At a community​ college, a survey was taken to determine where students study on campus. Of the 250 students​ surveyed, it was determined that
170 studied in the library
135 studied in the cafeteria
76 studied in both the library and the cafeteria
How many studied in library or cafeteria (including both)?

The probability that the sum of the numbers on the dice will be a square number is 1/6.

The area of mathematics known as probability explores potential outcomes of events as well as their relative probabilities and distributions.

We can first make a list of every conceivable result when rolling two dice in order to determine the likelihood that the sum of the numbers on the two dice will be a square number:

[tex](1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6[/tex])

[tex](2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)[/tex]

[tex](3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)[/tex]

[tex](4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)[/tex]

[tex](5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)[/tex]

[tex](6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)[/tex]

The sums can then be listed, and the square numbers can be identified:

2, 3, 4, 5, 6, 7

3, 4, 5, 6, 7, 8

4, 5, 6, 7, 8, 9

5, 6, 7, 8, 9, 10

6, 7, 8, 9, 10, 11

7, 8, 9, 10, 11, 12

The likelihood that the sum of the numbers on the dice will be a square number is 6 out of a total of 36 potential outcomes.

P(square sum) = 6/36 = 1/6.

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All rules towards each statement shown in table below.

The relationships between the sides and angles of triangles are the subject of the mathematical discipline of trigonometry. It entails the study of trigonometric functions like sine, cosine, and tangent, which connect a triangle's angles and side lengths.

Rules towards each statement:

                                                                  Statements

Rule1.     Cos x/ Sin x  (Sec 2x - 1)             Pythagorean identity

Rule2.    Cos x/Sin x  (Tan 2x)                  Algebra

Rule3.    Cos x/Sin x (Sin 2x/Cos 2x)       Reciprocal

Rule4.    Sin x/Cos x                                 Odd/ Even

Rule5.    Tan x                                           Quotient

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As per trigonometric functions, rules towards each statement are:

Rules towards each statement:

                                                                 Statements

Rule1.     Cos x/ Sin x (Sec 2x - 1)             Pythagorean identity

Rule2.    Cos x/Sin x (Tan 2x)                  Algebra

Rule3.    Cos x/Sin x (Sin 2x/Cos 2x)       Reciprocal

Rule4.    Sin x/Cos x                                 Odd/ Even

Rule5.    Tan x                                           Quotient

Trigonometry is a branch of mathematics that focuses on the relationships between triangles' sides and angles. It requires studying the trigonometric functions that link the angles and side lengths of a triangle, such as sine, cosine, and tangent.

Rules towards each statement:

                                                                 Statements

Rule1.     Cos x/ Sin x (Sec 2x - 1)             Pythagorean identity

Rule2.    Cos x/Sin x (Tan 2x)                  Algebra

Rule3.    Cos x/Sin x (Sin 2x/Cos 2x)       Reciprocal

Rule4.    Sin x/Cos x                                 Odd/ Even

Rule5.    Tan x                                           Quotient

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Answer:

Step-by-step explanation:

This is an example of a hypergeometric distribution problem, where we have a population of 10,000 clock radios with 500 defective ones, and we want to calculate the probability of getting at least one defective radio in a random sample of 10 without replacement.

The probability of getting no defective radios in the sample is:

(9500/10000) * (9499/9999) * (9498/9998) * ... * (9491/9992)

This is because, for the first radio, there are 9500 good radios out of 10,000, and for the second radio, there are 9499 good radios out of 9,999, and so on.

The probability of getting at least one defective radio in the sample is then:

1 - (9500/10000) * (9499/9999) * (9498/9998) * ... * (9491/9992)

which is approximately equal to 0.401.

Therefore, the probability that the entire batch will be rejected is 0.401.

The weight that corresponds to the highest 5% is also approximately 398.12 grams.

What is Z-Score?

A score's connection to the mean within a group of scores is statistically measured by a Z-Score.

To find the weight that corresponds to each event, we need to use the standard normal distribution and convert each value to a z-score using the formula:

z = (x - μ) / σ

Here are the calculations for each event:

The weight that corresponds to the 25th percentile:

-0.68 = (x - 385) / 8

Solving for x gives:

x = 379.44 grams (rounded to two decimal places)

Therefore, the weight that corresponds to the 25th percentile is approximately 379.44 grams.

The weight that corresponds to the 95th percentile. we find that the z-score is approximately 1.64 (rounded to two decimal places). Then we can use the formula above to solve for x:

1.64 = (x - 385) / 8

x = 398.12 grams (rounded to two decimal places)

Therefore, the weight that corresponds to the 95th percentile is approximately 398.12 grams.

The weight that corresponds to the highest 5%:

1.64 = (x - 385) / 8

x = 398.12 grams (rounded to two decimal places)

Therefore, the weight that corresponds to the highest 5% is also approximately 398.12 grams.

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Answer:

D

Step-by-step explanation:

when an angle is supplementary to another angle, it means that both of the angles added together equal 180 degrees.

120+60=180.

Answer:

A. To find the derivative of h(x), we can use the chain rule:

h(x) = a(-2x + 1)^5 - b

h'(x) = a * 5(-2x + 1)^4 * (-2) = -10a(-2x + 1)^4

To find the second derivative, we can again use the chain rule:

h''(x) = -10a * 4(-2x + 1)^3 * (-2) = 80a(-2x + 1)^3

B. To show that h is monotonic, we need to show that h'(x) is either always positive or always negative. Since h'(x) is a multiple of (-2x + 1)^4, which is always non-negative, h'(x) is always either positive or negative depending on the sign of a. If a > 0, then h'(x) is always negative, which means that h(x) is decreasing. If a < 0, then h'(x) is always positive, which means that h(x) is increasing.

C. To find the critical points, we need to find where h'(x) = 0:

h'(x) = -10a(-2x + 1)^4 = 0

-2x + 1 = 0

x = 1/2

Thus, the critical point is at x = 1/2. This value is independent of a and b, as neither a nor b appear in the calculation of the critical point.

Since we can't find an ordered pair (x, y) that satisfies all the conditions, there is no solution to this problem.

An equation is a mathematical statement that asserts the equality of two expressions. It typically contains variables, coefficients, and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. Equations can be used to model relationships between variables, solve real-world problems, and make predictions.

Here,

Let's start by defining the variables:

x: number of hot dogs

y: number of peanuts

We need to find an ordered pair (x, y) that satisfies the following conditions:

x and y are both integers

x is greater than or equal to 0

y is greater than or equal to 0

2x + y ≤ 7 (total cost of snacks can't exceed $7)

x ≥ 4 (at least 4 snacks)

We can use trial and error to find a suitable ordered pair. Let's start with x = 4 and see if we can find a corresponding y value that satisfies the conditions:

If x = 4, then the total cost of hot dogs is 4 * $2 = $8.

We need to spend no more than $7, so we have $7 - $8 = -$1 left for peanuts.

Since we can't spend a negative amount of money, there is no solution for x = 4.

Let's try x = 5:

If x = 5, then the total cost of hot dogs is 5 * $2 = $10.

We have $7 - $10 = -$3 left for peanuts, so there is no solution for x = 5 either.

Finally, let's try x = 6:

If x = 6, then the total cost of hot dogs is 6 * $2 = $12.

We have $7 - $12 = -$5 left for peanuts, so there is no solution for x = 6 either.

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Complete question:

Cody has $7 dollars. he wants to buy at least 4 snacks. Hot dogs (x) and $2 each. Peanuts (y) are $1 each. Find the solution for this question of equation?

Answer:

A linear function would be the best type of equation to model this situation. The total cost of renting the truck increases linearly with the number of miles driven. The initial charge of $60 can be considered as the y-intercept of the linear function, and the additional fee per mile driven can be considered as the slope of the line. Therefore, the equation that models this situation can be written in the form y = mx + b, where y is the total cost of renting the truck, x is the number of miles driven, m is the additional fee per mile driven (the slope of the line), and b is the initial charge of $60 (the y-intercept).

Answer:

A linear function would be the best type of equation to model this function.

Step-by-step explanation:

The total cost of renting the truck is composed of two parts:

The initial charge of $60 is the fixed charge, and the additional fee is the variable charge that is proportional to the number of miles driven.

Let "x" be the number of miles driven and "y" be the total cost of the rental (in dollars), then the linear equation is:

y = mx + 60

where "m" is the additional fee (in dollars) per mile driven.

Therefore, a linear function, in the form y = mx + b, where m represents the slope or rate of change, and b represents the initial fixed charge, is the most appropriate function to model this situation.

An adjusted R-squared value of 0 indicates that the model has no ability to explain the variation in the dependent variable using the independent variables included in the model.

In other words, the model does not fit the data well and cannot make accurate predictions. An adjusted R-squared value of 1 represents a perfect fit, where the model explains all of the variation in the dependent variable using the independent variables. However, it is important to consider other factors such as the sample size, the quality of the data, and the appropriateness of the model to make valid conclusions about the model's ability to explain the dependent variable.

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Answer:

k = -5

Step-by-step explanation:

According to the Remainder Theorem, when we divide a polynomial f(x) by (x − c), the remainder is f(c).

Therefore, if we divide polynomial f(x) = x³ + kx + 9 by (x - 2) and the remainder is 7 then:

To find the value of k, simply substitute x = 2 into the function, equate it to 7 and solve for k.

[tex]\begin{aligned}f(2)=(2)^3 + k(2) + 9 &= 7\\8+2k+9&=7\\2k+17&=7\\2k&=-10\\k&=-5\end{aligned}[/tex]

Therefore, the value of k is -5.

Thus, Opuwo received 470.4mm of precipitation annually in 2018.

Depth x Radius x Radius x 3.14 will give you the typical rainfall amount. The apex of the bucket's region can be located. To calculate the amount of rain, divide the capacity by this region.

To find the annual rainfall in 2018, we can use the fact that it was 12% more than in 2017.

Let R be the annual rainfall in 2017 (which we know to be 420mm). Then, the annual rainfall in 2018 can be expressed as:

R + 0.12R

Simplifying this expression, we get:

1.12R

Therefore, the annual rainfall in 2018 was:

1.12 x 420mm = 470.4mm

So the annual rainfall in 2018 in Opuwo was 470.4mm

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The net amount on the order if the bill is paid by the 22nd day is $16,340.

The terms of payment that a supplier offers to a buyer are referred to as rade credit terms. These conditions outline the deadline for payment as well as any early payment discounts. For instance, if a supplier offers "2/10 net 30" terms, the customer can choose to pay the whole amount up front or receive a 2% reduction if they pay within 10 days.

The parameters of a trade credit agreement can significantly affect a company's cash flow. If a company can benefit from an early payment discount, they can save expenses and increase cash flow.

The term of sale is given as 5/22 n/30.

Here, the discount is 5%, thus the invoice amount is:

Discount = 0.05 x $17,200 = $860

Now, the net amount is:

Net Amount = Invoice Amount - Discount

Net Amount = $17,200 - $860

Net Amount = $16,340

Hence, the net amount on the order if the bill is paid by the 22nd day is $16,340.

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For given function f(x)= x³ + 2x , the complete table is mentioned below.  f⁻¹(3)= 1, f⁻¹(-12) = -2.

In mathematics, a function is a rule that assigns a unique output value for every input value in a specified set. It is a fundamental concept in algebra, calculus, and other areas of mathematics.

A function is typically denoted by a symbol, such as f(x), where x is the input variable, and f(x) is the output variable. The set of all input values for which the function is defined is called the domain, and the set of all output values is called the range.

To complete the table of values, we simply plug in the given values of x into the expression for f(x) and evaluate:

x       f(x)

0        0

1         3

2        14

To find f⁻¹(3), we need to solve for x in the equation f(x) = 3:

x³ + 2x = 3

x³ + 2x - 3 = 0

We can use trial and error to find that x = 1 is a solution to this equation:

1³ + 2(1) - 3 = 0

Therefore, f⁻¹(3) = 1.

To find f⁻¹(-12), we need to solve for x in the equation f(x) = -12:

x³ + 2x = -12

x³ + 2x + 12 = 0

We can use trial and error to find that x = -2 is a solution to this equation:

(-2)³ + 2(-2) + 12 = 0

Therefore, f⁻¹(-12) = -2.

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The exact value of sin∅ if cos∅ =3/8 and in the fourth quadrant is 292⁰

We should recall that the the angles of the 4th quadrant range from 270° to 360°.  that is to say that in the fourth quadrant, 270≤x≤360

The given angle is cos∅ =3/8

cos∅ = 0.375

Find the angle we take the cos inverse

That ∅ = Cos⁻¹0.375

∅ = 68⁰

In the fourth quadrant, cos is positive that 360-68 = 292⁰

Then the exact value of sin is 292⁰

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To cover the house roof, as we only need 6,750 shingles and we have 7,200 shingles available.

What are arithmetic operations ?

Arithmetic operations are basic mathematical operations used to perform calculations involving numbers. The four basic arithmetic operations are:

According to the question:
To determine the amount of shingles needed to cover the roof of the house, we can use the fact that every 2 yards requires 3 shingles. Therefore, for 4500 yards, we need to divide by 2 and then multiply by 3 to get the total number of shingles needed.

(4500 yards) / (2 yards/2) * (3 shingles/2 yards) = 6,750 shingles

Since we have 60 boxes that contain 120 shingles each, we can calculate the total number of shingles we have:

60 boxes * 120 shingles per box = 7,200 shingles

Therefore, we have more than enough shingles to cover the roof, as we only need 6,750 shingles and we have 7,200 shingles available.

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Doing simple algebra with the two summations we can see that:

[tex]\Sigma x = 799\\\\\Sigma x^2 = 7,188[/tex]

First, we know that there are 97 passengers, and we know that the summation:

[tex]\Sigma (x - 5) = 314[/tex]

Where x represents the weights.

Then we can rewrite that sum as:

[tex]\Sigma (x - 5) = \Sigma x - 97*5[/tex]

And replace that in the original equation to get:

[tex]\Sigma x - 97*5 = 314\\\Sigma x = 314 + 5*97 = 799[/tex]

So that is the first summation, now let's get the second one,  we can rewrite the summation as:

[tex]\Sigma (x - 5)^2 = \Sigma x^2 - 10x +25 \\\\\Sigma x^2 - \Sigma 10x + \Sigma 25[/tex]

Where remember we have 97 terms, and the summation is equal to 1623, then:

[tex]\Sigma x^2 - \Sigma 10x + 25*97 = 1623[/tex]

Now we can replace the second term by the thing we found earlier:

[tex]\Sigma x^2 - 10\Sigma x + 25*97 = 1623\\\\\Sigma x^2 - 10*799 + 25*97 = 1623\\\\\Sigma x^2 = 1623 + 10*799 - 25*97\\\\\Sigma x^2 = 7,188[/tex]

That is the answer.

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Answer:

A right angled triangle is similar to triangle ABC

Step-by-step explanation:

If you tilt the triangle and put it straight, you'll see that angle C is equal to 90°

And if a triangle has one angle of 90° then it is a right angled triangle

Hope you understand :)

Due to the directed nature of the alternative hypothesis, a one-tailed test is being used (greater than).

The null hypothesis, which is an assertion or assumption that there is no significant difference or association between two or more variables or populations, is used in statistical hypothesis testing. It is frequently indicated by the letter H0 and is typically the hypothesis that is tested against a competing hypothesis. The objective of the hypothesis test is to either reject or fail to reject the null hypothesis based on the evidence or data seen. The null hypothesis serves as the default or baseline assumption. If the alternative hypothesis is supported by evidence, the null hypothesis is likely to be rejected.

given

The test's null and alternate hypotheses are as follows:

H0: p 0.025 (The percentage of American women with university educations who had never previously been married at the start of their 40s and are now 45 and married is less than or equal to 2.5%)

Ha: p > 0.025 (More than 2.5% of American women with college degrees who were unmarried at the start of their 40s and are now 45 and married are never before married).

Due to the directed nature of the alternative hypothesis, a one-tailed test is being used (greater than).

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28.5cm^2  is the area of shape B.

A solid object's surface area is a measurement of the total area that the surface of the object takes up.

The definition polyhedra of arc length for one-dimensional curves and the definition of surface area for (i.e., objects with flat polygonal faces), where the surface area is the sum of the areas of its faces, are both much simpler mathematical concepts than the definition of surface area when there are curved surfaces.

A smooth surface's surface area is determined using its representation as a parametric surface, such as a sphere.

This definition of surface area uses partial derivatives and double much simpler mathematical concepts than the definition of surface area integration and is based on techniques used in infinitesimal calculus.sought a general definition of surface area.

(3×7)+(1.5×5)

21+7.5

28.5cm^2

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Answer:

Anna had 23 sweets in her bag at the start of the day.

Step-by-step explanation:

Let's use working backwards to find out how many sweets were in the bag at the start of the day.

At the end of lesson 4, Anna had 1 sweet left in her bag. So, before she gave a sweet to her teacher in lesson 4, she had 2 sweets left in her bag.

In lesson 3, she gave out half of the sweets left in her bag and then gave one to the teacher. So, before she gave a sweet to her teacher in lesson 3, she had 2 x 2 + 1 = 5 sweets in her bag.

In lesson 2, she gave out half of the sweets left in her bag and then gave one to the teacher. So, before she gave a sweet to her teacher in lesson 2, she had 5 x 2 + 1 = 11 sweets in her bag.

In lesson 1, she gave out half of the sweets in her bag and then gave one to the teacher. So, before she gave a sweet to her teacher in lesson 1, she had 11 x 2 + 1 = 23 sweets in her bag.

Therefore, Anna had 23 sweets in her bag at the start of the day.

When three dices are rolled.

Total number of outcomes =  = 216

Sum of 13 can be achieved in the following ways:

From the digits 6,4,3

So, there are 3! ways =  = 6

From the digits 6,2,5

So, there are 3! ways =  = 6

From the digits 5,4,4

So, there are  ways = 3

From the digits 6,6,1

So, there are  ways = 3

From the digits 3,5,5

So, there are  ways = 3

So, total numbers whose sum is 13=

So, Probability = .

Therefore, the probability of getting sum as 21 on rolling three dice =   .

Step-by-step explanation:

To find the range, we need to subtract the smallest value from the largest value in the dataset:

Range = Largest value - Smallest value

Range = 92 - 3

Range = 89

To find the variance and standard deviation, we need to calculate the mean first:

Mean = (Sum of all values) / (Number of values)

Mean = (92+19+41+24+75+53+70+3+67+64+9) / 11

Mean = 45.09 (rounded to two decimal places)

Next, we need to calculate the variance:

Variance = (Sum of squared differences from the mean) / (Number of values - 1)

Variance = [(92-45.09)^2 + (19-45.09)^2 + (41-45.09)^2 + (24-45.09)^2 + (75-45.09)^2 + (53-45.09)^2 + (70-45.09)^2 + (3-45.09)^2 + (67-45.09)^2 + (64-45.09)^2 + (9-45.09)^2] / (11-1)

Variance = 1071.45 (rounded to two decimal places)

Finally, we can calculate the standard deviation by taking the square root of the variance:

Standard deviation = Square root of variance

Standard deviation = Square root of 1071.45

Standard deviation = 32.74 (rounded to two decimal places)

The range tells us the difference between the highest and lowest values in the dataset, which in this case is 89. The variance and standard deviation tell us how spread out the data is from the mean. The higher the variance and standard deviation, the more spread out the data is. In this case, the variance and standard deviation are both relatively high, indicating that the data is fairly spread out.

Answer:

If i = -1 then the answer is f = 0

0 = ( 1 + -1 ).

f equaling 0 makes the statement true 0=0

Step-by-step explanation:

Answer:

D, 12 cm.

Step-by-step explanation:

First, draw out the circle with a hexagon in it. I did this by dividing the circle into thirds and then halving each of those because it was easier to visualize. I ended up with something like the image below. Now, we know that the sum of all of the angles that come together at the center of the circle is going to be 360. We can divide 360 by 6 to get 60. The top angle of the triangle, the angle closest to the center, is 60 degrees. We also know that the length of all of these legs that are going out will be the same, because the radius of the circle is constant. This makes each triangle an isosceles triangle. Given the isosceles base angle theorem, both base angles of the triangle must be the same. The sum of all angles in a triangle is 180 degrees. So the equation for x, one of the angles, would be  60 (angle closest to center) + 2x = 180. Let's subtract 60 from both sides to get 2x = 120. Divide both sides by two to get x = 60. Okay, so now we know that all angles in the triangle are 60 degrees. By the equilateral triangle theorem, we know these triangles must be equilateral because all of their angles are 60 degrees. Equilateral triangles also have the same side lengths. Because 12 is one side length, then all side lengths = 12. Therefore, the length of one side of the hexagon is 12.

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The equation of the tangent line is y = (-1/49)x + 8/49. 2. The equation of the normal line is y = 49x - 342.

A line that meets a curve at a point and has the same slope as the curve there is said to be the tangent line to that curve. A line that is perpendicular to the tangent line at a given position is the normal line to a curve at that location. In other words, the normal line's slope equals the tangent line's slope's negative reciprocal. The normal line is helpful for determining the direction of greatest change of the curve at a place whereas the tangent line gives information about the instantaneous rate of change of the curve at that point.

The given function is f(x) = 1/x.

The slope of the function is the derivative of the function thus,

f(x) = 1/x

f'(x) = -1/x²

f'(7) = -1/49

The equation of the line is given as:

y - y1 = m(x - x1)

where, m is the slope of the equation.

The equation of the tangent line is:

y - f(7) = f'(7)(x - 7)

y - 1/7 = -1/49(x - 7)

y = (-1/49)x + 8/49

b. The equation of the normal line to the graph is given as:

The slope of the normal line is negative and opposite of the tangent line.

That is,

m = 49.

y - f(7) = 49(x - 7)

y - 1/7 = 49(x - 7)

y = 49x - 342

Hence, the equation of the tangent line is y = (-1/49)x + 8/49. 2. The equation of the normal line is y = 49x - 342.

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Therefore, the possible number of pens in the box is p, where p is greater than 135.

Inequality refers to a situation in which there is a difference or disparity between two or more things, usually in terms of value, opportunity, or outcome. Inequality can take many forms, including social, economic, and political inequality.

Inequalities are mathematical expressions that compare two values using the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). To solve an inequality, you need to isolate the variable (the unknown quantity) on one side of the inequality symbol and determine the range of values for which the inequality holds true.

Here are some general steps to solve an inequality:

by the question.

Let's say there are 'p' pens in the box. Each pack has more than 15 pens, so we can write the inequality:

p/9 > 15

Multiplying both sides by 9, we get:

p > 135

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In both cases, the sum of x and y is not divisible by 3, we have demonstrated that the proposition is true. and the proposition is false, and we have shown a counterexample where the sum of two integers, one of which is not divisible by 3 and the other is divisible by 3, can be divisible by 3.

(a) To prove that the sum of two integers, x and y, neither of which is divisible by 3, cannot be divisible by 3, we can use a case-based proof.

Case 1: x and y leave a remainder of 1 when divided by 3.

Let x = 3m + 1 and y = 3n + 1, where m and n are integers. Then, the sum of x and y is 3m + 3n + 2, which leaves a remainder of 2 when divided by 3. Therefore, x + y is not divisible by 3.

Case 2: x and y leave a remainder of 2 when divided by 3.

Let x = 3m + 2 and y = 3n + 2, where m and n are integers. Then, the sum of x and y is 3m + 3n + 4, which leaves a remainder of 1 when divided by 3. Therefore, x + y is not divisible by 3.

Since in both cases, the sum of x and y is not divisible by 3, we have  demonstrated that the proposition is true.

(b) To prove that the sum of two integers, x and y, where x is not divisible by 3 and y is divisible by 3, cannot be divisible by 3, we can use a counterexample.

Let x = 2 and y = 6. Then, x is not divisible by 3 and y is divisible by 3. However, x + y = 8, which is not divisible by 3.

Therefore, the proposition is false, and we have shown a counterexample where the sum of two integers, one of which is not divisible by 3 and the other is divisible by 3, can be divisible by 3.

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Answer:

23.8 cm

Step-by-step explanation:

17 * 140% = 17 * 1.4 = 23.8 cm

The image of the house would be 23.8 cm tall on the screen.

To calculate the height of the image of the house on the screen, we can use the given zoom factor of 140%.

The zoom factor of 140% means that the images on the screen are 140% as long and 140% as wide compared to when they are printed on a sheet of paper.

To calculate the height of the image on the screen, we need to multiply the printed height by the zoom factor (140% or 1.4).

Height on the screen = Printed height * Zoom factor

Height on the screen = 17 cm * 1.4

Height on the screen = 23.8 cm

Therefore, the image of the house would be 23.8 cm tall on the screen.

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The evaluation of sin(20)sin(70) - sin(14)cos(26) - cos(6)cos(84) is approximately -1.

Use the trigonometric identities to simplify the expression:

sin(20)sin(70) - sin(14)cos(26) - cos(6)cos(84)

= (sin(20)cos(20)) / 2 - (sin(14)sin(64)) / 2 - (cos(6)cos(6)) / 2

= [(sin(40) - sin(80)) / 2] - [(cos(50) - cos(78)) / 2] - [(1 + cos(12)) / 2]

= (sin(40) - cos(50) + cos(78) - sin(80) - 1 - cos(12)) / 2

Now, use a calculator to evaluate each trigonometric function:

sin(40) = 0.6428, cos(50) = 0.6428, cos(78) = 0.2079, sin(80) = 0.9848, cos(12) = 0.9781

Substituting these values:

= (0.6428 - 0.6428 + 0.2079 - 0.9848 - 1 - 0.9781) / 2

= -2.755 / 2

= -1.3775

Therefore, the value of the given expression is approximately -1.

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The required average density of the giant envelope is equal to 3.30 x 10^9  kg/m^3.

Average density of the giant's envelope, required the mass and volume of the envelope.

Let us assume that the giant is a sphere with a radius of 0.6 AU,

the volume of the envelope is ,

V = (4/3)πr³

  = (4/3)π(0.6 AU)³

  = (4/3)(3.14)(0.6 × 149.6 ×10⁶ )³

To convert this volume to units of cubic meters,

Multiply by the conversion factor by,

1 AU = 149.6 x 10⁶m

⇒1 AU³ = (149.6 x 10⁶ km)³

⇒1 AU³ = 3.35 x 10³³ m³

Now ,calculate the density of the giant's envelope using the formula,

density = mass / volume

Giant's envelope has a mass of 0.5 solar masses,

which is equivalent to,

M = 0.5 x 1.99 x 10³⁰ kg

   = 9.95 x 10²⁹ kg

Average density of the giant's envelope is equals to,

⇒ density = 9.95 x 10²⁹ kg / (4/3)(3.14)(0.6 × 149.6 ×10⁶ )³

                = 3.30 x 10^9 kg/m^3

Therefore, the average density of the giant's envelope is approximately 3.30 x 10^9  kg/m^3.

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