in each of the problems 7 through 9 find the inverse laplace transform of the given function by using the convolution theoremf(s)=1/(s +1)^2 (s^2+ 4)

Answer:

4th Order Derivative: 120

Step by sep solution:

To find the fourth-order derivative of the function f(x) = 5x^4, we can differentiate the third-order derivative f(3)(x) = d^3/dx^3 (5x^4) with respect to x:

f(3)(x) = d^3/dx^3 (5x^4) = 5 * d^3/dx^3 (x^4)

To find d^3/dx^3 (x^4), we differentiate the function x^4 three times:

d/dx (x^4) = 4x^3

d^2/dx^2 (x^4) = d/dx (4x^3) = 12x^2

d^3/dx^3 (x^4) = d/dx (12x^2) = 24x

Substituting this back into the expression for the third-order derivative, we get:

f(3)(x) = 5 * d^3/dx^3 (x^4) = 5 * 24x = 120x

Now we can differentiate f(3)(x) = 120x to find the fourth-order derivative:

f(4)(x) = d^4/dx^4 (f(x)) = d/dx (f(3)(x)) = d/dx (120x) = 120

Therefore, the fourth-order derivative of the function f(x) = 5x^4 is f(4)(x) = 120

The length of the arc s as required to be determined in the attached image is; 17.5 cm.

It follows from the task content that the length of the arc s is to be determined from the given information.

As evident in the task content, the angle subtended at the center of the two concentric circles is same for the 2cm and 5 cm radius circles.

On this note, it follows from proportion that the length of an arc is directly proportional to the radius of the containing circle.

Therefore, the ratio which holds is;

s / 5 = 7 / 2

s = (7 × 5) / 2

s = 17.5 cm.

Consequently, the length of the arc s is; 17.5 cm.

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The given problem does not have a minimum value as the constraint equation and the values of x, y, z obtained from the partial derivatives of the Lagrange equation are contradictory.

Equation is a mathematical statement that expresses the equality of two expressions on either side of an equal sign. It is used to solve problems or to find unknown values. An equation is usually composed of two or more terms that are separated by an equal sign. Each side of the equation must have the same value in order for the equation to be true.

The given problem is a constrained optimization problem which can be solved using the Lagrange multiplier method. According to the Lagrange multiplier method, the objective function and the constraint equation must be combined into a single equation. Thus, the Lagrange equation for the given problem is given by:

L(x,y,z,λ) = x² + y² + z² + λ(4x² + 2y² + z² - 4)

Now, the partial derivatives of the Lagrange equation with respect to x, y and z is given by:

∂L/∂x = 2x + 8λx

∂L/∂y = 2y + 4λy

∂L/∂z = 2z + 2λz

Setting the partial derivatives of the Lagrange equation equal to zero, we get:

2x + 8λx = 0

2y + 4λy = 0

2z + 2λz = 0

Solving the above equations, we get:

x = 0

y = 0

z = 0

Substituting these values in the constraint equation, we get:

4x² + 2y²+ z² = 4

4(0)² + 2(0)²+ (0)²= 4

0 = 4

Which is a contradiction. Hence, the given problem does not have a minimum value.

In conclusion, the given problem does not have a minimum value as the constraint equation and the values of x, y, z obtained from the partial derivatives of the Lagrange equation are contradictory.

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The formula for the nth term of the sequence (an)n≥0, which starts at 2, 5, 11, 21, 36, is an = n^4 - 3n^3 + 5n^2 - n + 2.

To use polynomial fitting to find the formula for the nth term of the sequence (an)n≥0 which starts at 2, 5, 11, 21, 36, follow these steps:

1. List the terms with their corresponding indices (n values): (0, 2), (1, 5), (2, 11), (3, 21), (4, 36).

2. Since there are 5 terms, assume a 4th-degree polynomial of the form: an^4 + bn^3 + cn^2 + dn + e.

3. Substitute the indices and corresponding terms into the polynomial and form a system of linear equations:

  e = 2
  a + b + c + d + e = 5
  16a + 8b + 4c + 2d + e = 11
  81a + 27b + 9c + 3d + e = 21
  256a + 64b + 16c + 4d + e = 36

4. Solve the system of linear equations:

  a = 1, b = -3, c = 5, d = -1, e = 2

5. Substitute these values back into the polynomial:

  a_n = n^4 - 3n^3 + 5n^2 - n + 2

So, the formula for the nth term of the sequence (an)n≥0, which starts at 2, 5, 11, 21, 36, is: an = n^4 - 3n^3 + 5n^2 - n + 2.

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To calculate the mean, median, q1, and q3, you will need a set of data. Once you have the data, you can find the mean by adding up all the numbers and dividing by the total number of values. The median is the middle value of the data set when it is arranged in order from lowest to highest. Q1 is the value that separates the bottom 25% of the data from the top 75%, while Q3 separates the top 25% from the bottom 75%.

The relationship between the mean and the median can tell you about the distribution of the data. If the mean is equal to the median, then the data is evenly distributed. If the mean is greater than the median, then the data is skewed to the right, meaning that there are a few high values that are affecting the overall average. If the mean is less than the median, then the data is skewed to the left, meaning that there are a few low values that are affecting the overall average.
To calculate the mean, median, Q1, and Q3, follow these steps:

1. Mean: Add all the values in your dataset and divide by the total number of values.
2. Median: Arrange the values in ascending order, then find the middle value. If there are two middle values, take their average.
3. Q1: Find the median of the lower half of the dataset, excluding the overall median if there's an odd number of values.
4. Q3: Find the median of the upper half of the dataset, excluding the overall median if there's an odd number of values.

The relationship between the mean and the median helps identify the skewness of the dataset. If the mean is greater than the median, the dataset is right-skewed, indicating more high-value outliers. If the mean is less than the median, the dataset is left-skewed, indicating more low-value outliers. If the mean and median are approximately equal, the dataset is likely symmetric with no skewness. This relationship helps understand the overall distribution of the data.

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The two triangles that are similar are triangles A and B

Similar triangles are triangles that have the same shape, but their sizes may vary. For two triangles to be equal, the corresponding angles must be equal and the ratio of corresponding sides must be equal.

Checking the three triangles, triangle A has the angles of 125° , 25° and the third angle can be calculated as 180-(125+25) = 180-150 = 30°

Triangle B has 125°, 30° and the Third angle can be calculated as 180-(125+30) = 180-155 = 25°

Triangle C has the angle 35°,25° and the third angle can be calculated as 180-(35+25) = 180-60 = 130°

Therefore,it is shown That triangles And B are similar to each other because they have thesame corresponding angles.

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The values of the numerical operations obtained using the number line indicates;

1. 20 groups

2. 3 1/3

A number line consists of a line marked at (regular) intervals, which can be used for performing numerical operations.

The number line indicates that each small marking is 1/5

1. The number of groups of 1/5 in 4, can be obtained by counting the number of small markings from the start of the number line to 4 as follows;

The number of small markings between 0 and 4 = 20

Therefore, the number of groups of 1/5 that are in 4 are 20 groups

2. The value of 4 ÷ 6/5, can be obtained from the number line as follows;

The number of groups of 6/5 that are in 4, from the number line = 3 groups

The fraction of a group of 6/5 remaining when the three groups are counted before 4 is 2/5, which is (2/5)/(6/5) = 1/3

Adding the remaining fraction to the whole number value, we get the value of 4 ÷ (6/5) as follows;

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W is not a subspace of the vector space of all matrices in Mn,n.

To determine if W is a subspace of the vector space:

We need to check if W meets the criteria of a subspace.
To be a subspace of a vector space, W must satisfy three conditions:
1. W must contain the zero matrix.
2. W must be closed under vector addition.
3. W must be closed under scalar multiplication.
Let's examine each condition for W:
1. W contains the zero matrix: The zero matrix has a determinant of 0, so it is included in W.
2. W is closed under vector addition: If A and B are matrices in W with zero determinants, their sum,

A + B, should also have a zero determinant to be in W.

The determinant property for sums of matrices doesn't guarantee that det(A+B) = det(A) + det(B), so we can't guarantee that W is closed under vector addition.
Since W fails to meet the second condition, it is not a subspace of the vector space of all matrices in Mn,n.

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A differential equation (2xy² − 5) dx + (2x²y + 7) dy = 0  is an exact differential equation

We know that a differential equation M dx + N dy = 0 is an exact differential equation when [tex]\partial N/\partial x=\partial M/\partial y[/tex]

Consider a differential equation (2xy² − 5) dx + (2x²y + 7) dy = 0

Comparing this equation with M dx + N dy = 0 we get,

M =  (2xy² − 5)

and N = (2x²y + 7)

The partial derivative of M with respect to y is:

[tex]\frac{\partial M}{\partial y} \\\\=\frac{\partial}{\partial y}(2xy^2 -5)[/tex]

= 4xy           ...........(1)

The partial derivative of N with respect to x is:

[tex]\frac{\partial N}{\partial x} \\\\=\frac{\partial}{\partial x}(2x^2y+7)[/tex]

= 4xy          ...........(2)

From (1) and (2),

[tex]\partial N/\partial x=\partial M/\partial y[/tex]

Therefore, the differential equation is an exact differential equation

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The sum of the series is 0.0901.

The formula for the sum of an infinite geometric series is:

S = a/(1-r)

where S is the sum of the series, a is the first term = 1/10, and r is the common ratio = -1/10

So,

S = (1/10)/(1-(-1/10)) = (1/10)/(11/10) = 1/11

To approximate the sum correct to four decimal places, we need to evaluate the series up to a certain number of terms that gives us an error of less than 0.00005. To do this, use the formula for error of an alternating series:

|E| <= |a_n+1|, where a_n+1 is the first neglected term

In this case:

a_n+1 = (-1)^n+1 (n+1)^2/10^(n+1)

To find the number of terms, we can use the inequality:

|a_n+1| < 0.00005

Solving for n gives:

(-1)^n+1 (n+1)^2/10^(n+1) < 0.00005

Taking the logarithm of both sides and simplifying gives:

n > 5.623

So we need to evaluate the series up to n=6 to get an error of less than 0.00005. Evaluating the series up to n=6 gives:

S = 1/10 - 4/100 + 9/1000 - 16/10000 + 25/100000 - 36/1000000 + 49/10000000

S = 0.090123

Therefore, the sum of the series correct to four decimal places is approximately 0.0901.

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Answer: 2.15g+0.75w=20.35

Step-by-step explanation:

Since we're creating an equation, we know it has to have an = sign. The total amount of money spent on g pounds of grapes and w pounds of watermelon is $2-.35, so we know that's going to be on the opposite side of the equal to sign. $2.15 is what a pound of g costs, so a g pounds of grapes would cost 2.15g. I used the same reasoning for the watermelons too to get 2.15g + 0.75w.

Answer:

2.15g+0.75w=20.35

Step-by-step explanation:

sorry im in a rush bye gtg :D

The expected outcome would be that approximately 225 players would draw hearts, 225 players would draw diamonds, 225 players would draw clubs, and 225 players would draw spades. This can be answered by the concept of Probability.

In this scenario, 900 players are each given a standard deck of 52 cards that has been well-shuffled. Each player will draw the top card from their deck and identify the suit, which could be hearts, diamonds, clubs, or spades.

To begin, each player is given a deck of 52 cards, which is the standard number of cards in a deck. These decks are well-shuffled, meaning the cards are randomly arranged to prevent any specific order or pattern. Each player will draw the top card from their deck, revealing the suit of that card, which could be hearts, diamonds, clubs, or spades. Since there are four suits in a standard deck, the probability of drawing any particular suit is 1/4 or 25%.

Therefore, in this scenario with 900 players, each drawing one card from their shuffled deck, there will likely be a distribution of suits that is relatively close to 25% for each suit, but with some natural variation due to the randomness of the shuffling process.

Therefore, the expected outcome would be that approximately 225 players would draw hearts, 225 players would draw diamonds, 225 players would draw clubs, and 225 players would draw spades. However, due to the random nature of shuffling, the actual distribution of suits among the players may deviate slightly from this expected outcome.

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Answer:

2 litres

Step-by-step explanation:

The capacity of a bottle of oil = 4000 ml

It is said that the bottle is half full so the half of 4000 is 2000.

Now, to convert ml to litre we need to divide 2000 by 1000

= 2000÷1000=2

Therefore, the answer is 2 litres

hope it helps! byeee

The area of the given hexagon is 419.1 square units

From the question, we are to determine the area of the given hexagon.

The area of a  hexagon is given by the formula,

Area = 1/2 Apothem × Perimeter

From the given information,

Apothem = 11

Now, we will determine the perimeter

First, we need to find the length of  a side

Let the length of a side be s and half the length be x

Then,

tan (30°) = x / 11

x = 11 × tan(30)

x = 6.35

Length of a side = 6.35 × 2

Length of a side = 12.70

Thus,

Area = 1/2 Apothem × Perimeter

Area = 1/2 × 11 × 6 × (12.70)

Area = 419.1 square units

Hence,

The area is 419.1 square units

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Okay, let's break this down step-by-step:

A1 = -100 - This means A1 has a value of -100

r = 1/5 - This means r is equal to 0.2 (one divided by 5)

So in summary:

A1 = -100

r = 0.2

Did I interpret those two lines correctly? Let me know if you need any clarification.

Answer:

Therefore, the integer that is congruent to 5 modulo 13 is 122.

Step-by-step explanation:

Answer:

k = 10

Step-by-step explanation:

given y varies inversely as x then the equation relating them is

y = [tex]\frac{k}{x}[/tex] ← k is the constant of variation

to find k use the condition that y = 5 when x = 2

5 = [tex]\frac{k}{2}[/tex] ( multiply both sides by 2 )

10 = k

Answer:

  a ≈ 6.8, B ≈ 50°, C ≈ 82°

Step-by-step explanation:

You want to solve the triangle with A=48°, b=7, c=9.

The relation given by the law of cosines is ...

  a² = b² +c² -2bc·cos(A)

  a² = 7² +9² -2·7·9·cos(48°) ≈ 45.6895

  a ≈ √45.6895 ≈ 6.76 ≈ 6.8

The law of sines can be used to find one of the other angles:

  sin(C)/c = sin(A)/a

  C = arcsin(c/a·sin(A)) ≈ arcsin(9/6.7594·sin(48°)) ≈ 81.68° ≈ 82°

The remaining angle can be found from the sum of angles in a triangle:

  B = 180° -A -C = 50°

The solution is a ≈ 6.8, B ≈ 50°, C ≈ 82°.

Answer:

  a ≈ 6.8, B ≈ 50°, C ≈ 82°

Step-by-step explanation:

You want to solve the triangle with A=48°, b=7, c=9.

The relation given by the law of cosines is ...

  a² = b² +c² -2bc·cos(A)

  a² = 7² +9² -2·7·9·cos(48°) ≈ 45.6895

  a ≈ √45.6895 ≈ 6.76 ≈ 6.8

The law of sines can be used to find one of the other angles:

  sin(C)/c = sin(A)/a

  C = arcsin(c/a·sin(A)) ≈ arcsin(9/6.7594·sin(48°)) ≈ 81.68° ≈ 82°

The remaining angle can be found from the sum of angles in a triangle:

  B = 180° -A -C = 50°

The solution is a ≈ 6.8, B ≈ 50°, C ≈ 82°.

The volume of the given solid is [tex]}V= \frac{32}{3} (π)[/tex]

To find the volume of the solid enclosed by the two paraboloids, we can set up a triple integral over the region of integration in xyz-space.

The paraboloids intersect where [tex]y = x^2 + z^2 = 8 -x^2 -z^2[/tex].

Solving for [tex]x^2 + z^2[/tex] we get:

[tex]x^2 + z^2 = 4[/tex]

This is the equation of a cylinder with radius 2, centered at the origin. Therefore, the region of integration is the volume enclosed between the two paraboloids within this cylinder.

To set up the triple integral, we need to choose an order of integration and determine the limits of integration for each variable.

Let's choose the order of integration as dz dy dx. Then the limits of integration are:

For z: from [tex]-\sqrt{4-x^{2} } to \sqrt{4-x^{2} }[/tex]

For y: from [tex]x^2 + z^2 to 8 - x^2 - z^2[/tex]

For x: from -2 to 2

Therefore, the triple integral to find the volume is:

integral from -2 to 2 [integral from [tex]x^2 + z^2 to 8 - x^2 - z^2[/tex] [integral from                          [tex]-\sqrt{4-x^{2} } to \sqrt{4-x^{2} }[/tex] dz] dy] dx

Evaluating this triple integral gives the volume of the solid enclosed by the two paraboloids within the cylinder to be:

[tex]V= \frac{32}{3} (π)[/tex]

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Answer:

f(n) = 4 x 2^(n-1)

Step-by-step explanation:

The general form of a geometric sequence is given by:

an = ar^(n-1)

where a is the first term, r is the common ratio, and n is the term number.

Using the given sequence, we can find the values of a and r:

a = 4

r = 8/4 = 2

Therefore, the function that generates this sequence is:

f(n) = 4 x 2^(n-1)

For example, when n = 1, f(1) = 4 x 2^(1-1) = 4 x 1 = 4, which is the first term of the sequence. When n = 2, f(2) = 4 x 2^(2-1) = 4 x 2 = 8, which is the second term of the sequence, and so on.

Answer:

Using the given recursive formula, we can find the value of H(3) as follows:

H(3) = H(1) x H(2)

H(3) = 9 x 3

H(3) = 27

Therefore, H(3) = 27.

Step-by-step explanation:

Answer:

The sequence you provided is a recursive sequence where each term is defined using the two previous terms. Given that H(1) = 9 and H(2) = 3, we can find H(3) by multiplying H(1) and H(2): H(3) = H(1) x H(2) = 9 x 3 = 27.

The total amount of money earned over 12 years would be $483,732.

Amount is a word used to describe a numerical value or quantity. It is commonly used in mathematics, finance, and economics in order to identify the size or magnitude of something. Within those contexts, it is often used to refer to the total sum of money, goods, or services that are available or being exchanged.

To calculate this, we can use the formula for compound interest:
A = [tex]P(1 + r/n)^{(nt)[/tex]
Where A is the total amount, P is the principal (initial amount), r is the interest rate (4% per year in this case), n is the number of times the interest is compounded per year (1 for annually) and t is the time (12 years in this case).
Plugging in the values, we get:
A = [tex]\$36,500 (1 + 0.04/1)^{(1\times 12)[/tex]
A = $483,732.
Therefore, the total amount of money earned over 12 years would be $483,732.

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The total amount of money earned over 12 years would be $483,732.

Amount is a word used to describe a numerical value or quantity. It is commonly used in mathematics, finance, and economics in order to identify the size or magnitude of something. Within those contexts, it is often used to refer to the total sum of money, goods, or services that are available or being exchanged.

To calculate this, we can use the formula for compound interest:
A = [tex]P(1 + r/n)^{(nt)[/tex]
Where A is the total amount, P is the principal (initial amount), r is the interest rate (4% per year in this case), n is the number of times the interest is compounded per year (1 for annually) and t is the time (12 years in this case).
Plugging in the values, we get:
A = [tex]\$36,500 (1 + 0.04/1)^{(1\times 12)[/tex]
A = $483,732.
Therefore, the total amount of money earned over 12 years would be $483,732.

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The t-value that would be used to construct a 95% confidence interval with a sample size of n=24 is c. 2.069.

To explain why, consider the idea of a t-distribution. We utilize the t-distribution instead of the usual normal distribution when working with small sample sizes (less than 30) and unknown population standard deviations. The t-distribution is more variable than the usual normal distribution, and this difference is compensated for by using a t-value rather than a z-value.

The t-value we select is determined by two factors: the desired level of confidence and the degrees of freedom (df) for our sample. We have 23 degrees of freedom for a 95% confidence interval with n=24 (df=n-1). We can calculate the t-value for a 95% confidence interval with 23 df using a t-table or calculator. This implies we can be 95% certain that the real population means is inside our estimated confidence zone.

It's worth noting that as the sample size grows larger, the t-distribution approaches the regular normal distribution, and the t-value approaches the z-value. So, for large sample sizes (more than 30), the ordinary normal distribution and a z-value can be used instead of the t-distribution and a t-value.

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The amount that each will get from the given fraction of amount is :

Kenneth's share = $236.842

Joy's share = 2/3 x = $157.895

Arlyn's share = 4/9 x = $105.263

Given that,

Total amount = 500

Let the fraction of amount of money Kenneth gets = x

The fraction of amount of money Joy gets = 2/3 of Kenneth's share

                                                                       = 2/3 x

The fraction of amount of money Arlyn gets = 2/3 of joy's share

                                                                         = 2/3 (2/3 x)

                                                                         = 4/9 x

Now,

x + 2/3x + 4/9 x = 500

(9x + 6x + 4x) / 9 = 500

9x + 6x + 4x = 4500

19x = 4500

x = 236.842

Kenneth's share = $236.842

Joy's share = 2/3 x = $157.895

Arlyn's share = 4/9 x = $105.263

Hence each will get $236.842, $157.895 and $105.263.

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An algorithm that takes as input a list of n integers and finds the number of negative integers in the list:

1. Initialize a variable called count to 0.
2. Loop through the list of n integers:
  a. If the current integer is negative, increment the count variable by 1.
  b. Otherwise, continue to the next integer.
3. Return the count variable as the number of negative integers in the list.

This algorithm iterates through each integer in the list and checks if it's negative. If it is, it increments a count variable. At the end of the loop, the count variable contains the total number of negative integers in the list, which is returned as the output of the algorithm.

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An algorithm that takes as input a list of n integers and finds the number of negative integers in the list:

1. Initialize a variable called count to 0.
2. Loop through the list of n integers:
  a. If the current integer is negative, increment the count variable by 1.
  b. Otherwise, continue to the next integer.
3. Return the count variable as the number of negative integers in the list.

This algorithm iterates through each integer in the list and checks if it's negative. If it is, it increments a count variable. At the end of the loop, the count variable contains the total number of negative integers in the list, which is returned as the output of the algorithm.

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The intersection, union, and complement sets based on the universal set U = {2, 4, 6, 8}, are;

a. A ∩ B = {∅}

b. A ∪ B = {2, 4, 6, 8}

c. (A ∪ B)' = {∅}

The universal set is the set to which the other sets are subsets, and one which contains all the elements.

The Universal set is; U = (2, 4, 6, 8)

The set A = (2, 4)

The set B = (6, 8)

Therefore;

a. The set A ∩ B is the set of elements common to both sets A and B

Therefore no elements common to both sets A and B, therefore;

A ∩ B = {∅}

b. The set A ∪ B is the set that contains elements in set A and elements in set B as well as elements in set A ∩ B

The set A ∪ B = {2, 4, 6, 8}

c. The set of the complement of the union of the set A and B is the set that contains elements that are not in the union of set A and B

A ∪ B = {2, 4, 6, 8} = U,

U' = {∅}

Therefore (A ∪ B)' = {∅}

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The function f(x) = -9x^2 + 8x + 10 is increasing on the interval (-∞, 4/9) and decreasing on the interval (4/9, ∞)

To find the open intervals on which the function f(x) = -9x^2 + 8x + 10 is increasing or decreasing, we need to find its first derivative and determine its sign over different intervals.

f(x) = -9x^2 + 8x + 10

f'(x) = -18x + 8

Setting f'(x) = 0, we get:

-18x + 8 = 0

x = 8/18 = 4/9

The critical point of the function is x = 4/9.

Now, we can determine the sign of f'(x) for x < 4/9 and x > 4/9 by testing a value in each interval.

For x < 4/9, let's choose x = 0:

f'(0) = -18(0) + 8 = 8 > 0

This means that f(x) is increasing on the interval (-∞, 4/9).

For x > 4/9, let's choose x = 1:

f'(1) = -18(1) + 8 = -10 < 0

This means that f(x) is decreasing on the interval (4/9, ∞).

Therefore, the function f(x) = -9x^2 + 8x + 10 is increasing on the interval (-∞, 4/9) and decreasing on the interval (4/9, ∞).

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The percentage increase of the function from g(x) over the interval x = 8 to x = 9, is 100%. The correct option is therefore;

Increase by 100%

A percentage increase is the representation of the increase of a quantity over an interval as a percentage.

Whereby the function is expressed as follows;

g(x) = 2ˣ

The value of the function at the values x = 8, and x = 9, are;

g(x) = 2ˣ

g(8) = 2⁸ = 256

g(9) = 2⁹ = 512

The percentage increase is therefore;

Percentage increase = ((g(9) - g(8))/g(8)) × 100

Percentage increase = ((2⁹ - 2⁸)/(2⁸)) × 100

2⁸ × ((2 - 1)/(2⁸)) × 100 = 100%

Therefore, the change of g(x) over the interval from x = 8 to x = 9 is an increase of 100%

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b. normal distribution.  As the sample size becomes larger, the sampling distribution of the sample mean approaches a normal distribution.

Explanation:

As the sample size becomes larger, the sampling distribution of the sample mean approaches a normal distribution. This concept is known as the Central Limit Theorem, which states that the distribution of sample means approximates a normal distribution as the sample size increases, regardless of the population's distribution.

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution from which the samples are drawn. This is true for any population distribution, including those that are not normally distributed.

The binomial distribution, chi-square distribution, and Poisson distribution are all probability distributions with specific characteristics and are not necessarily related to the sampling distribution of the sample mean. However, the normal distribution is often observed as an approximation to the sampling distribution of the sample mean when the sample size is large, making option b, "normal distribution," the correct answer.

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the correct option is [tex]11.2 inch/sec[/tex] , as it represents the rate of change of the hypotenuse with the correct sign. Thus, option C is correct.

Let's denote the short leg by 'x' and the long leg by 'y'. The given information states that   [tex]dx/dt = -2[/tex] in/sec (since the short leg is decreasing by 2 in/sec) and dy/dt = 5 in/sec (since the long leg is growing at 5 in/sec).

We can use the Pythagorean theorem to relate the short leg, long leg, and hypotenuse of the right triangle:

[tex]x^2 + y^2 = h^2[/tex]

where 'h' represents the length of the hypotenuse.

Differentiating both sides of the equation with respect to time 't', we get:

[tex]2x(dx/dt) + 2y(dy/dt) = 2h(dh/dt)[/tex]

Substituting the given values for [tex]dx/dt, dy/dt, x,[/tex] and  [tex]y,[/tex] we have:

[tex]2(12)(-2) + 2(16)(5) = 2h(dh/dt)[/tex]

Simplifying, we get:

[tex]-48 + 160 = 2h(dh/dt)[/tex]

[tex]112 = 2h(dh/dt)[/tex]

Dividing both sides by 2h, we get:

[tex](dh/dt) = 112/(2h)[/tex]

We can now plug in the given values for x and y to find h:

[tex]x = 12 in[/tex]

[tex]y = 16 in[/tex]

Using the Pythagorean theorem, we can solve for h:

[tex]h^2 = x^2 + y^2[/tex]

[tex]h^2 = 12^2 + 16^2[/tex]

[tex]h^2 = 144 + 256[/tex]

[tex]h^2 = 400[/tex]

[tex]h = \sqrt400[/tex]

h = 20 in

Now, substituting the value of h into the equation for  [tex](dh/dt),[/tex] we get:

[tex](dh/dt) = 112/(2\times 20)[/tex]

[tex](dh/dt) = 112/40[/tex]

[tex](dh/dt) = 2.8 in/sec[/tex]

So, the rate of change of the hypotenuse is 2.8 in/sec. However, note that the question asks for the rate of change of the hypotenuse with the correct sign, indicating whether it is increasing or decreasing.

Since the long leg is growing at 5 in/sec and the short leg is decreasing at 2 in/sec.

the hypotenuse must be increasing at a rate of 2.8 in/sec (as the change in the long leg is dominating over the change in the short leg).

Therefore, the correct option is  [tex]11.2 inch/sec,[/tex] as it represents the rate of change of the hypotenuse with the correct sign.

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