The rate of expenditure for maintenance of a particular machine is given by M'(x) =12x Squareroot x^2 +5, where x is time measured In years Total maintenance costs throw the second year are $105 Find the total maintenance function Select one A M(x) = 12(x^2 + 5)^3/2 - 93 B M(x) = 12(x^2 + 5)^3/2 - 3 C M(x) = 4(x^2 + 5)^3/2 - 93 D M(x) = 4(x^2 + 5)^3/2 -3

The 99% confidence interval for estimating the average monthly purchases of lower-income players is A = $5.74 and B = $6.12.

To construct a confidence interval, we can use the formula:

Confidence interval = sample mean ± (critical value * standard error)

Given that the sample size is 404, the mean monthly purchase amount is $5.93, and the standard deviation is $1.05, we can calculate the standard error using the formula:

Standard error = standard deviation / √(sample size)

Substituting the values, we find that the standard error is approximately $0.052.

To determine the critical value, we need to consider the desired confidence level. For a 99% confidence interval, we have 1 - (0.99) = 0.01, and dividing this by 2 gives us 0.005 for each tail. Consulting a t-distribution table or using a statistical software, we find that the critical value is approximately 2.626.

Substituting the values into the confidence interval formula, we get:

Confidence interval = $5.93 ± (2.626 * $0.052)

Calculating the lower and upper limits, we find that A = $5.74 and B = $6.12. Therefore, we can estimate with 99% confidence that the average monthly purchases of lower-income players fall between $5.74 and $6.12.

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a) The proportion of male Internet users who use social networking is approximately 0.6747, and the proportion of female Internet users who use social networking is approximately 0.7358.

b) The difference in proportions is approximately -0.0611.

c) The standard error of the difference is approximately 0.0181.

d) The 95% confidence interval for the difference between these proportions is (-0.096, -0.026).

To calculate the standard error of the difference and find a 95% confidence interval for the difference between the proportions, we can use the formulas for proportions and their differences.

Let [tex]p_1[/tex] be the proportion of male Internet users who use social networking, and [tex]p_2[/tex] be the proportion of female Internet users who use social networking.

a) Proportion for male Internet users: [tex]p_1[/tex] = 517/766 = 0.6747

Proportion for female Internet users: [tex]p_2[/tex] = 692/941 = 0.7358

b) Difference in proportions: [tex]p_1 - p_2[/tex] = 0.6747 - 0.7358 = -0.0611

c) The standard error of the difference (SE) can be calculated using the formula:

[tex]SE = \sqrt{(p_1(1-p_1)/n_1) + (p_2(1-p_2)/n_2)}[/tex]

Where [tex]n_1[/tex] and [tex]n_2[/tex] are the sample sizes for male and female Internet users, respectively.

For male Internet users: [tex]n_1[/tex] = 766

For female Internet users: [tex]n_2[/tex] = 941

Plugging in the values, we have:

[tex]SE = \sqrt{(0.6747(1-0.6747)/766) + (0.7358(1-0.7358)/941)}[/tex]

d) To find the 95% confidence interval for the difference between these proportions, we can use the formula:

[tex]CI = (p_1 - p_2) \± (Z * SE)[/tex]

Where Z is the critical value corresponding to a 95% confidence level. For a large sample size, Z is approximately 1.96.

CI = (-0.0611) ± (1.96 * SE)

CI = -0.0611 ± 0.0355

CI = (-0.096, -0.026)

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S is not orientable.

Given a regular surface S in R given by x² + y² = 2022, we need to find out whether S is orientable or not.

The surface is given by, x² + y² = 2022.

Rearranging the terms, we get, y² = 2022 - x²

Let the differentiable function g(x, y) = y, and the set U be the upper hemisphere (upper half) of the surface S.

Then, U = {(x, y, z) : x² + y² = 2022, z ≥ 0}

We know that the partial derivatives of the above function are continuous in U and it follows that U is a regular surface.

We now compute the partial derivatives of g(x, y) :∂g/∂x = 0, and ∂g/∂y = 1

Taking the cross-product of the two partial derivatives, we get : (0i - j + 0k) which is -j.

Now, if we define the positive normal to U to be the upward-pointing unit normal, then we see that (-j) points downward at all points on U.

Thus, U is not orientable.

Therefore, we conclude that the given surface S in R is not orientable.

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Answer:

x = 27.2 or 27 1/5

Step-by-step explanation:

cos 54° = 16/x

x = 27.2 or 27 1/5

Answer:

Circumference = 2(pi)r

C = 2*3.14*4.0= 25.12 round 25.10 m

Step-by-step explanation:

The vector v can be written as the sum of a vector in Span{u} and a vector orthogonal to u as follows: v = (1/5)u + (-4/5)[1 -3].

The main answer can be obtained by decomposing the vector v into two components: one component lies in the span of vector u, and the other component is orthogonal to u. To find the vector in the span of u, we scale the vector u by the scalar (1/5) since v = - [3 1] can be written as (-1/5)[2 1]. This scaled vector lies in the span of u and can be denoted as (1/5)u.

To find the vector orthogonal to u, we subtract the vector in the span of u from v. This can be calculated by multiplying the vector u by the scalar (-4/5) and subtracting the result from v. The orthogonal component is obtained as (-4/5)[1 -3].

Thus, we have successfully decomposed vector v as v = (1/5)u + (-4/5)[1 -3], where (1/5)u lies in the span of u and (-4/5)[1 -3] is orthogonal to u.

In linear algebra, vector decomposition is a fundamental concept that allows us to express a given vector as a sum of vectors that have specific properties. The decomposition involves finding a vector in the span of a given vector and another vector that is orthogonal to it. This process enables us to analyze the behavior and properties of vectors more effectively.

In the context of this problem, the vector v is decomposed into two components. The first component, (1/5)u, lies in the span of the vector u. The span of a vector u is the set of all vectors that can be obtained by scaling u by any scalar value. Therefore, (1/5)u represents the part of v that can be expressed as a linear combination of u.

The second component, (-4/5)[1 -3], is orthogonal to u. Two vectors are orthogonal if their dot product is zero. In this case, we subtract the vector in the span of u from v to obtain the orthogonal component. By choosing the scalar (-4/5), we ensure that the resulting vector is orthogonal to u.

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a. the point estimate of the true driver cell phone use rate is 0.3 or 30%.

b. the 90% confidence interval for the true driver cell phone use rate is approximately (21.5%, 38.5%).

c. The practical interpretation of the confidence interval is that we are 90% confident that the true driver cell phone use rate falls within the range of 21.5% to 38.5%

a. The point estimate of the true driver cell phone use rate (population proportion) can be calculated by dividing the number of drivers using their cell phone by the total sample size. In this case, the sample size is 100, and 30 drivers were using their cell phone.

Point estimate = Number of drivers using their cell phone / Total sample size

Point estimate = 30/100 = 0.3 (or 30%)

Therefore, the point estimate of the true driver cell phone use rate is 0.3 or 30%.

b. To compute a 90% confidence interval for the true driver cell phone use rate, we can use the formula for a confidence interval for a proportion. The formula is:

Confidence interval = Point estimate ± (Critical value × Standard error)

The critical value depends on the desired level of confidence. For a 90% confidence interval, the critical value is typically obtained from the standard normal distribution and is approximately 1.645.

The standard error can be calculated using the formula:

Standard error = sqrt((point estimate * (1 - point estimate)) / sample size)

In this case, the point estimate is 0.3, and the sample size is 100.

Standard error = sqrt((0.3 * (1 - 0.3)) / 100) ≈ 0.048

Plugging in the values, we can calculate the confidence interval:

Confidence interval = 0.3 ± (1.645 * 0.048)

Confidence interval = (0.215, 0.385)

Therefore, the 90% confidence interval for the true driver cell phone use rate is approximately (21.5%, 38.5%).

c. The practical interpretation of the confidence interval is that we are 90% confident that the true driver cell phone use rate falls within the range of 21.5% to 38.5%. This means that based on the sample data, we can estimate with 90% confidence that the proportion of drivers using their cell phone while driving in the entire population lies between these two percentages. It provides a range of likely values for the true population proportion.

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The approximate percentage of a 10c sample left after the time it took to walk one lap around the gym is 100 - 25c.

Let x be the time it takes to walk one lap around the gym.

We know that 5 laps take 200 seconds.

Therefore, x can be found by dividing 200 by 5:

x = 200/5 = 40 seconds.

Now, let's find the percentage of the sample left after walking one lap around the gym.

Since x is the time it takes to walk one lap around the gym, we know that the sample decreases at a rate of 10c/x per second.

Therefore, after x seconds, the percentage of the sample remaining is given by: 100(1 - 10c/x)

Substituting x = 40, we get:

100(1 - 10c/40) = 100(1 - 0.25c) = 100 - 25c

So the approximate percentage of a 10c sample left after the time it took to walk one lap around the gym is 100 - 25c.

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B. 0.14 mg will be present after 9 hours.

The given function is D(h) = 5e^(-0.4h), which can be used to determine the milligrams D of a certain drug in a patient's bloodstream h hours after the drug has been given.

To find the milligrams after 9 hours, we need to plug in h = 9 in the function D(h) = 5e^(-0.4h).

D(h) = 5e^(-0.4h)

D(9) = 5e^(-0.4(9))

D(9) = 5e^(-3.6)

D(9) = 5 × 0.024419

D(9) = 0.1220 ≈ 0.12 mg

Hence, the answer is option (c) 1.22 mg.

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Answer:

13

Step-by-step explanation:

Answer:

13

Step-by-step explanation:

Hello There!

We can calculate the distance between two points using the distance formula

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

where the x and y values are derived from the coordinates in which you are trying to find the distance between

The points we need to find the distance between is (3,-8) and (8,4)

So we plug the x values and y values into the formula

[tex]d=\sqrt{(8-3)^2+(4-(-8)^2} \\8-3=5\\4-(-8)=12\\d=\sqrt{5^2+12^2} \\5^2=25\\12^2=144\\144+25=169\\\sqrt{169} =13[/tex]

so we can conclude that the distance between the two points is 13 units

a) The probability of finding   a meter stick with a length of more than 100.04cm is 0.44013.

b) The probability of finding a group of 42 meter sticks with a mean length of less than 99.82 cm is 0.65866.

c) The probability of finding a group of 50 meter sticks with a mean length of more than 99.87 cm is 0.44013.

d) The probability of finding   a group of 28 meter sticks with a mean length of between 99.82and 99.86 cm is 0.11974.

e) The mean length that would be at the 92nd percentile for a random sample of 32 meter sticks is 99.89714 cm.

a) The probability of finding a meter stick   with a length of more than 100.04cm is

P(X >   100.04 ) =1 - P(X <= 100.04)

= 1- Φ((100.04   - 99.84) / 0.2)

= 1 - Φ(0.12)

= 1 - 0.55987

= 0.44013

b) The probability of finding a group of 42 meter sticks with a mean length of less than 99.82 cm is

P(¯X < 99.82) = 1 - P(X >= 99.82)

= 1 - Φ((99.82 - 99.84) / 0.2 / √42)

= 1 - Φ(-0.1)

= 1 - 0.34134

= 0.65866

c) The probability of finding a group of 50 meter sticks with a mean length of more than 99.87 cm is

P(X > 99.87) = 1 - P(X <= 99.87)

= 1 - Φ((99.87 - 99.84) / 0.2 / √50)

= 1 - Φ(0.15)

= 0.44013

d) The   probability of finding a group of 28 meter sticks with a mean length of between 99.82 and 99.86 cm    is

P(99.82 < X  < 99.86) = Φ  ((99.86 - 99.84)/ 0.2 / √28) - Φ((99.82 - 99.84) / 0.2/ √28)

= Φ(0.15) - Φ(0.12)

= 0.55987 - 0.44013

= 0.11974

e) The mean length that would be at the 92nd percentile for a random sample of 32-meter sticks is

X₉₂ = μ + z₉₂ σ / √n

= 99.84 + z₉₂ (0.2) / √32

= 99.84 + 1.85 (0.2) / √32

= 99.84 + 0.05714

= 99.89714

Therefore, the mean length that would be at the 92nd percentile for a random sample of 32 meter sticks is 99.89714 cm.

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To compute the area of a parallelogram determined by two vectors U = (4, -1) and V = (-6, -2), we can use the formula that states the area of a parallelogram is equal to the magnitude of the cross product of the two vectors. Therefore, the area of the parallelogram determined by U = (4, -1) and V = (-6, -2) is 28 square units.

The formula to compute the area of a parallelogram determined by two vectors U and V is given by:

Area = |U x V|

To calculate the cross product U x V, we can use the following determinant:

| i j k |

| 4 -1 0 |

|-6 -2 0 |

Expanding the determinant, we get:

i * (0 * -2 - 0 * -2) - j * (4 * -2 - 0 * -6) + k * (4 * -2 - (-1) * -6)

= -12i + 24j + 8k

Taking the magnitude of the cross product, we have:

|U x V| = √((-12)^2 + 24^2 + 8^2) = √(144 + 576 + 64) = √784 = 28

Therefore, the area of the parallelogram determined by U = (4, -1) and V = (-6, -2) is 28 square units.

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To determine the standard error of the estimated slope coefficient and its statistical significance, more information is needed, such as the t-statistic or the p-value associated with the estimated slope coefficient. The options provided do not include the necessary details to make a conclusion.

The standard error of the estimated slope coefficient measures the precision or variability of the estimated coefficient. It provides information about how much the estimated slope coefficient could vary across different samples.

The t-statistic and the p-value, on the other hand, are used to assess the statistical significance of the estimated slope coefficient. The t-statistic measures the number of standard errors the estimated coefficient is away from zero, while the p-value indicates the probability of observing a coefficient as extreme as the estimated one under the null hypothesis that the true coefficient is zero.

Without the t-statistic or p-value, it is not possible to determine the statistical significance of the estimated slope coefficient at the 5% level.

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The volume of the prism with an equilateral triangular base of side S and altitude h is (S * h^2 * sqrt(3))/4.

To find the volume of a prism with an equilateral triangular base using integration, we can divide the prism into infinitesimally small slices parallel to the base and integrate their volumes.

Consider an infinitesimally thin slice located at a distance "y" from the base. The length of this slice is equal to the length of the base, S. The width of the slice at distance "y" can be determined by considering the height of the equilateral triangle at that distance, which is given by h - (h/S) * y.

The volume of this slice is then given by the product of its length, width, and infinitesimal thickness dy, which is S * [h - (h/S) * y] * dy.

To find the total volume, we integrate this expression from y = 0 to y = h:

V = ∫[0,h] S * [h - (h/S) * y] dy.

Evaluating this integral gives us the volume of the prism:

V = S * [h * y - (h/S) * (y^2/2)] evaluated from y = 0 to y = h.

Simplifying this expression yields:

V = (S * h^2 * sqrt(3))/4.

Therefore, the volume of the prism with an equilateral triangular base of side S and altitude h is (S * h^2 * sqrt(3))/4.

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The probability that the  proportion will be less than 0.05 is approximately 0.1056, rounded to four decimal places.

We have,

To calculate the probability that the sample proportion will be less than 0.05, we can use the sampling distribution of the sample proportion.

Given that the true proportion is 0.07 and a sample of size 259 is taken, we can assume that the distribution of the sample proportion follows a normal distribution with a mean equal to the true proportion (0.07) and a standard deviation equal to the square root of (p(1-p)/n), where p is the true proportion and n is the sample size.

In this case, the mean is 0.07 and the standard deviation is:

= √((0.07 x (1 - 0.07)) / 259).

To find the probability that the sample proportion will be less than 0.05, we can standardize the value using the z-score formula:

z = (x - mean) / standard deviation

In this case, we want to find P(X < 0.05), which is equivalent to finding P(z < (0.05 - mean) / standard deviation).

Calculating the z-score and using a standard normal distribution table or a calculator, we can find the corresponding probability.

Substituting the values into the formula:

z = (0.05 - 0.07) / √((0.07 x (1 - 0.07)) / 259)

Now, we can find the probability by looking up the corresponding

z-value in the standard normal distribution table or using a calculator.

The probability that the sample proportion will be less than 0.05 is the probability corresponding to the calculated z-value.

Round the answer to four decimal places to get the final result.

Therefore,

The probability that the proportion will be less than 0.05 is approximately 0.1056, rounded to four decimal places.

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The proportion of mature pecan trees between 43 and 46 feet tall can be calculated using the normal distribution with a mean of 42 feet and a standard deviation of 7.5 feet.

To find the proportion, we need to calculate the z-scores corresponding to the given heights and then find the area under the normal curve between those z-scores.

First, we calculate the z-score for 43 feet:

z1 = (43 - 42) / 7.5 = 0.1333

Next, we calculate the z-score for 46 feet:

z2 = (46 - 42) / 7.5 = 0.5333

Using a standard normal distribution table or a calculator, we can find the area between these two z-scores. The area corresponds to the proportion of trees between 43 and 46 feet tall.

The explanation would involve using a standard normal distribution table or a calculator to find the area under the normal curve between the z-scores of 0.1333 and 0.5333. This area represents the proportion of mature pecan trees between 43 and 46 feet tall.

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Answer:

Step-by-step explanation:

lol true

{{{ THE BOLDED CHARACTERS SHOULD BE SMALL. }}}

A sequence can be generated by using an = a(n-1) - 5, where a1 = 100 and n is a whole number greater than 1.

a1 = 100 (given)

a2 = a1 - 5 = 100 - 5 = 95

a3 = a2 - 5 = 95 - 5 = 90

a4 = a3 - 5  = 90 - 5 = 85

a5 = a4 - 5 = 85 - 5 = 80

ANSWER for PART (a): 100, 95, 90, 85, 80

-----------------------------------------------------------------------------

an = a1 + d(n - 1)

an = 100 + -5(n - 1)

an = 100 + -5n + 5

an = 105 - 5n

ANSWER for PART (b): an = 105 - 5n

Answer:

I agree with the person above me and here give him brainliest

Step-by-step explanation:

Two possible representations of vector w as a linear combination of V1, V2, and V3 are: w = 2V1 + V2 + 3V3 and w = -3V1 + 2V2 - 4V3.

The vector w can be represented as a linear combination of V1, V2, and V3 in the following ways:

w = 2V1 + V2 + 3V3

w = -3V1 + 2V2 - 4V3

Explanation:

To find the possible representations of w as a linear combination of V1, V2, and V3, we need to determine the coefficients that satisfy the equation w = aV1 + bV2 + cV3, where a, b, and c are scalars.

We can set up a system of equations to solve for a, b, and c. Using the given vectors and coefficients, we get:

2a + 2b + c = 2

a + c = 3

-3a + 2b - 4c = -5

By solving this system of equations, we find the values of a, b, and c that satisfy the equation. In this case, we have two possible solutions: a = 2, b = 1, and c = 3 for the first representation, and a = -3, b = 2, and c = -4 for the second representation.

Therefore, the possible representations of w as a linear combination of V1, V2, and V3 are w = 2V1 + V2 + 3V3 and w = -3V1 + 2V2 - 4V3.

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Answer:

20.25

Step-by-step explanation:

Just multiply the base and height then divide the results in half.

Hope this helps!

The length function f(w) of a string w = w₁w₂...W), where n ∈ N and Vi ∈ W: € 9, is equal to n.

The length function f(w) is defined recursively based on the structure of the string w. In the base case, if w is an empty string (ε), the length is defined as 0. In the recursive case, if w can be written as au, where a is a character from the alphabet and u is a string over the alphabet Σ, then the length is defined as 1 plus the length of u.

To prove that for any string w =w₁w₂...wy, where n ∈ N and Vi ∈ W: € 9, the length function f(w) is equal to n, we will use a proof by induction.

For w = ε (an empty string), we have f(ε) = 0, which satisfies the condition when n = 0.

Assume that for any string w = w₁w₂...wn, where n ∈ N and Vi ∈ W: € 9, the length function f(w) = n.

Now, consider a string w' = w₁w₂...wn+1. By the recursive definition, we can write w' as au, where a is the last character wn+1 and u is the string w₁w₂...wn. From our assumption, we know that f(u) = n.

Therefore, f(w') = 1 + f(u) = 1 + n = n + 1.

Since we have established that for any string w = w₁w₂...wy, where n ∈ N and Vi ∈ W: € 9, the length function f(w) = n, we can conclude that f(w) = n.

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Answer: 12.5

Step-by-step explanation:

Answer:

450 cm ^3

Step-by-step explanation:

The mean difference between Process A and Process B is 3.0 (rounded to 1 decimal place).

To calculate the mean difference between two processes, we subtract the average of Process B from the average of Process A.

Mean difference = Average of Process A - Average of Process B

Mean difference = 277 - 274 = 3.0

Therefore, the mean difference between Process A and Process B is 3.0.

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(a) To compute the cumulative distribution function (CDF) of W, denoted as FW(w), we integrate the joint probability density function (PDF) over the appropriate region. The region is defined by the inequalities x ≥ 0, y ≥ 0, and 2x + 3y ≤ 6. The CDF is given by: FW(w) = P(W ≤ w) = ∫∫[fX,Y(x, y)] dy dx

To find the PDF fw(w), we differentiate FW(w) with respect to w.

(b) To compute E[W], we integrate the product of w and the PDF fw(w) over the range of W. The variance V ar[W] is calculated by finding E[W^2] and subtracting (E[W])^2.

(c) To find the minimum and maximum values of Z, we need to determine the range of Y - X. We consider the range of x and y that satisfy the given conditions. By substituting the limits of x and y, we can calculate the minimum and maximum values of Z.

(d) The cumulative distribution function FZ(z) can be written as a double integral over the joint PDF fX,Y(x, y). We consider two cases: w ≥ 0 and w < 0. For each case, we determine the appropriate region and integrate the PDF accordingly.

(e) To find the PDF fZ(z), we differentiate FZ(z) with respect to z.

(f) To calculate E[Z], we integrate the product of z and the PDF fZ(z) over the range of Z. The variance V ar[Z] is computed by finding E[Z^2] and subtracting (E[Z])^2.

Please note that without the specific range or shape of the region defined by the inequalities, it is not possible to provide detailed numerical calculations for each part.

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Answer:

2a°+2a°=180°[opposite angle of cyclic quadrilateralis 180°]

4a°=180

a=180°/4

a=45°

4b+2b=180°[similarly]

6b=180°

b=180°/6

b=30

1) The term a_3 of this sequence a3 = -362/37.2

2) By the principle of mathematical induction, for all n ∈ N, aₙ < 0.

1) We are given the recursive formula:

a₁ = -36, aₙ₊₁ = aₙ/2 + 72/aₙ.

We need to find the term a₃ of this sequence. a₂ is given by the recursive formula as:

a₂ = a₁/2 + 72/a₁a₂ = -36/2 + 72/(-36) = -37/2

a₃ is given by the recursive formula as:

a₃ = a₂/2 + 72/a₂= (-37/2)/2 + 72/(-37/2)= -74/37 + (-288/37) = -362/37

Therefore, a₃ = -362/37.2

2) We need to prove by induction that for all n ∈ N, aₙ < 0.

Base case:

For n = 1, we have a₁ = -36 < 0. So, the base case is true.

Inductive step:

Let's assume that for some arbitrary n = k, aₖ < 0.

We need to show that aₖ₊₁ < 0.

Using the recursive formula: aₖ₊₁ = aₖ/2 + 72/aₖ

Since aₖ < 0, -aₖ > 0 and aₖ/2 < 0.Hence, aₖ/2 + 72/aₖ < 0

Therefore, aₖ₊₁ < 0.So, the statement that for all n ∈ N, aₙ < 0 is true for n = 1 and if it's true for n = k, then it's true for n = k + 1.

Therefore, by the principle of mathematical induction, for all n ∈ N, aₙ < 0.

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Answer:

$2,725

Step-by-step explanation:

The maximum number of days is 7.

The cost per days is $375.

The cost for 7 days is 7 * $375 = $2,625.

The owner applies a $100 fee for cleaning which must be added to the cost of the days.

$2,625 + $100 = $2,725

The greatest value for the range is the greatest cost there can be which is $2,725.

The standard deviation of the test marks is 6.5

To calculate the standard deviation of the test marks, we need to follow a few steps. Let's go through them:

Step 1: Calculate the midpoint for each interval.

The midpoint is calculated by adding the lower and upper limits of each interval and dividing by 2.

Midpoint for 41-50: (41 + 50) / 2 = 45.5

Midpoint for 51-60: (51 + 60) / 2 = 55.5

Midpoint for 61-70: (61 + 70) / 2 = 65.5

Midpoint for 71-80: (71 + 80) / 2 = 75.5

Midpoint for 81-90: (81 + 90) / 2 = 85.5

Step 2: Calculate the deviation for each midpoint.

The deviation is calculated by subtracting the mean (average) from each midpoint.

Mean = ((45.5 * 5) + (55.5 * 0) + (65.5 * 10) + (75.5 * 8) + (85.5 * 2)) / (5 + 0 + 10 + 8 + 2)

= (227.5 + 0 + 655 + 604 + 171) / 25

= 1657.5 / 25

= 66.3

Deviation for 45.5: 45.5 - 66.3 = -20.8

Deviation for 55.5: 55.5 - 66.3 = -10.8

Deviation for 65.5: 65.5 - 66.3 = -0.8

Deviation for 75.5: 75.5 - 66.3 = 9.2

Deviation for 85.5: 85.5 - 66.3 = 19.2

Step 3: Square each deviation.

(-20.8)^2 = 432.64

(-10.8)^2 = 116.64

(-0.8)^2 = 0.64

(9.2)^2 = 84.64

(19.2)^2 = 368.64

Step 4: Calculate the squared deviation sum.

Sum of squared deviations = 432.64 + 116.64 + 0.64 + 84.64 + 368.64 = 1003.2

Step 5: Calculate the variance.

Variance = (Sum of squared deviations) / (Number of data points - 1) = 1003.2 / (25 - 1) = 1003.2 / 24 = 41.8

Step 6: Calculate the standard deviation.

Standard deviation = √(Variance) ≈ √(41.8) ≈ 6.5 (rounded to one decimal place)

Therefore, the standard deviation of the test marks is approximately 6.5 (to one decimal place).

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(a) We get the result as:

to.005 = 2.539

(b) The required value is to.

10 = 1.345

(c) From the distribution we get:

to.975 = 2.086.

Given: v=19, α= 0.005

For finding to.005 when v= 19, we need to follow the below steps:

The t-distribution table has two tails and it is symmetric about the mean.

So, the area in one tail is (α/2), and in the second tail is also (α/2).

Step 1:  First of all we need to find the row of the t-distribution table and this will be equal to the degree of freedom (v) which is given to be 19.

In this case, we will find the value in row 19 in the table of critical values of the t-distribution which is shown below:

Step 2: Now, look for the value of α at the top of the table (at 0.005).

Step 3: Since the table is showing the area in the right-hand tail, the value of to.005 will be a positive value.

Therefore, we have to use the positive row of the table and for this, we can find the to.005 by looking at the intersection of row 19 and the column corresponding to α=0.005.

Therefore, to.005 = 2.539 (approximately) (Rounded to three decimal places)

Hence, the correct option is to.005 = 2.539

(b) v=14, α= 0.10

For finding to.10 when v= 14, we need to follow the same steps that we followed in part (a).

The table of critical values of the t-distribution is shown below:

Step 1: Find the row corresponding to the v=14 in the t-distribution table.

Step 2: Look for the α=0.10 at the top of the table.

Since the area in one tail is (α/2) which is equal to 0.05, therefore we need to find the critical values that will cut off the top 5% of the curve.

Step 3: Since the table is showing the area in the right-hand tail, the value of to.10 will be a positive value.

Therefore, we have to use the positive row of the table and for this, we can find the to.10 by looking at the intersection of row 14 and the column corresponding to α=0.10 .

Therefore, to.10 = 1.345 (approximately) (Rounded to three decimal places)

Hence, the correct option is to.10 = 1.345

(c) v = 20, α = 0.025

For finding to.025 when v= 20, we need to follow the same steps that we followed in part (a).

The table of critical values of the t-distribution is shown below:

Step 1: Find the row corresponding to the v=20 in the t-distribution table.

Step 2: Look for the α=0.025 at the top of the table.

Since the area in one tail is (α/2) which is equal to 0.0125, therefore we need to find the critical values that will cut off the top 1.25% of the curve.

Step 3: Since the table is showing the area in the right-hand tail, the value of to.975 will be a positive value.

Therefore, we have to use the positive row of the table and for this, we can find the to.975 by looking at the intersection of row 20 and the column corresponding to α=0.025 .

Therefore, to.025 = 2.086 (approximately) (Rounded to three decimal places)

Hence, the correct option is to.975 = 2.086.

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