To plumbers charge an initial fee and an hourly rate.
The equation y 100+30z models plumber A's fee, where y is the total charge, in dollars, and z is the number of
hours worked.
Plumber B
Hours Total Charges (5)
1
105
2
160
215
The table shown represents plumber B's total charge for different numbers of hours.
Which statement about the plumbers'charges is true?
The two plumbers have equal hourly rates.
Plumber A has a greater initial fee.
Plumber A has a greater hourly rate.
The two plumbers have equal initial fees.

We need at least 27 terms of the series to ensure that the remainder is less than 0.0005.

We need to find the number of terms required to satisfy the following inequality:

| R | < 0.0005

where R is the remainder after truncating the series to n terms.

The nth term of the series is given by:

[tex]an = 5n^5[/tex]

The sum of the first n terms can be expressed as:

[tex]Sn = 5(1^5 + 2^5 + ... + n^5)[/tex]

Using the formula for the sum of the first n natural numbers, we can simplify this to:

[tex]Sn = 5(n(n+1)/2)^2(n^2 + n + 1)[/tex]

We can now express the remainder R as:

[tex]R = 5((n+1)^5 + (n+2)^5 + ...)[/tex]

Using the inequality (n+1[tex])^5[/tex] > [tex]n^5[/tex], we can simplify this to:

R < [tex]5((n+1)^5 + (n+1)^5 + ...)[/tex] = [tex]5/(1-(n+1)^(-5))[/tex]

We want R to be less than 0.0005, so we can set up the inequality:

[tex]5/(1-(n+1)^{(-5))[/tex] < 0.0005

Solving for n, we get:

n ≥ 26.86

Since n must be an integer, the smallest value of n that satisfies this inequality is:

n = 27

Therefore, we need at least 27 terms of the series to ensure that the remainder is less than 0.0005.

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Answer:

The bicycle will be worth less than 1000 after 4 years.

Step-by-step explanation:

Answer: R2.64

Step-by-step explanation:

To calculate the difference in cost that Mr Jones would have to pay from 2014 to 2015, we need to find the cost of using 32 kl of water per month in 2014 and 2015, respectively, and then find the difference between the two costs.

From the table given, we can see that in 2014, the cost of using 32 kl of water per month would fall in the fourth category, where the price per kilolitre is R27.74. Therefore, the total cost of using 32 kl of water per month in 2014 would be:

32 kl x R27.74/kl = R887.68

In 2015, the water tariff has changed, and the cost of using 32 kl of water per month would fall in the fourth category, where the price per kilolitre is R30.38. Therefore, the total cost of using 32 kl of water per month in 2015 would be:

32 kl x R30.38/kl = R972.16

The difference in cost between 2014 and 2015 would be:

R972.16 - R887.68 = R84.48

Therefore, Mr Jones would have to pay R84.48 more in 2015 than in 2014.

To calculate the cost difference between 2014 and 2015 for using 32 kl of water per month, we need to find the price per kl for the relevant tiers in both years and then multiply by 32.

In 2014, the price per kl for usage between 25 and 30 kl was R17.99. Since Mr Jones used 32 kl of water, he exceeded this tier and would have been charged the price per kl for usage between 30 and 32 kl, which was R27.74. Therefore, the total cost for 32 kl of water in 2014 would have been:

25 kl x R17.99 = R449.75

7 kl x R27.74 = R193.18

Total = R642.93

In 2015, the price per kl for usage between 30 and 32 kl was R30.38. Therefore, the total cost for 32 kl of water in 2015 would have been:

32 kl x R30.38 = R973.76

The difference in cost between 2014 and 2015 for using 32 kl of water per month is:

R973.76 - R642.93 = R330.83

Therefore, Mr Jones would have to pay R330.83 more in 2015 compared to 2014 for using 32 kl of water per month.

Answer: a

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

before mean = 4.75

after adding 7 the mean = 5

Step-by-step explanation:

16 P 7 = 57 657 600 combos

Answer:

x=3.5

Step-by-step explanation:

 3x+5=x+12

collect like terms

3x-x=12-5

2x=7

x=7÷2

x=3.5

Answer:

X is equal to 7/2 (3.5)

Step-by-step explanation:

Bring the x terms to one sides and the constants to the other. It would be preferable to make the x term positive.

3x - x = 12 - 5

2x = 7

x = 7/2 or 3.5

The volume of the cone with a height of 9 inches and a diameter of 14 inches is 147π cubic inches. So, the correct answer is D).

To calculate the volume of a cone, we use the formula

V = (1/3)πr²h

where "r" is the radius of the base and "h" is the height of the cone.

In this problem, we are given the diameter of the base, which is 14 inches. To find the radius, we divide the diameter by 2

r = 14/2 = 7 inches

We are also given the height, which is 9 inches.

Now we can substitute these values into the formula

V = (1/3)π(7²)(9)

V = (1/3)π(49)(9)

V = (1/3)(441π)

V = 147π

So the volume of the cone is 147π cubic inches.

So the answer is option (D) 147π.

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Answer:

Step-by-step explanation:To find the inverse of the function f(x), we can follow these steps:

Replace f(x) with y:

y = 3x - 4

Swap x and y:

x = 3y - 4

Solve for y:

x + 4 = 3y

y = (x + 4)/3

Replace y with f^-1(x):

f^-1(x) = (x + 4)/3

Therefore, the inverse of the function f(x) is f^-1(x) = (x + 4)/3.

Note that to find f^-1(x), we swapped x and y in step 2, and solved for y in step 3. The resulting expression for y gives us the inverse function f^-1(x).

The recursive formulas for each sequence are listed below:

Case 1: aₙ = 4 · aₙ₋₁ + 6

Case 2: aₙ = aₙ₋₁ · 2ⁿ

Case 3: aₙ = aₙ₋₁ + 99

Case 4: aₙ = aₙ₋₁ + n

Case 5: aₙ = aₙ₋₁ · (- 14)

Case 6: aₙ = aₙ₋₁ · n²

In this problem we find six sequences, whose recursive formulas must be determined. This can be done by a trial-and-error approach, this is, using the first element of the sequence and any of the six given sequences.

Case 1: 10, 46, 190, 766

aₙ = 4 · aₙ₋₁ + 6

a₁ = 10

a₂ = 4 · 10 + 6

a₂ = 46

a₃ = 4 · 46  + 6

a₃ = 184 + 6

a₃ = 190

a₄ = 4 · 190 + 6

a₄ = 766

Case 2: 4, 16, 128, 2048, 65536

aₙ = aₙ₋₁ · 2ⁿ

a₁ = 4

a₂ = 4 · 2²

a₂ = 16

a₃ = 16 · 2³

a₃ = 128

a₄ = 128 · 2⁴

a₄ = 2048

a₅ = 2048 · 2⁵

a₅ = 65536

Case 3: - 100, - 1, 98, 197, 296

aₙ = aₙ₋₁ + 99

a₁ = - 100

a₂ = - 100 + 99

a₂ = - 1

a₃ = - 1 + 99

a₃ = 98

a₄ = 98 + 99

a₄ = 197

a₅ = 197 + 99

a₅ = 296

Case 4: 17, 19, 22, 26, 31

aₙ = aₙ₋₁ + n

a₁ = 17

a₂ = 17 + 2

a₂ = 19

a₃ = 19 + 3

a₃ = 22

a₄ = 22 + 4

a₄ = 26

a₅ = 26 + 5

a₅ = 31

Case 5:

aₙ = aₙ₋₁ · (- 14)

a₁ = - 7

a₂ = (- 7) · (- 14)

a₂ = 98

a₃ = 98 · (- 14)

a₃ = - 1372

a₄ = (- 1372) · (- 14)

a₄ = 19208

Case 6: 7, 28, 252, 4032

aₙ = aₙ₋₁ · n²

a₁ = 7

a₂ = 7 · 2²

a₂ = 28

a₃ = 28 · 3²

a₃ = 252

a₄ = 252 · 4²

a₄ = 4032

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To find P_w(W), we need to determine the probability that W takes on each possible value. Since W is defined as the minimum of X and Y, we can see that W can take on any value between 1 and 100.

To find P_w(W), we need to sum the joint probabilities for all pairs (X, Y) that give us a minimum of W. For example, if we want to find P_w(1), we need to add up all the joint probabilities where either X=1 or Y=1 (since the minimum of X and Y must be 1).

P_w(1) = P(X=1, Y=1) = 1/10000

For P_w(2), we need to add up all the joint probabilities where either X=1 or Y=1 (since the minimum of X and Y must be 2), and so on:

P_w(2) = P(X=1, Y=2) + P(X=2, Y=1) = 2/10000

P_w(3) = P(X=1, Y=3) + P(X=2, Y=3) + P(X=3, Y=1) = 3/10000

Continuing this pattern, we can see that

P_w(w) = w/10000

for w=1, 2, ..., 100.

Therefore, the probability distribution of W is given by

P_w(W) = {1/10000 for W=1, 2, ..., 100; 0 otherwise.}

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The minimum marginal cost occurs at q = 30.

The lowest average cost is 7.

The marginal cost at q = 30 is 16.

what is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

To find the total cost of producing q goods, we need to multiply the average cost by the number of goods produced:

C(q) = a(q) * q

Substituting a(q) = 0.01q² - 0.6q + 13, we get:

C(q) = (0.01q² - 0.6q + 13) * q

= 0.01q³ - 0.6q² + 13q

To find the minimum marginal cost, we need to take the derivative of the average cost function:

a'(q) = 0.02q - 0.6

Setting a'(q) = 0 to find the critical point, we get:

0.02q - 0.6 = 0

q = 30

Therefore, the minimum marginal cost occurs at q = 30.

To find the production level at which the average cost is a minimum, we need to find the minimum point of the average cost function. We can do this by taking the derivative of the average cost function and setting it equal to zero:

a'(q) = 0.02q - 0.6 = 0

q = 30

Therefore, the production level at which the average cost is a minimum is q = 30.

To find the lowest average cost, we can substitute q = 30 into the average cost function:

a(30) = 0.01(30)² - 0.6(30) + 13

= 7

Therefore, the lowest average cost is 7.

To compute the marginal cost at q = 30, we need to take the derivative of the total cost function:

C(q) = 0.01q³ - 0.6q² + 13q

C'(q) = 0.03q² - 1.2q + 13

Substituting q = 30, we get:

C'(30) = 0.03(30)² - 1.2(30) + 13

= 16

Therefore, the marginal cost at q = 30 is 16.

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The statement "f(w ∩ x) = f(w) ∩ f(x)" is false in general for a function f: a → b and subsets w, x ⊆ a.

To see why, consider the following counterexample:

Let f: {1,2} → {1} be the constant function defined by f(1) = f(2) = 1.

Let w = {1} and x = {2}. Then w ∩ x = ∅, the empty set. Therefore, f(w ∩ x) = f(∅) = ∅, the empty set.

However, f(w) = {1} and f(x) = {1}, so f(w) ∩ f(x) = {1} ∩ {1} = {1}.

Since ∅ ≠ {1}, we can see that the equation f(w ∩ x) = f(w) ∩ f(x) does not hold in this case. Therefore, the statement is false in general.

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The statement "f(w ∩ x) = f(w) ∩ f(x)" is false in general for a function f: a → b and subsets w, x ⊆ a.

To see why, consider the following counterexample:

Let f: {1,2} → {1} be the constant function defined by f(1) = f(2) = 1.

Let w = {1} and x = {2}. Then w ∩ x = ∅, the empty set. Therefore, f(w ∩ x) = f(∅) = ∅, the empty set.

However, f(w) = {1} and f(x) = {1}, so f(w) ∩ f(x) = {1} ∩ {1} = {1}.

Since ∅ ≠ {1}, we can see that the equation f(w ∩ x) = f(w) ∩ f(x) does not hold in this case. Therefore, the statement is false in general.

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The vertices of the resulting image, Figure C’D’E’F’ are; C'(-6, 6); D'(-8, 6); E'(-10, 8); F'(-8, 10)

WE are given that coordinates of the vertices of quadrilateral CDEF are C(6, 6), D(6, 8), E(8, 10), and F(10, 8). The figure is rotated 90° about the origin.

WE can take that point in any of the two surrounding quadrants. Example, if the point is on positive x axis, then it can taken as of first quadrant or fourth quadrant.

On origin, No effect as we assumed rotation is being with respect to origin.

If the figure is rotated clockwise as

C'(6, -6); D'(8, -6); E'(10,-8); F'(8, -10)

If the figure is rotated counterclockwise as

C'(-6, 6); D'(-8, 6); E'(-10, 8); F'(-8, 10)

Since clockwise rotation 90 degrees about the origin transforms a point (x, y) to (y, -x).

Also, counterclockwise rotation 90 degrees about the origin transforms a point (x, y) to (-y, x).

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The vertices of the resulting image, Figure C’D’E’F’ are; C'(-6, 6); D'(-8, 6); E'(-10, 8); F'(-8, 10)

WE are given that coordinates of the vertices of quadrilateral CDEF are C(6, 6), D(6, 8), E(8, 10), and F(10, 8). The figure is rotated 90° about the origin.

WE can take that point in any of the two surrounding quadrants. Example, if the point is on positive x axis, then it can taken as of first quadrant or fourth quadrant.

On origin, No effect as we assumed rotation is being with respect to origin.

If the figure is rotated clockwise as

C'(6, -6); D'(8, -6); E'(10,-8); F'(8, -10)

If the figure is rotated counterclockwise as

C'(-6, 6); D'(-8, 6); E'(-10, 8); F'(-8, 10)

Since clockwise rotation 90 degrees about the origin transforms a point (x, y) to (y, -x).

Also, counterclockwise rotation 90 degrees about the origin transforms a point (x, y) to (-y, x).

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The eigenvalues and eigenspaces of A are: λ1 = 1 - sqrt(7), eigenspace span{(6 - sqrt(7))/5, 1} and λ2 = 1 + sqrt(7), eigenspace span{(6 + sqrt(7))/5, 1}

To find the eigenvalues and eigenspaces of the matrix A, we need to solve the characteristic equation det(A - λI) = 0, where I is the 2x2 identity matrix.

det(A - λI) = det(leftbracket2.gif 1 5 rightbracket2.gif6 0 - λleftbracket1.gif 0 0 1 rightbracket)

= (2 - λ)(-λ) - (1)(6)

= λ² - 2λ - 6

Using the quadratic formula, we get:

λ = (2 ± sqrt(2² - 4(1)(-6))) / 2

λ = 1 ± sqrt(7)

Therefore, the eigenvalues are λ1 = 1 - sqrt(7) and λ2 = 1 + sqrt(7).

Next, we find the eigenvectors for each eigenvalue by solving the system of equations (A - λI)x = 0.

For λ1 = 1 - sqrt(7), we have:

(A - λ1I)x = leftbracket2.gif 1 5 rightbracket2.gif6 0 - (1 - sqrt(7))leftbracket1.gif 0 0 1 rightbracketx = leftbracket0.gif 0 5 6 - sqrt(7) rightbracketx = 0

Reducing the augmented matrix to row echelon form, we get:

leftbracket0.gif 0 5 6 - sqrt(7) rightbracket --> leftbracket0.gif 0 1 6/(5 + sqrt(7)) rightbracket --> leftbracket0.gif 0 0 0 0 rightbracket

So, the eigenvector corresponding to λ1 is any non-zero solution to the equation 5x2 + (6 - sqrt(7))x1 = 0. We can choose x2 = 1, which gives x1 = (-6 + sqrt(7))/5. Therefore, the eigenspace corresponding to λ1 is span{(6 - sqrt(7))/5, 1}.

For λ2 = 1 + sqrt(7), we have:

(A - λ2I)x = leftbracket2.gif 1 5 rightbracket6 0 - (1 + sqrt(7))leftbracket1.gif 0 0 1 rightbracketx = leftbracket0.gif 0 5 6 + sqrt(7) rightbracketx = 0

Reducing the augmented matrix to row echelon form, we get:

leftbracket0.gif 0 5 6 + sqrt(7) rightbracket --> leftbracket0.gif 0 1 (6 + sqrt(7))/5 rightbracket --> leftbracket0.gif 0 0 0 0 rightbracket

So, the eigenvector corresponding to λ2 is any non-zero solution to the equation 5x2 + (6 + sqrt(7))x1 = 0. We can choose x2 = 1, which gives x1 = (-6 - sqrt(7))/5. Therefore, the eigenspace corresponding to λ2 is span{(6 + sqrt(7))/5, 1}.

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The function y=f(x)=2x−ln(2x) has a global minimum at x=1 and a global maximum at x=2.5 within the interval 1<x<2.5, and there are no local non-global maximum points within the interval.

To find the points where y = f(x) = 2x - ln(2x) has a global maximum, global minimum, and local, non-global maximum on the interval 1 < x < 2.5, we need to find the critical points and analyze the behavior of the function.

1. Find the first derivative: f'(x) = 2 - (1/x)
2. Set f'(x) to zero and solve for x: 2 - (1/x) = 0 => x = 1/2 (but it's outside the interval, so discard it)

So, the critical point of f(x) is at x= 1/2. However, we need to check if this critical point is within the given interval 1<x<2.5. Since 1/2​ is not within that interval, we can conclude that f(x) does not have any critical points within the given interval.

Since there's no critical point within the interval, we need to check the endpoints of the interval:

1. f(1) = 2(1) - ln(2(1)) = 2 - ln(2)
2. f(2.5) = 2(2.5) - ln(2(2.5)) = 5 - ln(5)

Since f(1) < f(2.5), we can conclude that:
Global minimum: At x = 1, f(x) ≈ 2 - ln(2) ≈ 0.31
Global maximum: At x = 2.5, f(x) ≈ 5 - ln(5) ≈ 3.39

So, we can see that f( 1 ) is the global minimum point and f( 2.5 ) is the global maximum point within the given interval.

Local, non-global maximum: Not present within the interval 1 < x < 2.5

In summary, the function y=f(x)=2x−ln(2x) has a global minimum at x=1 and a global maximum at x=2.5 within the interval  1<x<2.5, and there are no local non-global maximum points within the interval.

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Derivative of r(t) =(e^(-t), 8t - t^3, ln(t)) is (-e^(-t), 8 - 3t^2, 1/t).

Explanation: -

The derivative of the given vector function r(t) = (e^(-t), 8t - t^3, ln(t)) first find the derivative for each component separately and the following formulas.

d/dt (e^(t)) = e^(t)

d/dt (x^(n)) = n x^(n-1)

d/dt (ln(t)) = 1/t

1. For the first component by the use of chain rule, e^(-t), take the derivative with respect to t:
d/dt (e^(-t)) = -e^(-t)

2. For the second component, 8t - t^3, take the derivative with respect to t:
d/dt (8t - t^3) = 8 - 3t^2

3. For the third component, ln(t), take the derivative with respect to t:
d/dt (ln(t)) = 1/t

Now, combine the derivatives of each component to form the derivative vector r'(t):
r'(t) = (-e^(-t), 8 - 3t^2, 1/t)

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For the given graphs of a linear equation and a quadratic equation. There are 2 number of solutions to the system.

The coordinates of a ordered pair(s) which satisfy all of the system's equations make up the solution set. In other words, the equations will be true for certain x and y numbers. As a result, when a system of equations is graphed, all of the places at which the graphs cross are the solution.

Depending on how many solutions a system of linear equations has, it can be classified. Systems of equations fall into one of two categories:

For the question:

As, there are two intersecting points for the graphs of a linear equation and a quadratic equation. Thus, there are 2 number of solutions to the system.

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If a quadrantal angle 0 is coterminal with 0° or 180°, then the trigonometric functions tangent and cotangent are undefined.

In trigonometry, a quadrantal angle is an angle whose terminal side lies on either the x-axis or the y-axis, such as 0°, 90°, 180°, or 270°.

When a quadrantal angle is coterminal with 0° or 180°, the angle lies entirely on the x-axis, and its tangent is undefined because the x-coordinate is zero.

Similarly, when a quadrantal angle is coterminal with 90° or 270°, the angle lies entirely on the y-axis, and its cotangent is undefined because the y-coordinate is zero. The other trigonometric functions, such as sine and cosine, are well-defined for all angles, including quadrantal angles.

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If the research question doesn't involve exploring the relationship between these two variables or if they are not nominal variables, then a cross-tabulation analysis may not be appropriate or useful.

What is cross tabulation?

Cross tabulation, also known as contingency table analysis or simply "crosstabs," is a statistical tool used to analyze the relationship between two or more categorical variables.

in general, whether or not it makes sense to conduct a cross-tabulation analysis using MLTSRV and MFPAY depends on the research question and the nature of the variables.

If the research question involves exploring the relationship between these two variables and they are both nominal variables with two values each, then conducting a 2x2 cross-tabulation analysis could be appropriate. This analysis would allow you to examine the frequencies and percentages of the different categories of each variable and explore any potential associations between them.

However, if the research question doesn't involve exploring the relationship between these two variables or if they are not nominal variables, then a cross-tabulation analysis may not be appropriate or useful. In any case, it is always important to carefully consider the nature of the variables and the research question before deciding on a statistical analysis method.

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Answer:

B,C

Step-by-step explanation:

B and C work.

A and D do not work.

Answer:

2

Step-by-step explanation:

Since line m is parallel to line n, any line that is perpendicular to line n will also be perpendicular to line m. Therefore, we just need to determine the number of lines perpendicular to line n that pass through point p.

If we draw a diagram, we can see that there are two such lines: one that is perpendicular to line n and passes through the endpoint of line segment p on line m, and another that is perpendicular to line n and passes through the other endpoint of line segment p on line m. These two lines are the only ones that are perpendicular to line n and pass through point p, so the answer is 2.

(a) R is an equivalence relation, we need to prove that it satisfies the following three properties: reflexivity, symmetry, and transitivity.

(b) Reflexivity: For any a ∈ R+, we have aRa, since atb = Vab is equivalent to [tex]a^2 = a^2[/tex], which is true for any positive real number a.

a. Symmetry: For any a, b ∈ R+, if aRb, then bRa. This is because if atb = Vab, then bt a = Vab, which can be rearranged as atb = Vab, showing that bRa.

Transitivity: For any a, b, c ∈ R+, if aRb and bRc, then aRc. This is because if atb = Vab and btc = Vbc, then we can multiply these equations to get atb btc = Vab Vbc, which simplifies to atc = Vabbc. But by the commutativity of multiplication, Vabbc =  [tex]Vabc^2[/tex]. , so we have atc = [tex]Vabc^2[/tex]. Taking the square root of both sides gives atc = Vabc, which shows that aRc.

(b) The distinct equivalence classes resulting from R are the sets of positive real numbers whose arithmetic mean equals their geometric mean. Let us denote one such equivalence class as [a], where a is a positive real number that belongs to the class. Then, for any b ∈ [a], we have atb = Vab, which implies that b =  [tex]a^2/t[/tex]. Thus, every element of [a] is of the form [tex]a^2/t[/tex], where t is a positive real number.

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To prove a closed rectangular box of maximum value with the surface area s is a cube we need to maximize volume V with respect to the surface area which is S.

To show that a closed rectangular box of maximum volume having a prescribed surface area (S) is a cube, we can use the following steps:
1. Let's denote the dimensions of the rectangular box as length (L), width (W), and height (H).

2. The surface area (S) of a closed rectangular box can be expressed as:
S = 2(LW + LH + WH)

3. The volume (V) of a closed rectangular box can be expressed as:
V = LWH

4. To find the maximum volume, we need to express one dimension in terms of the others using the surface area equation. For example, let's express H in terms of L and W:
H = (S - 2LW) / (2L + 2W)

5. Substitute H in the volume equation:
V = LW[(S - 2LW) / (2L + 2W)]

6. To find the maximum volume, we need to find the critical points of V by taking the partial derivatives with respect to L and W, and setting them to 0:
∂V/∂L = 0
∂V/∂W = 0

7. Solving these equations simultaneously, we obtain:
L = W
W = H

8. Since L = W = H, the dimensions are equal, and the rectangular box is a cube.

In conclusion, a cube is a closed rectangular box of maximum volume with a prescribed surface area (S).

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Thus, the value of the give composite function is found as: f(-9) = 38.

Typically, a composite function is a function that is embedded within another function. The process of creating a function involves replacing one function for another. For instance, the composite function of f (x) with g is called f [g (x)] (x). You can read the composite function f [g (x)] as "f of g of x." In contrast to the function f (x), the function g (x) is referred to as an inner function.

Given that:

f(x) = x² + 6x + 11

g(x) = -5x + 1

To find: f(g(2)) , Input x = 2 at  in the function g(x).

g(2) = -5(2) + 1

g(2) = -10 + 1

g(2) = -9

Now,

f(g(2)) = f(-9) = (-9)² + 6(-9) + 11

f(-9) = 81 - 54 + 11

f(-9) = 38

Thus, the value of the give composite function is found as: f(-9) = 38.

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0.47474747 can be expressed as the ratio of integers 47/33.

We can write it as 47/100.

To express 0.47474747 as a ratio of integers, we can write it as 47/99. This is because the repeating decimal can be represented as an infinite geometric series:

0.47474747 = 0.47 + 0.0047 + 0.000047 + ...

The sum of this infinite series can be found using the formula S = a/(1-r), where a is the first term (0.0047) and r is the common ratio (0.01).

S = 0.0047/(1-0.01) = 0.0047/0.99 = 47/9900

Simplifying this fraction by dividing both numerator and denominator by 100 gives 47/990, which can be further simplified by dividing both numerator and denominator by 3 to get 47/33.

Therefore, 0.47474747 can be expressed as the ratio of integers 47/33.

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The corresponding points (x, y) for the given parameter values t = -2, -1, 0, 1, 2 are:

(-ln(33), 11), (ln(9), 8), (0, 9), (ln(17), 10), (ln(33), 7).

To find the corresponding points (x, y) for the given parameter values, we substitute the values of t into the given parametric equations:

For t = -2:

x = ln(8(-2)^2 + 1) = ln(33)

y = -2 + 9 = 7

So, the point is (ln(33), 7).

For t = -1:

x = ln(8(-1)^2 + 1) = ln(9)

y = -1 + 9 = 8

So, the point is (ln(9), 8).

For t = 0:

x = ln(8(0)^2 + 1) = ln(1) = 0

y = 0 + 9 = 9

So, the point is (0, 9).

For t = 1:

x = ln(8(1)^2 + 1) = ln(17)

y = 1 + 9 = 10

So, the point is (ln(17), 10).

For t = 2:

x = ln(8(2)^2 + 1) = ln(33)

y = 2 + 9 = 11

So, the point is (-ln(33), 11).

Therefore, the corresponding points (x, y) for the given parameter values t = -2, -1, 0, 1, 2 are:

(-ln(33), 11), (ln(9), 8), (0, 9), (ln(17), 10), (ln(33), 7).

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g'(4) = 18.75.

To find g'(4) given that f(4)=3, f'(4)=9, and g(x)=sqrt(xf(x)), follow these steps:

1. Write down the given information: f(4) = 3, f'(4) = 9, and g(x) = sqrt(xf(x)).
2. Differentiate g(x) using the product rule and chain rule: g'(x) = d(sqrt(xf(x)))/dx.
3. Apply the product rule: g'(x) = (d(sqrt(x))/dx) * (f(x)) + (sqrt(x)) * (df(x)/dx).
4. Differentiate sqrt(x) using the chain rule: d(sqrt(x))/dx = (1/2) * (x^(-1/2)).
5. Plug in the given values of f(4) and f'(4) into the equation: g'(4) = (1/2) * (4^(-1/2)) * (3) + (sqrt(4)) * (9).
6. Simplify the expression: g'(4) = (1/2) * (1/2) * (3) + (2) * (9).
7. Calculate the final result: g'(4) = (3/4) + 18.

So, g'(4) = 18.75.

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The maximum of a - b, given the values of a and b, would be 78.785.

The maximum difference between a and b can be found by looking for the difference between the largest possible value for a and the smallest possible value for b.

Maximum value of a because it was rounded off would be:

80. 0 + 0. 05 = 80. 05

Smallest possible value of b would then be:

1. 27 - 0. 005 = 1. 265

The maximum difference between a and b is:

= 80. 05 - 1. 265 = 78. 785

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