/2 is approximately 0.05. Therefore, 3/5 is closer to 0 than to -1/2 or 1/2.

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Answer: #3 = 22/50 (reduced version is 11/25

Step-by-step explanation:

Answer:

Step-by-step explanation:

Predicted probabilities are different then experimental probabilities.  Experimental probabilities use the actual data.

P(red)= red/(total)= 22/(12+15+22)  = 22/50 = .44

P(hot cocoa = hot cocoa/total =5/(7+5+8) = 1/4 = .25

The slope of the line representing the rope is -1/4.

To find the slope of the line representing the rope, we need to use the formula for slope, which is:

slope = rise / run

In this case, the rise is the height of the wall, which is 6 feet, and the run is the horizontal distance between the wall and the point where the rope is attached to the ground, which is 24 feet. However, we need to be careful about the sign of the slope, since the rope is going down from the wall to the ground.

To take this into account, we can use the convention that a positive slope means a line is going up from left to right, and a negative slope means a line is going down from left to right. In this case, since the rope is going down, we know the slope will be negative.

So, we can calculate the slope as follows:

slope = - rise / run
slope = - 6 / 24
slope = - 1/4

Therefore, the line representing the rope has a slope of -1/4.

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Question: "Identify the correct description for the formula g'(x) ≈ g(x)/h – g(x – h)/h from the following options: FFD1: forward finite difference with stepsize h for the first derivative of g at a BFD1: backward finite difference with stepsize h for the first derivative of g at a CFD1: central finite difference with stepsize h for the first derivative of g at x CFD2: central finite difference with stepsize h for the second derivative of g at x None of the Above"

The correct description for the formula g'(x) ≈ g(x)/h – g(x – h)/h is BFD1: backward finite difference with stepsize h for the first derivative of g at a.

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Therefore, the area of the trapezoid is 1338 3/4 square units.

A quadrilateral with at least one set of parallel sides is known in geometry as a trapezium, or trapezium in British and other versions of English. In Euclidean geometry, a trapezium is invariably a convex quadrilateral. The trapezoid's parallel sides are referred to as its bases.

To find the area of a trapezoid, we use the formula:

Area = (1/2) × (sum of parallel sides) × (height)

In this case, we have the following information:

The two parallel sides are 12 1/4 and 24 1/4.

The height is 73 1/2.

First, we need to add the two parallel sides to find the sum:

12 1/4 + 24 1/4 = 36 1/2

Next, we can plug these values into the formula:

Area = (1/2) × 36 1/2 × 73 1/2

Area = 1338.75

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Correct Question:

Given sides are 12(1/4), 24 (1/4), 73 (1/2) and 134(1/2), then What is the area of this trapezoid?

Answer:

Step-by-step explanation:

im pretty sure it is 5:2

The standard deviation of this sample of counts is approximately 1.83.

To find the standard deviation of the sample of red blood cell counts, we can use the formula:

s = sqrt [ Σ(x - X)2 / (n - 1) ]

where Σ(x - X)2 is the sum of the squared deviations from the mean, n is the sample size, and X is the sample mean.

First, we need to find the sample mean:

X = (53 + 49 + 54 + 51 + 48 + 51) / 6 = 51

Next, we can calculate the squared deviations from the mean for each observation:

(53 - 51)2 = 4

(49 - 51)2 = 4

(54 - 51)2 = 9

(51 - 51)2 = 0

(48 - 51)2 = 9

(51 - 51)2 = 0

Then we can sum these squared deviations:

Σ(x - X)2 = 4 + 4 + 9 + 0 + 9 + 0 = 26

Finally, we can plug in these values into the formula for the standard deviation:

s = sqrt [ Σ(x - X)2 / (n - 1) ] = sqrt [ 26 / (6 - 1) ] ≈ 1.83

Therefore, the standard deviation of this sample of counts is approximately 1.83.

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The general solution to the original differential equation is:  

y(t) = [tex]C1 e^t cos t + C2 e^t sin t + (1/2)ex sin t + (1/4)ex sin(2t) + (1/4)ln|[/tex]

We first solve the associated homogeneous differential equation:

[tex]2y'' - 4y' + 4y[/tex] = 0

The characteristic equation is[tex]r^2[/tex] - 2r + 2 = 0, which has roots r = 1 ± i. Therefore, the general solution to the homogeneous equation is:

[tex]y_h(t) = e^t([/tex]C1 cos t + C2 sin t)

To use the method of variation of parameters to find the particular solution to the original equation, we assume that the solution has the form:

[tex]y_p(t) = u(t)e^t cos t + v(t)e^t sin t[/tex]

where u(t) and v(t) are functions to be determined.

[tex]y_p''(t) \\\\2u'(t)e^t cos t + 2v'(t)e^t sin t + 2u(t)e^t cos t - 2v(t)e^t sin t - 2u(t)e^t sin t - 2v(t)e^t cos t[/tex]

[tex]y_p'(t) = u'(t)e^t cos t + v'(t)e^t sin t + u(t)e^t cos t + v(t)e^t sin t[/tex]

Substituting these into the original equation and simplifying, we get:

[tex]2u'(t)e^t cos t + 2v'(t)e^t sin t = ex sec x[/tex]

We need to find u'(t) and v'(t) such that this equation holds for all t. To do this, we take the derivative of the assumed solution with respect to t and equate coefficients of cos t and sin t separately:

[tex]u'(t)e^t cos t + v'(t)e^t sin t + u(t)e^t cos t + v(t)e^t sin t = 0 (1)\\v'(t)e^t cos t - u'(t)e^t sin t + u(t)e^t sin t - v(t)e^t cos t = ex sec x (2)[/tex]

Solving equation (1) for u'(t) and v'(t) and substituting into equation (2), we get:

[tex]v(t) = ∫ [ex sec x / (e^(2t))] dt\\u(t) = -∫ [ex sec x / (e^(2t))] tan t dt[/tex]

Evaluating the integrals, we get:

[tex]v(t) = (1/2)ex tan x - (1/2)ln|cos x| + C1\\u(t) = (1/4)ex [sin(2t) - 2cos(2t)] + (1/4)ln|cos x| tan x + C2[/tex]

where C1 and C2 are arbitrary constants.

The general solution to the original differential equation is:  

y(t) = [tex]C1 e^t cos t + C2 e^t sin t + (1/2)ex sin t + (1/4)ex sin(2t) + (1/4)ln|[/tex]

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T3 = S(a, 1) = {x ∈ M : d(x,a) < 1} = {a} U (M{a}), using the same argument as for S3 in part (a).

In a discrete metric space, any subset of the space is an open set, since every point has a neighborhood of radius 1 that contains only that point. Therefore, for any point a in the discrete metric space, we have:

(a) Si = B(a, j) = {x ∈ M : d(x,a) < j} = {a}, since the only point within a distance of j from a is a itself.

S2 = B(a, g) = {x ∈ M : d(x,a) < g} = {a}, since the only point within a distance of g from a is a itself.

S3 = B(a, 1) = {x ∈ M : d(x,a) < 1} = {a} U (M{a}), since every point in M except a is within a distance of 1 from a, so the open ball of radius 1 centered at a contains all points in M except a, as well as a itself.

(b) Ti = S(0,7) = {x ∈ M : d(x,0) < 7} = M, since every point in the discrete metric space is within a distance of 7 from 0.

T2 = S(a, 1) = {x ∈ M : d(x,a) < 1} = {a} U (M{a}), using the same argument as for S3 in part (a).

T3 = S(a, 1) = {x ∈ M : d(x,a) < 1} = {a} U (M{a}), using the same argument as for S3 in part (a).

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The frog's probability of being on lily pad B after an hour (or in the long run) is around 0.2975 or 29.75%.

To calculate the probability that the frog will be on lily pad B after an hour, we can start by calculating the long-run distribution or steady-state probabilities of the frog's movement between the two lily pads.

Let p represent the probability that the frog is on lily pad B at any given time. We may develop two equations based on the probabilities given in the problem:

p = 0.85(1-p) (the frog moves from point A to point B with a probability of 0.85 and stays on point B with a probability of 1-p).

1-p = 0.65p (the frog moves from B to A with the probability of 0.65 and stays on A with the probability of 1-p).

Simplifying the equation, we get:

p = 0.85 - 0.85p + 0.65p

p = 0.85(1 - 0.65) = 0.2975

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Check the picture below.

The number of individuals in the sample with the specified characteristics is 258.

Based on the given information, we can answer the following:

1. The point estimate of the population proportion is 0.215. This value is given in the problem statement.

2. The margin of error is 0.123. This value is also given in the problem statement. Now, let's determine the number of individuals in the sample with the specified characteristics.

3. We know the sample size (n) is 1,200. We can use the point estimate of the population proportion (0.215) to calculate the number of individuals with the specified characteristics:

Number of individuals = n * point estimate

Number of individuals = 1,200 * 0.215 (Round to the nearest integer as needed)

Number of individuals = 258

So, the number of individuals in the sample with the specified characteristics is 258.

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Answer:

[tex] {9}^{x} \div {81}^{ \frac{2}{x} } = 1[/tex]

[tex]{9}^{x} \times {81}^{ \frac{ - 2}{x} } = 1[/tex]

[tex]{9}^{x} \times { ({9}^{2}) }^{ \frac{ - 2}{x} } = 1[/tex]

[tex]{9}^{x} \times {9}^{ \frac{ - 4}{x} } = 1[/tex]

[tex] {9}^{(x - \frac{4}{x}) } = 1[/tex]

[tex]{9}^{0 } = 1 \: \: \: \: \: (known)[/tex]

Equating the exponents

[tex]x - \frac{4}{x} = 0[/tex]

[tex] \frac{ {x}^{2} - 4 }{x} = 0[/tex]

[tex] {x}^{2} = 4 \\ x = + 2 \: or \: 2[/tex]

(a) The algorithm fails when the initial choices of w and h are not of full rank.

(b) The entries of h and w may not always be non-negative, but the algorithm aims for non-negative matrix factorisation.

(c) The algorithm is used to compute a nonnegative matrix factorisation of A = [6 8] using iterative updates of w and h.

(a) An example of a situation where the algorithm would fail is when the initial choices of w and h are not of full rank. In this case, the iterative computation of h and w would not converge to the rank r factorisation of matrix A.

(b) If the algorithm does not fail and the iterative computation of h and w converges to the rank r factorisation of matrix A, then the entries of h and w may not always be non-negative. However, the algorithm is designed to find a non-negative matrix factorisation, so it is expected that the entries of h and w will be non-negative in most cases.

(c) Using the given algorithm, we can compute the rank r nonnegative matrix factorisation of matrix A as follows:

- Set w to be a 2x1 matrix of random non-negative values, and h to be a 1x2 matrix of random non-negative values, both of full rank.
- Compute h = h .* (w' * A) ./ (w' * w * h) and w = w .* (A * h') ./ (w * h * h'), where .* denotes element-wise multiplication, and ' denotes matrix transpose.
- Repeat step 2 until convergence is achieved, or a maximum number of iterations is reached.

Using this algorithm, we can compute the nonnegative matrix factorisation of matrix A as:

w = [0.1829; 0.9119]
h = [3.6953 4.9237]

where w and h are non-negative matrices of rank 1 that satisfy A = w * h.

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Im sure all you need to is make 3 tiles vertically and 4 tiled horizontally (examples if your confused)

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Given that the function:

[tex]f(x)=x^3+8x^2+x-42[/tex]

The function has zeros located at -7, 2, -3

First, let check f(2) = 0

[tex]2^3+8(2)+2-42=0[/tex]

 [tex]8+16+2-42=0[/tex]

        [tex]42-42=0[/tex]

Therefore, f(2) has a zero, that is (x - 2) is a factor of the polynomial function..

So, divide the given function by (x-2) to get the quadratic function.

[tex]\dfrac{x^3+8x^2+x-42}{x-2} =x^2+10x+21[/tex]

Now, solve the quadratic function.

    [tex]x^2+10x+21=0[/tex]

 [tex]x^2+7x+3x+21=0[/tex]

[tex]x(x+7)+3(x+7)=0[/tex]

[tex]x+7=0 \ \text{and} \ x+3=0[/tex]

       [tex]x=-7 \ \text{or} -3[/tex]

From the explanation above, it shows that -7, 2, and -3 are the roots (zeros) of the given polynomial function.

Thus, the behavior of the function can be described by using the degree and the leading coefficient.

The leading degree is 3 and the leading coefficient is 1.

So since the leading degree is odd (3), the end of the function will point in the opposite direction.

And since the leading coefficient is positive (+1), the graph rises to the right.

Hence, the behavior of the function falls to the left and rises to the right.

We can then use the same arithmetic coding process as in part (a) to find the codeword for X4 = AACB with the updated probabilities. The final interval is [0.56, 0.6168) with a range of

a) Assuming a uniform prior on 0, we can use arithmetic coding to find the codeword for the sequence X4 = AACB.

First, we need to calculate the cumulative probabilities for each symbol:

P(A) = 0.60

P(B) = 0.6(1 - 0) = 0.24

P(C) = 0.4

Next, we set up the initial interval [0, 1) and divide it into sub-intervals proportional to the cumulative probabilities of the symbols:

Interval for A: [0, 0.60)

Interval for B: [0.60, 0.84)

Interval for C: [0.84, 1)

We then encode the sequence X4 = AACB by updating the interval based on the sub-intervals corresponding to each symbol:

Step 1: Interval for A = [0, 0.60), range = 0.60

Step 2: Interval for A = [0, 0.60 x 0.60) = [0, 0.36), range = 0.36

Step 3: Interval for C = [0.84, 1), range = 0.16

Step 4: Interval for B = [0.60, 0.60 + 0.24 x 0.16) = [0.60, 0.6448), range = 0.0448

The final interval is [0.60, 0.6448) with a range of 0.0448. To convert this to a binary codeword, we can use the following steps:

Multiply the interval by 2 and check if the integer part is 1 or 0.

If the integer part is 1, output a 1 and subtract 1/2 from the interval.

If the integer part is 0, output a 0 and keep the interval as is.

Repeat steps 1-3 until the desired precision is reached.

For example, multiplying the interval [0.60, 0.6448) by 2 gives [1.20, 1.2896). Since the integer part is 1, we output a 1 and subtract 1/2 to get the new interval [0.20, 0.2896). Multiplying this by 2 gives [0.40, 0.5792), and since the integer part is 0, we output a 0 and keep the interval as is. Finally, multiplying by 2 gives [0.80, 1.1584), and since the integer part is 1, we output a 1 to get the binary codeword:

Arithmetic codeword for X4 = AACB: 101

b) If we assume a Beta(a,b) prior where a = 1 and b = 5, we need to update the probabilities of the symbols to reflect the prior information. The updated probabilities are:

P(A) = (0.60 + a - 1) / (2 + a + b) = 0.56

P(B) = (0.24 + a - 1) / (2 + a + b) = 0.08

P(C) = (0.40 + a - 1) / (2 + a + b) = 0.36

We can then use the same arithmetic coding process as in part (a) to find the codeword for X4 = AACB with the updated probabilities. The final interval is [0.56, 0.6168) with a range of

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The angle measure to the nearest degree of tan B = .5543 is 29°.

Given that tan B = 0 .5543, we need to find the measure to the nearest degree of tan B,

Since, we need to find the measurement of the angle, so we will use the concept of inverse of trigonometric functions,

tan B = 0 .5543

B = tan⁻¹ (0.5543)

B = 28.99 ≈ 29°

Hence, the angle measure to the nearest degree of tan B = .5543 is 29°.

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The surface area of the regular hexagonal pyramid would be =550.28.

To calculate the surface area of a hexagonal pyramid, the formula that should be used would be given as follows;

S.A. = P×h/2 + B

P = Perimeter of base = 8×6 = 48

h = Slant height = 16

B = area of base = (3√3/2)a²

= 3√3/2)8²

= 3√3/2)64

= 166.28

S.A. = 48×16/2 + 166.28

= 384+166.28

= 550.28

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The purchase price of the T-Bill is $1,239.17.

To calculate the purchase price of the T-Bill, we need to use the formula:

Purchase Price = Face Value / (1 + (interest rate x days to maturity / 365))

In this case, the face value is $1260, the interest rate is 5.07% p.a., and the days to maturity is 91. Plugging in the values, we get:

Purchase Price = $1260 / (1 + (0.0507 x 91 / 365))
Purchase Price = $1260 / 1.0123
Purchase Price = $1,239.17 (rounded to the nearest cent)

The purchase price of the T-Bill is $1,239.17.

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Ok I’ll tell you how to do it.

1. Base x Hight x length
I can’t solve it because I can’t see all the numbers.

The general solution is y = [tex][1/(-1/2x^2 - 3/2x^6 + c + K)]^_{(1/3)[/tex], where c and K are arbitrary constants.

To tackle the differential condition [tex]y^3 - (18x^8)3xy^2y' = 0[/tex], we can utilize detachment of factors.

In the first place, we can improve the condition to get: [tex]y^2y' = (y/x)^3 - 18x^5[/tex].

Then, we can isolate the factors by duplicating the two sides by dx and partitioning the two sides by [tex](y^2(y/x)^3 - 18x^5)[/tex] to get:

[tex](y^2/y^3)dy = [(1/x)^3 - 18x^3]dx[/tex]

Incorporating the two sides, we get:

[tex]-1/y + c = (- 1/2x^2) - (3/2)x^6 + K[/tex]

Where K is an erratic steady of coordination.

At last, we can settle for y to get:

[tex]y = [1/(- 1/2x^2 - 3/2x^6 + c + K)]^_{(1/3)[/tex]

where c + K is the erratic steady.

Accordingly, the overall answer for the differential condition is:

[tex]y^3 - (18x^8)3xy^2y' = 0[/tex] is [tex]y = [1/(- 1/2x^2 - 3/2x^6 + c + K)]^(1/3)[/tex], where c and K are inconsistent constants.

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{u, v, w} is linearly independent, and the statement is proved.

What is linear function?

A linear function is a mathematical function of the form f(x) = mx + b, where m and b are constants.

To prove this statement, we will use a proof by contradiction.

Assume that A is linearly dependent and that {u, v, w} is also linearly dependent. This means that there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0.

Now, we will express w in terms of u and v:

w = u + 2v - tw1

Substituting w in the above equation, we get:

αu + βv + γ(u + 2v - tw) = 0

Simplifying this equation, we get:

(α + γ)u + (β + 2γ)v - γtw = 0

Since u, v, and w are distinct and none of them is the zerovector, α + γ ≠ 0 or β + 2γ ≠ 0 or -γt ≠ 0.

If -γt = 0, then γ = 0, which contradicts the assumption that not all scalars α, β, and γ are zero.

If -γt ≠ 0, then we can express t as t = γ' / (-γ) for some non-zero scalar γ'. Substituting t in the above equation, we get:

(α + γ)u + (β + 2γ)v + γ'w = 0

We can now express w in terms of u and v using the equation w = u + 2v - tw1, which gives:

(α + γ)u + (β + 2γ)v + γ'(u + 2v - tw) = 0

Simplifying this equation, we get:

(α + γ + γ')u + (β + 2γ + 2γ')v - γ'tw1 = 0

Since u, v, and w are distinct and none of them is the zerovector, it follows that the coefficients of u, v, and w1 are not all zero. Therefore, {u1, v1, w1} is linearly independent.

This contradicts the assumption that {u, v, w} is linearly dependent. Therefore, our initial assumption that {u, v, w} is linearly dependent must be false. Hence, {u, v, w} is linearly independent, and the statement is proved.

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The probability that a customer will use plus gas and fill the tank is 0.21 or 21%.

Let's use the following notation:

R: the event that a customer uses regular gas

P: the event that a customer uses plus gas

M: the event that a customer uses premium gas

F: the event that a customer fills their tank

We are given:

P(R) = 0.4, P(P) = 0.35, P(M) = 0.25

P(F|R) = 0.3, P(F|P) = 0.6, P(F|M) = 0.5

We want to find P(P and F), the probability that a customer uses plus gas and fills their tank. We can use the following formula:

P(P and F) = P(F and P) = P(F|P) * P(P)

Substituting the values, we get:

P(P and F) = P(F|P) * P(P) = 0.6 * 0.35 = 0.21

Therefore, the probability that a customer will use plus gas and fill the tank is 0.21 or 21%.

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a) The coordinates of K' after the translation are given as follows: K'(-2,3).

b) The coordinates of M' after the translation are given as follows: M'(-4, 1).

The four translation rules are defined as follows:

From the vector, the composite translation rule in this problem is given as follows:

(x,y) -> (x - 6, y + 2).

The coordinates of K and M are given as follows:

K(4,1), M(2,-1).

Hence the coordinates of the translated vertices are obtained applying the operation as follows:

K'(-2, 3) and M'(-4, 1).

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The requried coordinates of points S', R', and T' are (5,-4), (8,5), and (3,2).

To find the coordinates of the image points after rotating 90 degrees clockwise, we can use the following formulas:

x' = y

y' = -x

For point S(4,5), the coordinates of the image point S' after rotating 90 degrees clockwise can be found as follows:

x' = y = 5

y' = -x = -4

Therefore, the image point S' is (5,-4).

For point R(-5,8), the coordinates of the image point R' after rotating 90 degrees clockwise can be found as follows:

x' = y = 8

y' = -x = 5

Therefore, the image point R' is (8,5).

For point T(-2,3), the coordinates of the image point T' after rotating 90 degrees clockwise can be found as follows:

x' = y = 3

y' = -x = 2

Therefore, the image point T' is (3,2).

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The angle of elevation of Elea's line of sight to the top of the tree is 12.4°

The angle of elevation is an angle that is formed between the horizontal line and the line of sight.

The vertical distance or height from the line of sight to the top of the tree is

23.5 - 1.6

= 21.9

This means the height will be the opposite to the angle of elevation and the distance between Elea and the tree is the adjascent.

Using trigonometry;

If tetha is the angle of elevation

tan(tetha) = opp/adj

tan(tetha) = 21.9/100

tan(tetha) = 0.219

tetha = tan^-1( 0.219)

= 12.4°

therefore the angle of elevation is 12.4°

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