How many 3 digit numbers are there which leave a reminder 4 and division by 7​

B(71,w) = 0.013812w gives a 71-inch person's body mass index when w is their weight in pounds (w ≥ 0).

To find the cross-sectional model B(71,w), you need to substitute h = 71 inches in the given equation B(h,w) = 697.5wh^{-2}.

Step 1: Substitute h = 71 in the given equation:
B(71,w) = 697.5 * (71)^(-2) * w

Step 2: Calculate the coefficient by evaluating (71)^(-2) and multiplying by 697.5:
Coefficient = 697.5 * (1/(71^2)) ≈ 0.013812

Step 3: Write the cross-sectional model using the calculated coefficient (rounded to six decimal places):
B(71,w) = 0.013812 * w

B(71,w) = 0.013812w gives a 71-inch person's body mass index when w is their weight in pounds (w ≥ 0).

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It is important to remember that the total probability rule refers to the sum of probabilities of intersections between A and two mutually exclusive and exhaustive events (B and BC), while the concept of independence relates to how the occurrence of one event affects the probability of another.

The total probability rule states that if we have two mutually exclusive and exhaustive events B and BC (B complement), then the probability of event A can be calculated as the sum of the probabilities of the intersections of A with both B and BC. Mathematically, this can be expressed as:

P(A) = P(A ∩ B) + P(A ∩ BC)

Now, let's discuss the term "independent". Two events are considered independent if the occurrence of one event does not affect the probability of the other event. In this case, if events A and B are independent, we can say:

P(A ∩ B) = P(A) * P(B)
P(A ∩ BC) = P(A) * P(BC)

However, the total probability rule is not dependent on whether events A and B are independent or not. It is important to remember that the total probability rule refers to the sum of probabilities of intersections between A and two mutually exclusive and exhaustive events (B and BC), while the concept of independence relates to how the occurrence of one event affects the probability of another.

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The value of c is 3.

To find the value of c if the given equation is ∞Σn=2 (1c)⁻ⁿ = 8, we need to first understand that this is a geometric series with a common ratio of (1/c) and starting from n=2. The sum of an infinite geometric series can be found using the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.

In this case, the first term (a) is (1/c)⁻², which simplifies to c², and the common ratio (r) is 1/c. We plug these values into the formula and get:

8 = c²/ (1 - (1/c))

Solving for c, we find that c = 3. Therefore, the value of c in the given equation is 3.

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The value of x after the following statements execute will be 4.

In the given code, there are two statements. First, an integer variable x is declared without being initialized, which means it will have an unspecified value. Then, x is assigned a value based on the result of a conditional (ternary) operator.

The conditional operator has the following syntax: (condition) ? value_if_true : value_if_false. It evaluates the condition, and if the condition is true, it returns value_if_true, otherwise it returns value_if_false.

In this case, the condition being evaluated is (5 <= 3 & 'a' < 'f'). Let's break it down:

5 <= 3 is a comparison between 5 and 3 using the less than or equal to operator. This evaluates to false, because 5 is not less than or equal to 3.

'a' < 'f' is a comparison between the ASCII values of 'a' and 'f'. In ASCII, the value of 'a' is less than the value of 'f'. So this comparison evaluates to true.

& is the bitwise AND operator, which performs a bitwise AND operation on the individual bits of the operands. In this case, it performs a bitwise AND operation on the result of the two previous comparisons. However, since the result of the first comparison is false (0), the bitwise AND operation will also result in false (0).

So, the overall result of the condition (5 <= 3 & 'a' < 'f') is false (0), because the first comparison is false. As a result, the value_if_false branch of the conditional operator is executed, which is 4. Therefore, the value of x will be assigned as 4 after the statements execute.

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As sample variance increases, the likelihood of rejecting the null hypothesis and the effect on measures of effect size such as r2 and Cohen's d can be described by the likelihood increases and measures of effect size increase. So, the correct option is A.

As sample variance increases, the data points are more spread out, making it more likely to detect a significant difference between groups, thus increasing the likelihood of rejecting the null hypothesis. Additionally, the larger variance may also lead to larger effect sizes, as r2 and Cohen's d both consider the magnitude of differences in the data. Hence Option A is the correct answer.

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The rate at which water is draining is 72 liters per second.

The slope of a graph is a measure of how steep the graph is, or how much the dependent variable changes in relation to the independent variable.

The rate at which water is draining is equal to the slope of the graph;

Mathematically, the slope is defined as the ratio of the change in the vertical or y-axis value (the dependent variable) to the change in the horizontal or x-axis value (the independent variable) between two points on the graph. It represents the rate of change or the steepness of the graph.

The slope is usually denoted by the letter "m" and is calculated using the following formula:

Slope (m) = (change in y-axis value)/(change in x-axis value)

rate = slope = (0 L - 360 L )/( 5 s - 0 s )

rate = -360 L / 5 s

rate = -72 L/s

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(a) The statement "In a small county, there are 110 people on any given day who are eligible for jury duty. of the 110 eligible people, 90 are women This is an example of sampling without replacement " is true.

(b) The probability that all four potential jurors excused for medical reasons are women can be calculated using the hypergeometric probability distribution.

(a) True. Sampling without replacement means that once a person is selected for a sample, they cannot be selected again. In this case, once a person is selected for jury duty, they cannot be selected again for another jury duty, which is an example of sampling without replacement.

b) There are 90 women out of 110 eligible people, so the probability of selecting a woman for the first potential juror is 90/110.

Since the sample size is decreasing with each selection, the probability of selecting a woman for the second potential juror is 89/109, for the third potential juror is 88/108, and for the fourth potential juror is 87/107

Therefore, the probability that all four potential jurors excused for medical reasons are women is (90/110) x (89/109) x (88/108) x (87/107) = 0.4324 (rounded to four decimal places).

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The code for Gaussnaive.m is given below:

(c) % I write the following math code for the above method in Matlab and running its came...I gave the file name as GaussNaive.m ..

Code from "Gauss elimination and Gauss Jordan methods using MATLAB"

a = [1 1 -1 -3

6 2 2 2

-3 4 1 1];

Here a=(AIb) augumented matrix

%Gauss elimination method [m,n)=size(a);

[m,n]=size(a);

for j=1:m-1

for z=2:m

if a(j,j)==0

t=a(j,:);a(j,:)=a(z,:);

a(z,:)=t;

end

end

for i=j+1:m

a(i,:)=a(i,:)-a(j,:)*(a(i,j)/a(j,j));

end

end

x=zeros(1,m);

for s=m:-1:1

c=0;

for k=2:m

c=c+a(s,k)*x(k);

end

x(s)=(a(s,n)-c)/a(s,s);

end

disp('Gauss elimination method:');

a

x' % solution in gauss jordan

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Gauss-Jordan method

[m,n]=size(a);

for j=1:m-1

for z=2:m

if a(j,j)==0

t=a(1,:);a(1,:)=a(z,:);

a(z,:)=t;

end

end

for i=j+1:m

a(i,:)=a(i,:)-a(j,:)*(a(i,j)/a(j,j));

end

end

for j=m:-1:2

for i=j-1:-1:1

a(i,:)=a(i,:)-a(j,:)*(a(i,j)/a(j,j));

end

end

for s=1:m

a(s,:)=a(s,:)/a(s,s);

x(s)=a(s,n);

end

disp('Gauss-Jordan method:');

a

x' % solution in Gauss elimination

The Gauss elimination is a popular numerical technique employed to solve linear equation systems. Its method includes applying row operations to an augmented matrix, bringing it to the row echelon form, and finally deriving the solution through back-substitution.

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the complete set of zeros of f(x) is:

x = 1, x = -11, and x = 10

To find the reasonable zeros of the polynomial capability[tex]f(x) = x^3 - 111x + 110[/tex], we can utilize the Normal Root Hypothesis.

Any rational zero of a polynomial function is, in accordance with this theorem, of the form p/q, where p is a factor of the constant term (in this case, 110) and q is a factor of the leading coefficient (which is 1).

So, the possible rational zeros of f(x) are:

p/q = ±1, ±2, ±5, ±10, ±11, ±22, ±55, ±110

We can now use synthetic division or long division to check which of these possible rational zeros actually are zeros of f(x). We start with p/q :

So, x - 1 is a factor of f(x), and we can write:

[tex]f(x) = (x - 1)(x^2 + x - 110)[/tex]

To find the other zeros of f(x), we need to solve the quadratic equation x^2 + x - 110 = 0. We can use the quadratic formula:

[tex]x = (-1 ± \sqrt{ (1^2 - 4(1)(-110)))} / 2(1)[/tex]

[tex]x = (-1 ± \sqrt{441}) / 2[/tex]

x = (-1 ± 21) / 2

So, the other two zeros of f(x) are:

x = -11 and x = 10

Therefore, the complete set of zeros of f(x) is:

x = 1, x = -11, and x = 10

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The coordinates of the matrix m=[337−6] with respect to the standard basis of the space of 2x2 matrices, b={[1000],[0100],[0010],[0001]}, are [337, -6, 0, 0].

To find the coordinates of a matrix m=[a b; c d] with respect to the standard basis b, we need to express m as a linear combination of the basis vectors.

So we have to solve the equation

m = x[1000] + y[0100] + z[0010] + w[0001]

where [1000], [0100], [0010], and [0001] are the standard basis vectors.

Expanding the equation gives

[a b; c d] = x[1 0; 0 0] + y[0 1; 0 0] + z[0 0; 1 0] + w[0 0; 0 1]

Equating the corresponding entries of the matrices gives

a = x

b = y

c = z

d = w

Therefore, the coordinates of the matrix m=[337 -6] with respect to the standard basis are

x = 337

y = -6

z = 0

w = 0

So the coordinates of m are (337, -6, 0, 0) with respect to the standard basis.

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The probability of at most 4 out of the next 5 patients surviving is 0.40941.

We are not given the value of p, so we cannot calculate the expected value and variance.

This question seems to be a combination of two unrelated problems. Here are the solutions to both problems:

Probability of at most 4 out of the next 5 patients surviving:

Let X be the number of patients out of 5 who survive the operation. X is according to a binomial distribution with n=5 and p=0.9. The probability mass function of X is:

[tex]P(X=k) = (5 choose k) * 0.9^k * 0.1^{5-k} for k = 0, 1, 2, 3, 4, 5[/tex]

To find the probability of at most 4 patients surviving, we can sum the probabilities for k=0 to 4:

P(X<=4) = P(X=0) + P(X=1) + P(X=2) + P(X=3[tex]P(X=k) = (5 choose k) * 0.9^k * 0.1^{5-k} {for} k = 0, 1, 2, 3, 4, 5) + P(X=4)[/tex][tex]) + P(X=4)[/tex]

[tex]= (5 choose 0) * 0.9^0 * 0.1^5 + (5 choose 1) * 0.9^1 * 0.1^4 + (5 choose 2) * 0.9^2 * 0.1^3 + (5 choose 3) * 0.9^3 * 0.1^2 + (5 choose 4) * 0.9^4 * 0.1^1[/tex]

= 0.00001 + 0.00045 + 0.0081 + 0.0729 + 0.32805

= 0.40941

Therefore, the probability of at most 4 out of the next 5 patients surviving is 0.40941.

Expected value and variance of the number of trucks that have blowouts out of the next 15 trucks tested:

Let X be the number of trucks out of 15 that have blowouts. X follows a binomial distribution with n=15 and some probability of success p. We are not given the value of p, so we cannot calculate the expected value and variance.

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Based on the mentioned values and the provided informations, the quotient of 5 ÷ 1/4 is calculated to be  20 [tex]\frac{2}{3}[/tex] . So, option D is correct.

To solve this problem, we need to divide 5 by 1/4. We can do this by multiplying 5 by the reciprocal of 1/4.

The reciprocal of 1/4 is 4/1, so we can rewrite the expression as 5 x 4/1, which simplifies to 20.

Therefore, the quotient of 5 ÷ 1/4 is 20 [tex]\frac{2}{3}[/tex]

To elaborate further, 1/4 represents one part of the large rectangle, which has been divided into five equal parts. When we divide 5 by 1/4, we are essentially asking how many times 1/4 goes into 5.

Multiplying 5 by the reciprocal of 1/4, which is 4/1, is the same as dividing 5 by 1/4. This gives us a quotient of 20, which can also be expressed as a mixed number, 20 ²/₃.

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The complete question is :

large rectangle is divided into five equal parts. What is the quotient of 5 ÷ 1/4? The possible answers are A) 1/20, B) 5/4, C) 4/5, and D) 20 2/3.

a. The bottom-up minimal-change algorithm generates all permutations by iteratively generating the next permutation with minimal change from the previous one.

b. The Johnson-Trotter algorithm generates all permutations by iteratively swapping adjacent elements that have different directions until no more swaps are possible.

c. The lexicographic-order algorithm generates all permutations by iterating through the permutations in lexicographic order.

The bottom-up minimal-change algorithm generates all permutations by iteratively generating the next permutation with minimal change from the previous one. Starting with the initial permutation, it finds the rightmost element that is smaller than the element to its right.

It then finds the smallest element to the right of this element that is greater than it, swaps them, and reverses the sequence to the right of the original element. This process is repeated until all permutations have been generated.

b. The Johnson-Trotter algorithm generates all permutations by iteratively swapping adjacent elements that have different directions until no more swaps are possible. The direction of an element is determined by its relative size to its adjacent elements.

The algorithm starts with the initial permutation and repeatedly finds the largest mobile element (an element that is smaller than its adjacent element in its direction) and swaps it with its adjacent element in the opposite direction. This process is repeated until all permutations have been generated.

c. The lexicographic-order algorithm generates all permutations by iterating through the permutations in lexicographic order. It starts with the initial permutation and repeatedly finds the largest index i such that a[i] < a[i+1].

If no such index exists, the permutation is the last one. Otherwise, it finds the largest index j such that a[i] < a[j], swaps a[i] and a[j], and reverses the sequence from a[i+1] to the end. This process is repeated until all permutations have been generated.

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The given question is incomplete, the complete question is:

Explain a. the bottom-up minimal-change algorithm. b. the Johnson-Trotter algorithm. c. the lexicographic-order algorithm.

The given sequence [tex]an=(5n+1)^2/(4n−1)^2[/tex] converges to the limit 25/16.

To determine whether the sequence [tex]an=(5n+1)^2/(4n−1)^2[/tex] converges, we can use the limit comparison test. This involves comparing the given sequence to a known convergent or divergent sequence.

Let [tex]bn=1/n^2[/tex]. This is a known convergent sequence, as it is a p-series with p=2. Using algebraic manipulation, we can rewrite an as follows:

[tex]an=(5n+1)^2/(4n−1)^2= (25n^2 + 10n + 1)/(16n^2 - 8n + 1)= (25 + 10/n + 1/n^2)/(16 - 8/n + 1/n^2)= (25/n^2 + 10/n + 1)/(16/n^2 - 8/n + 1/n^2)[/tex]

Now, taking the limit as n approaches infinity of the ratio of an to bn gives:

lim(n→∞) [tex]an/bn[/tex]
= lim(n→∞) [tex][(25/n^2 + 10/n + 1)/(16/n^2 - 8/n + 1/n^2)] / (1/n^2)[/tex]
= lim(n→∞) [tex](25 + 10n + n^2)/(16 - 8n + n^2)[/tex]
= 25/16

Since this limit is finite and nonzero, and bn converges, then an also converges by the limit comparison test. Thus, the sequence converges to the same limit as the limit of the ratio of an to bn, which is 25/16.

In summary, the given sequence [tex]an=(5n+1)^2/(4n−1)^2[/tex] converges to the limit 25/16.

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The equation in slope-intercept form of the line that passes through (8,-8) and (4,7) is y = -2x + 9

The line that is parallel to this line is 4x - 5y = -2 (option A).

The rate at which Mario rides his bike is 6 feet per second. The correct answer is A.

The equation in slope-intercept form of the line that passes through (-4,-19) and (3,-14) is:

y = (1/7)x - (117/7)

The equation in slope-intercept form of the line that passes through (5.5, 9) and (5,2) is:

y = 14x - 63

The equation in slope-intercept form of the line that passes through (2.14, 5) and (5, 9) is:

y = (8/3)x - (2/3)

The equation in slope-intercept form of the line that passes through (8,-8) and (4,7) is:

y = -2x + 9

So the correct answer is y = -2x + 9.

To find an equation that is parallel to 8x - 10y = -2, we need to find the slope of this line.

We can rearrange the equation into slope-intercept form (y = mx + b) by solving for y:

8x - 10y = -2

-10y = -8x - 2

y = (4/5)x + (1/5)

So the slope of this line is 4/5. Any line that is parallel to this line will also have a slope of 4/5.

We can now use the point-slope form of the equation of a line to find the equation of the line that is parallel to 8x - 10y = -2 and passes through (1,-2):

y - (-2) = (4/5)(x - 1)

y = (4/5)x + 6/5

Multiplying both sides by 5, we get:

4x - 5y = -2

So the correct answer is 4x - 5y = -2 (option A).

Mario rides past one block every 50 seconds, and each block is 300 feet long. This means that he rides 300 feet every 50 seconds, or:

300 feet/50 seconds = 6 feet/second

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The surface area of the right prism with dimensions 3m, 8m, 9.1m is 126.60 m².

A prism is a three-dimensional shape that has two parallel congruent bases that are both polygons, and lateral faces that connect these bases. The shape of the lateral faces can vary, but they are typically parallelograms. Examples of prisms include rectangular prisms (such as a box), triangular prisms, and hexagonal prisms.

To find the surface area of a right prism, we need to find the area of each face and add them up.

In this case, we have a rectangular base with dimensions of 3 m and 8 m, so the area of the base is:

Area of base = length x width = 3 m x 8 m = 24 m²

The height of the prism is 9.1 m, so the area of the two rectangular faces is:

2 x (length x height) = 2 x (3 m x 9.1 m) = 54.6 m²

The area of the top and bottom faces, which are also rectangles, are the same as the base, so we add that twice:

2 x 24 m² = 48 m²

Now we can add up all the areas to find the surface area:

Surface area = area of base + area of two rectangular faces + area of top and bottom faces

Surface area = 24 m² + 54.6 m² + 48 m²

Surface area = 126.6 m²

Rounding to the nearest hundredth, the surface area is about 126.60 square meters.

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The equation in standard form for the circle with diameter endpoints (1,17) and (1,-1) is (x - 1)^2 + (y - 8)^2 = 81.

To write the equation of a circle in standard form, we need to use the formula: (x - h)^2 + (y - k)^2 = r^2 Where (h,k) is the center of the circle and r is the radius.

We can use the midpoint formula to find the center of the circle, which is the midpoint of the diameter: Midpoint = ((x1 + x2)/2 , (y1 + y2)/2) Substituting the given endpoints, we get: Midpoint = ((1 + 1)/2 , (17 + (-1))/2) = (1, 8) So the center of the circle is (1,8).

Now we need to find the radius, which is half the length of the diameter: Length of diameter = sqrt((1-1)^2 + (17-(-1))^2) = sqrt(18^2) = 18 Radius = 18/2 = 9 Substituting the center and radius in the standard form equation, we get: (x - 1)^2 + (y - 8)^2 = 9^2 Simplifying, we get: (x - 1)^2 + (y - 8)^2 = 81

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1. Determine what mathematics are being used. This looks like a trigonometry question, so we'll be using sin, cos, or tan.
2. Are we finding an angle or a side? We're finding an angle, which means we will be using the inverse of sin, cos, or tan.
3. Sin, cos, or tan? Sin means opposite/hypotenuse, cos means adjacent/hypotenuse, and tan means opposite/adjacent. If x° is our theta, then we will be using tan since the problem only supplies us with our opposite and adjacent side.
4. Write down the equation. x = tan⁻¹(opposite/adjacent)
5. Fill in the blanks. x = tan⁻¹(19/22)
6. Input into a calculator. x = 40.81508387

It might look daunting, but just follow the rules of trigonometry, and you'll finish these questions within seconds. This means your final answer is 40.82°.

The perameter to find a cartesian equation of the curve is y^2 = 1 + x.

We are given that;

x=tan^2(theta)

y=sec(theta)

Now,

We need to solve for t in one equation and substitute it into the other equation. In this case, we have:

x = tan^2(t) y = sec(t)

Solving for t in the first equation, we get:

t = arctan(sqrt(x))

Substituting this into the second equation, we get:

y = sec(arctan(sqrt(x)))

Using the identity sec^2(t) = 1 + tan^2(t),

we can simplify this equation as:

y^2 = 1 + x

Therefore, by the given equation the answer will be y^2 = 1 + x

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The value of the car 20 years after it was purchased is $5,300.

What is purchase price?

Purchase price refers to the amount of money that a buyer pays to purchase a product, service, or asset from a seller. It is the price that is agreed upon between the buyer and the seller at the time of the transaction. The purchase price may be influenced by various factors, such as the demand and supply of the product, the quality of the product, the competition in the market, and the negotiation skills of the buyer and seller. In short, the purchase price is the cost of acquiring the item being purchased.

Here,the car depreciates by one half every 3.5 years.

After 3.5 years the car will be worth half of its original value, or $14,500. Again after 3.5 years, it will be worth half of 14,500, or 7,250. This process can be continued until the 20-year mark.

20 years is equal to 20/3.5 = 5.71 periods of 3.5 years. Since the car's value is halved every period, its value after 5.71 periods will be [tex]29000 \times ( \frac{ 1}{2})^{5.71}[/tex] = $5,258.22

Rounding to the nearest hundred dollars, the value of the car 20 years after it was purchased is $5,300.

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By  exponential growth , In light of this, it will take roughly 12 years for the population to reach 6300.

A process called exponential growth sees a rise in quantity over time. It happens when a quantity's derivative, or instantaneous rate of change with respect to time, is proportionate to the amount itself1. A quantity that is increasing exponentially is referred to as a function, and the exponent, which stands in for time, is the variable that represents time. (in contrast to other types of growth, such as quadratic growth)¹. If the proportionality constant is negative, exponential decline happens instead

We may utilise the exponential growth formula to resolve this issue:

A = P(1 + r)ⁿ

where: A = total sum

P = starting sum

Annual Growth Rate is r.

N equals how many years.

We are aware that P is the initial population and A is the end population, both of which are 5000. The yearly growth rate, r, is 3.5%, as well. Solving for n using these values as inputs results in:

6300 = 5000(1 + 0.035)ⁿ

If we simplify this equation, we get:

1.26 = 1.035ⁿ

When you take the natural logarithm of both sides, you get:

ln(1.26) =  ln(1.035)

To find n, solve for:

12 yearsⁿ = ln(1.26) / ln(1.035).

In light of this, it will take roughly 12 years for the population to reach 6300.

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Answer:

All whole numbers are rational numbers.

Step-by-step explanation:

The limit expression sin(x sin(x)) when evaluated by continuity does not exist

The limit expression is given as

sin(x sin(x))

Where, x tends to infinity

By examining the function sin(x sin(x)), we can see that the function is a divergent series

This means that the limits diverges or the limit do not exist (DNE)

Hence, the limit expression sin(x sin(x)) where x tends to infinity does not exist

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The limit lim x→ 8 sin(x sin(x)) can be evaluated using continuity. The answer is sin(8 sin(8)), which can be calculated approximately using a calculator.

To evaluate the limit lim x→ 8 sin(x sin(x)), we can use the fact that the composition of continuous functions is continuous. Since sin(x) is continuous for all real numbers, and x sin(x) is continuous at x = 8, we can conclude that sin(x sin(x)) is also continuous at x = 8. Therefore, the limit is equal to sin(8 sin(8)).

Using a calculator, we can calculate sin(8 sin(8)) approximately to three decimal places.

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( A )-  We use the inclusion-exclusion principle to find the total number of integers in the set that are divisible by at least one of 2, 3, 5, or 7. The result is 104.

( B-) There are 48 primes in the set of integers that are divisible by at least one of 2, 3, 5, or 7.

(C-) n must be greater than 120, which means that all composite numbers in the set 1, 2,..., 120 that were not counted in part (a) must be divisible by at least one of 2, 3, 5, or 7.

(a) The number of integers in the set 1, 2,..., 120 that are divisible by at least one of 2, 3, 5, and 7 can be found using the principle of inclusion-exclusion. We first find the number of integers that are divisible by each individual prime factor:

Number of integers divisible by 2: 60

Number of integers divisible by 3: 40

Number of integers divisible by 5: 24

Number of integers divisible by 7: 17

Next, we find the number of integers that are divisible by each pair of prime factors:

Number of integers divisible by 2 and 3: 20

Number of integers divisible by 2 and 5: 12

Number of integers divisible by 2 and 7: 8

Number of integers divisible by 3 and 5: 8

Number of integers divisible by 3 and 7: 5

Number of integers divisible by 5 and 7: 3

We continue in this way to find the number of integers that are divisible by three prime factors, four prime factors, and so on. Finally, we use the inclusion-exclusion principle to find the total number of integers in the set that are divisible by at least one of 2, 3, 5, or 7. The result is 104.

(b) To find the number of primes in the set of integers that are divisible by at least one of 2, 3, 5, or 7, we need to exclude all composite numbers. We can do this by subtracting the number of integers that are divisible by two or more of 2, 3, 5, and 7 from the total number of integers found in part (a):

Number of integers divisible by 2 and 3: 20

Number of integers divisible by 2 and 5: 12

Number of integers divisible by 2 and 7: 8

Number of integers divisible by 3 and 5: 8

Number of integers divisible by 3 and 7: 5

Number of integers divisible by 5 and 7: 3

Number of integers divisible by 2, 3, and 5: 4

Number of integers divisible by 2, 3, and 7: 2

Number of integers divisible by 2, 5, and 7: 2

Number of integers divisible by 3, 5, and 7: 1

Therefore, there are 48 primes in the set of integers that are divisible by at least one of 2, 3, 5, or 7.

(c) Of the integers in 1, 2,..., 120 that were not counted in part (a), the only one that is not prime is 1. To see why all of the others are primes, consider any composite number n that is not divisible by 2, 3, 5, or 7. By the fundamental theorem of arithmetic, n can be written as a product of primes, none of which are 2, 3, 5, or 7. But since n is composite, it must have at least one prime factor other than 2, 3, 5, or 7. Therefore, n must be greater than 120, which means that all composite numbers in the set 1, 2,..., 120 that were not counted in part (a) must be divisible by at least one of 2, 3, 5, or 7.

(d) Using the results from parts (b) and (c), we can find the total number

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