Homework Problems Problem 9.12. Here is a game you can analyze with number theory and always beat me. We start with two distinct, positive integers written on a blackboard. Call them a and b. Now we take turns. (I'll let you decide who goes first.) On each turn, the player must write a new positive integer on the board that is the difference of two numbers that are already there. If a player cannot play, then they lose. For example, suppose that 12 and 15 are on the board initially. Your first play must be 3, which is 15 – 12. Then I might play 9, which is 12 – 3. Then you might play 6, which is 15 – 9. Then I can't play, so I lose. (a) Show that every number on the board at the end of the game is a multiple of gcd(a, b). (b) Show that every positive multiple of ged(a, b) up to max(a, b) is on the board at the end of the game. (c) Describe a strategy that lets you win this game every time.

Answer:

The zeroes are x = -4 and x = 2.

Answer: 62

Step-by-step explanation:

top row has 12

the next row has 1 more =13

and it alternates

12+13+12+13+12=62

a)

The events X and Y are not independent.

b)

(1)The value of P(B) is 1/124.

(ii) The value of P(A ∩ B') is 9/16.

We have,

a)

We have:

P(X) = 0.6

P(Y) = 0.2

P[(X Y) ∪ (X Y')] = 0.45 (using the distributive property of set operations)

We want to find P(X ∩ Y), which we can do using the formula:

P(X ∩ Y) = P(X) + P(Y) - P(X ∪ Y)

To find P(X ∪ Y), we can use the formula:

P(X ∪ Y) = P(X) + P(Y) - P(X ∩ Y)

Using the values given, we have:

P(X ∪ Y) = 0.6 + 0.2 - P(X ∩ Y) (substituting in P(X) and P(Y))

P[(X Y) ∪ (X Y')] = 0.45 (given)

We can rewrite the left-hand side as:

P[(X Y) ∪ (X Y')] = P(X Y) + P(X Y') (using the distributive property of set operations)

Since X and Y are disjoint events, we have:

P(X Y') = P(X) - P(X ∩ Y) (using the formula for the probability of the complement of an event)

Substituting these values into the expression for P[(X Y) ∪ (X Y')], we get:

P(X Y) + (P(X) - P(X ∩ Y)) = 0.45

Simplifying, we get:

P(X Y) = 0.45 + P(X ∩ Y) - P(X)

Substituting in the values of P(X) and P(Y) given, we get:

P(X Y) = 0.45 + P(X ∩ Y) - 0.6

P(X Y) = -0.15 + P(X ∩ Y)

Since probabilities cannot be negative, we know that P(X Y) ≤ P(X ∩ Y). Therefore, we can conclude that X and Y are not independent.

b)

i. We have:

P(A) = 7/8

P(B) = 1/12

P(A ∩ B) = 5/16

We want to find P(B). We can use the formula:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Since A and B are disjoint events, we have:

P(A ∪ B) = P(A) + P(B)

Substituting in the values given, we get:

P(A) + P(B) = 7/8 + P(B) = 1 - P(B') = 11/12

Solving for P(B), we get:

P(B) = 11/12 - 7/8 = 1/24

ii.

We want to find P(A ∩ B'). We can use the formula:

P(A ∩ B') = P(A) - P(A ∩ B)

Substituting in the values given, we get:

P(A ∩ B') = 7/8 - 5/16 = 9/16

Thus,

a)

X and Y are not independent.

b)

(i) P(B) = 1/124

(ii) P(A ∩ B') = 9/16

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Answer: 0.3

Step-by-step explanation:

3/10 = 30/100 = 0.3

Answer:

0.3

Step-by-step explanation:

The length of segment d is 8 when the value segment a is  4 cm, b is 12 cm , c is 6 cm

If two segments intersect inside or outside the circle then ab=cd

Given values of a is 4 cm, b is 12 cm , c is 6 cm and d is x

ab=cd

Plug in the values of a, b , c and d

4×12=6×d

48=6d

Divide both sides by 6

8=d

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Answer:

The answer to your problem is, D. If the data is categorical

Step-by-step explanation:

What a pictograph is:

Remember that a pictographs are less useful for comparing change over time or for presenting large sets of numerical data, as they can become cluttered and difficult to read.

Here is a little example, if you want to show how the population of a city has changed over the years, a line graph or a bar graph would be a better choice than a pictograph

Thus the answer to your problem is, D. If the data is categorical

Answer:5

explanation:

you are doing great finding the slopes:

The length of BD is 53 units.

We are given here that BD is (8x - 27) units and EC is (2x + 33) units. BD and EC are the diagonals of a trapezoid. As we know that diagonals of a trapezoid are equal. Therefore, we will equate the given diagonals of a trapezoid.

BD  =  EC

Substituting the given values of BD and EC

8x - 27  =  2x + 33

combining the like terms

8x - 2x  =  33 + 27

6x  =  60

x = 60/6

x = 10

Now, as we have to find BD, we will substitute the value of x in the given equation for BD.

BD = 8x - 27

BD = 8(10) - 27

BD = 80 - 27

BD = 53

Therefore, BD is 53 units.

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The complete question is " If BD = 8x - 27 and EC-2x + 33, find BD​ with respect to the image shown."

The value of n is 5.

Given that, the Coordinate of P = (n,3)

R is on y-axis & the y-coordinate of P & R are equal.

So coordinate of R = (3,0)

Coordinate of Q = (n,-2)

Using distance formula,

Distance between P & Q =

[tex]=\sqrt{(n-n)^2+(-3-(-2)^2} \\\\=\sqrt{(3+2)^2} \\\\= \sqrt{25} = 5[/tex]

Distance between P & R =

[tex]\sqrt{(n-0)^2+(3-3)^2}[/tex]

= n

According to the question it is given that distance between P & Q is equal to the distance between P & R. So, n = 5.

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Answer:

height = 9 in

Step-by-step explanation:

The formula for volume (V) of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height

In the question, we're given the volume and in the diagram, we're given the length (8.5 in.) and the width (2 in.).  We can solve for l by plugging in our volume, length, and width into the formula and solving for x (the length):

[tex]153=(8.5)(2)x\\153=17x\\9=x[/tex]

Answer:

303.375 or 303 3/8

Step-by-step explanation:

First, we can split the polygon into three smaller shapes: a triangle, a big rectangle, and a small rectangle. We will call them A, B, and C.

A:

Length: 28.25 - (3 + 13) = 28.25 - 16 = 12.25 (12 1/4)

Height: 10 + 9 = 19

Area: (12.25 * 19) / 2 = 232.75 / 2 = 116.375 ft

B:

Length: 3 + 13 = 16

Width: 10

Area: 16 * 10 = 160 ft

C:

Length: 3

Width: 9

Area: 3 * 9 = 27 ft

Now we have to add all three digits:

116.375 + 160 + 27 = 303.375 or 303 3/8

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Hope this helps :)

Sure, I can help with that. The problem is asking for the area of a polygon composed of two rectangles and a right triangle.

The area of a rectangle is given by the formula length * width, and the area of a right triangle is given by the formula 1/2 * base * height.

Let’s calculate the areas:

For the first rectangle with dimensions 9ft by 13ft, the area is 9 * 13 = 117 square feet.

For the second rectangle with dimensions 10ft by 8ft, the area is 10 * 8 = 80 square feet.

For the right triangle with dimensions 28ft by 4ft, the area is 1/2 * 28 * 4 = 56 square feet.

Adding these areas together gives the total area of the polygon:

117 + 80 + 56 = 253 square feet

So, the area of the polygon is 253 square feet.

As per the given values, the area of the hemisphere is 45065.41 m³

Volume = 450000 m³

Calculating the volume of the hemisphere -

Volume = 2/3 πr³

Substituting the values -

450,000 =2/3 x 3.14 x r³

Solving for r³

r³ = 450000 x 3/(2 x 3.14)

r³ = 214968.1

r = √214968.1

r = 59.9

Calculating the area of the hemisphere -

Area = 4πr²

Substituting the values -

Area = 4 x 3.14 x (59.9)²

= 12.56 x (59.9)²

= 45065.41

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Answer:

48.8 square inches

Step-by-step explanation:

To find the area of the triangle, we can use the formula:

Area = 1/2 * base * height

In this case, the base of the triangle is the longer side, which is 13 inches, and the height is the shorter side, which is 7.5 inches. However, we need to make sure that the angle provided is the angle between the base and the height, and not one of the other angles of the triangle.

Assuming that the angle provided is indeed the angle between the base and the height, we can proceed with the calculation:

Area = 1/2 * 13 inches * 7.5 inches

Area = 48.75 square inches

Rounded to the nearest tenth, the area of the triangle is 48.8 square inches.

The correct statement is "Find half the difference of 14 and 6, multiply by 6, then add 4".

The solution of the fraction expression is calculated as follows;

1/[2 x (14 - 6) x 6 + 4]

To solve the above expression, we will apply the rule of BODMAS as shown below;

The difference of 14 and 6 = 14 - 6 = 8

The next is to divide 8 by 2

= 8/2

= 4

The next step is to multily the result by 6

= 4 x 6

= 24

The final step is to add 4 to it;

= 24 + 4

= 28

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To calculate the accuracy of the test, we need to create a 2x2 contingency table:

Actual Positive (Meningitis)Actual Negative (No Meningitis)Test PositiveTrue Positive (TP) = 6False Positive (FP) = 3Test NegativeFalse Negative (FN) = 2True Negative (TN) = 69

From the information given, we know that there are 8 actual positive cases (people with meningitis) and 72 actual negative cases (people without meningitis). We also know that there were 3 false positive results (people who tested positive for meningitis but did not have it) and 2 false negative results (people who tested negative for meningitis but actually had it).

Using this information, we can calculate the accuracy of the test as:

Accuracy = (TP + TN) / (TP + TN + FP + FN)

Accuracy = (6 + 69) / (6 + 69 + 3 + 2)

Accuracy = 75 / 80

Accuracy = 0.9375 or 93.75%

Therefore, the accuracy of the meningitis test is 93.75%.

The expected value of Y is: 1.8

The student made a mistake by using the formula for the variance of XY instead of the expectation of XY^2.

(a) E(Y) = (0.2)(2) + (0.5)(1) + (0.3)(4) = 1.8

(b) Since X and Y are independent, the distribution of X + Y is the convolution of their respective distributions. That is, if Z = X + Y, then for any real number z,

F(z) = P(Z ≤ z) = P(X + Y ≤ z) = ∑P(X = i, Y ≤ z − i) for i = 0, 1, 2, ...

Now, since X has a Poisson distribution with parameter 2 and Y takes values 2, 3, and 4 with probabilities 0.2, 0.5, and 0.3 respectively, we have

P(X = i, Y = j) = P(X = i)P(Y = j) = e^(-2) (2^i/i!)p_j for i = 0, 1, 2, ... and j = 2, 3, 4

where p_2 = 0.2, p_3 = 0.5, and p_4 = 0.3. Then, for z ≥ 2,

F(z) = P(Z ≤ z) = ∑_{i=0}^{z-2} P(X=i, Y≤z-i) = ∑_{i=0}^{z-2} e^(-2) (2^i/i!) ∑_{j=2}^{min(z-i, 4)} p_j

(c) P(X > Y) = ∑_{i=0}^∞ ∑_{j=0}^{i-1} P(X=i, Y=j) = ∑_{i=0}^∞ ∑_{j=0}^{i-1} e^(-2) (2^i/i!) p_j

(d) The calculation is incorrect. It should be E[XY^2] = E[X]E[Y^2] = 2(0.2)(2^2) + 2(0.5)(3^2) + 2(0.3)(4^2) = 11.6. The student made a mistake by using the formula for the variance of XY instead of the expectation of XY^2.

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When you are given an equation for a function, it is important to know whether the function is increasing or decreasing. A function is said to be increasing if the value of the function increases as the input increases. Conversely, a function is said to be decreasing if the value of the function decreases as the input increases.

To determine whether a function is increasing or decreasing, you need to look at the sign of its first derivative. The first derivative of a function is the rate of change of the function with respect to its input. If the first derivative is positive, the function is increasing, and if it is negative, the function is decreasing. If the first derivative is zero, the function may have a local maximum or a local minimum.

To explain this in more detail, let's take the example of the function f(x) = x^2. To find the first derivative of this function, we need to differentiate it with respect to x. This gives us f'(x) = 2x. We can see that f'(x) is positive for x > 0, which means that the function f(x) = x^2 is increasing for x > 0. Similarly, f'(x) is negative for x < 0, which means that the function f(x) = x^2 is decreasing for x < 0.

In summary, to determine where a function is increasing or decreasing, you need to look at the sign of its first derivative. If the first derivative is positive, the function is increasing, and if it is negative, the function is decreasing. If the first derivative is zero, the function may have a local maximum or a local minimum.

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Therefore, the length of a side of the square is x/4 inches by the equation.

In this context, we have a wire that we need to bend into a square. A square has four equal sides, so if we let s be the length of one side of the square, then the total length of the wire must be 4s.

The equation 4s = x represents this relationship, where x is the total length of the wire.

To solve for s, we can isolate s on one side of the equation by dividing both sides by 4. This gives us:

4s / 4 = x / 4

Simplifying, we get:

s = x / 4

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The design described in the first scenario uses a related sample with matched subjects.

In the first scenario, Suzanne Thomas is interested in comparing alcoholics with social phobia to alcoholics without social phobia. She matches participants from both groups on several variables, such as age and gender, in order to create comparable groups. This indicates the use of matched subjects design, where participants in one group are matched with participants in another group based on certain criteria to create comparable groups for comparison. Suzanne Thomas then collects data on the severity of alcohol dependence from each group. Therefore, the design described in the first scenario uses a related sample with matched subjects.

In the second scenario, the design described does not involve the use of a related sample. It is mentioned that the researcher wants to compare kleptomaniacs who are taking antidepressants to the population average, without matching or pairing participants.

Therefore, the design in the second scenario does not involve the use of a related sample.

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Reverse logistics involves designing a customer-focused supply management system and understanding transportation options, but it does not necessarily require knowledge of e-procurement systems or triage.

Reverse logistics is the process of managing the flow of products, materials, and information from the end-user back to the point of origin. It involves activities such as returns, refurbishment, recycling, and disposal of products.

The options given are:

a. triage - This refers to the process of determining the priority of patients' treatments based on the severity of their condition. While triage is an important concept in healthcare, it is not directly related to reverse logistics.

b. designing a supply management system from the customer's perspective - This is a key aspect of reverse logistics. A successful reverse logistics system requires a customer-focused approach to ensure that products can be easily returned and that customers have a positive experience with the returns process.

c. understanding of e-procurement systems - While e-procurement systems can be helpful in managing the reverse logistics process, it is not a necessary component of reverse logistics. E-procurement systems are primarily used for purchasing and procurement activities.

d. understanding of transportation options - Transportation is a critical component of reverse logistics, as it is necessary to move returned products from the point of origin back to the manufacturer or retailer. Understanding transportation options and selecting the most cost-effective and efficient method of transportation is essential to managing the reverse logistics process.

In summary, reverse logistics involves designing a customer-focused supply management system and understanding transportation options, but it does not necessarily require knowledge of e-procurement systems or triage.

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The solution to the given set of equations is Infinite solutions.

We have the equation

R: -3y = -3x -9

S: y = x + 3

Now, solving the equation R and S as

-3y = -3x - 9

3y =  3x + 9

_________

   0 = 0 + 0

   0 = 0

Also, -3/1 = 3/(-1) = 9/(-3)

-3/1 = -3/1 = -3/1

Thus, the equation have Infinite many solutions.

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Answer:

The answer to your problem is, B. Sample space = lemon-lime, lemon-lime, lemon-lime, watermelon, watermelon, grape

Step-by-step explanation:

We are given that, Julia had a bag filled with gumballs. There were 3 lemon-lime, 4 watermelon, and 6 grape gumballs.

There are 3 lemon-lime: Elements in sample space are lemon-lime, lemon-lime, lemon-lime

There are also 4 watermelon: Elements in sample space are watermelon, watermelon, watermelon, watermelon

Lastly there are 6 grape: Elements in sample space are grape, grape, grape, grape, grape, grape

Which can lead to the problem of:

3 + 4 + 6 = 13. Same as option B

Thus the answer to your problem is, B. Sample space = lemon-lime, lemon-lime, lemon-lime, watermelon, watermelon, grape

Answer:

k<-10

Step-by-step explanation:

-3k+5>35

-3k+5-5>35-5

simplifica la expresión

k<-10

Based on the newspaper report, it can be concluded that Candidate A is leading with 52% of the vote while Candidate B has 48% of the vote. However, it is important to note that the margin of error is +/- 4%. This means that there is a chance that the actual percentage of votes for each candidate could be 4% higher or lower than what the poll suggests.

For example, Candidate A's actual percentage of votes could be anywhere between 56% (52% + 4%) and 48% (52% - 4%), while Candidate B's actual percentage of votes could be anywhere between 52% (48% + 4%) and 44% (48% - 4%).

It is also important to note that opinion polls are not always accurate predictors of election results. They are simply a snapshot of public opinion at a particular moment in time and can be influenced by various factors such as the wording of the questions, the sample size, and the demographic makeup of the respondents.

In summary, based on the newspaper report, Candidate A has a slight lead over Candidate B with 52% of the vote, but there is a margin of error of +/- 4% which means that the actual percentage of votes for each candidate could be different. It is important to keep in mind that opinion polls are not always accurate predictors of election results.

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The probability that a srs of 20 students will spend an average of between 600 and 700 dollars is 0.92081.

We need to find the probability that a simple random sample of 20 students will spend an average of between 600 and 700 dollars.

To solve this problem, we will use the central limit theorem, which states that the sampling distribution of the sample means will be approximately normally distributed with a mean of μ and a standard deviation of σ/√(n), where n is the sample size.

Thus, the mean of the sampling distribution is μ = 650 and the standard deviation is σ/sqrt(n) = 120/√(20) = 26.83.

We need to find the probability that the sample mean falls between 600 and 700 dollars. Let x be the sample mean. Then:

Z1 = (600 - μ) / (σ / √(n)) = (600 - 650) / (120 / √t(20)) = -1.77

Z2 = (700 - μ) / (σ / √(n)) = (700 - 650) / (120 / √(20)) = 1.77

Using a standard normal distribution table or calculator, we can find the area under the standard normal distribution curve between these two Z-scores as:

P(-1.77 < Z < 1.77) = 0.9208

Therefore, the probability that a simple random sample of 20 students will spend an average of between 600 and 700 dollars is 0.9208, or approximately 0.92081 when rounded to five decimal places.

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False.

The Cp and Cpk statistics are both used to assess the capability of a process to meet specifications, but they have different purposes.

The Cp statistic is a measure of how well the process spread (variation) fits within the specification limits. It assumes that the process mean is centered on the target value. It is calculated as the ratio of the specification width to the process spread (six times the process standard deviation).

The Cpk statistic, on the other hand, takes into account both the process spread and the deviation of the process mean from the target value. It is calculated as the minimum of two values: the difference between the process mean and the closest specification limit divided by three times the process standard deviation (assuming the process is centered within the specification limits), or the Cp value adjusted for the deviation of the process mean from the target value.

Both statistics have their uses and limitations depending on the situation. If the process mean is not centered on the target value, the Cpk statistic may be more useful than the Cp statistic since it takes into account both the spread and centering of the process.

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The 99% confidence interval for the mean score, μ, of all students taking the test is approximately (78.603, 87.397)

To construct a 99% confidence interval for the mean score, μ, of all students taking the calculus test, we need to follow these steps:

Step 1: Identify the sample mean, sample standard deviation, and sample size.
Sample mean (X) = 83
Sample standard deviation (σ) = 8.70
Sample size (n) = 26

Step 2: Determine the appropriate z-score for a 99% confidence interval.
For a 99% confidence interval, the z-score (z) is 2.576.

Step 3: Calculate the standard error of the mean (SEM).
[tex]SEM= \frac{σ }{\sqrt{n} } = \frac{8.70}{\sqrt{26} } = 1.706[/tex]

Step 4: Compute the margin of error (ME).
ME = z * SEM = 2.576 * 1.706 = 4.397

Step 5: Construct the 99% confidence interval.
Lower limit = X - ME = 83 - 4.397 = 78.603
Upper limit = X + ME = 83 + 4.397 =  87.397

Your answer: The 99% confidence interval for the mean score, μ, of all students taking the test is approximately (78.603, 87.397).

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The equation of the plane tangent to the surface z = x²y + xy² + ln x + R at (1, 0, R) is z = x + y + (R - 1).

To find the equation of the plane tangent to the surface z = x²y + xy^2 + ln x + R at the point (1, 0, R), we'll need to find the partial derivatives with respect to x and y, and then use the point-slope form of the tangent plane equation. Here are the steps:

1. Find the partial derivatives of the surface function with respect to x and y:
∂z/∂x = 2xy + y² + (1/x)
∂z/∂y = x² + 2xy

2. Evaluate the partial derivatives at the given point (1, 0, R):
∂z/∂x(1, 0) = 2(1)(0) + (0)² + (1/1) = 1
∂z/∂y(1, 0) = (1)² + 2(1)(0) = 1

3. Use the point-slope form of the tangent plane equation:
z - z₀ = a(x - x₀) + b(y - y₀)

4. Substitute the point (x₀, y₀, z₀) = (1, 0, R) and the partial derivative values a = ∂z/∂x = 1, b = ∂z/∂y = 1:
z - R = 1(x - 1) + 1(y - 0)

5. Simplify the equation:
z - R = x - 1 + y

6. Rearrange the equation to the standard form:
z = x + y + (R - 1)

The equation of the plane tangent to the surface z = x²y + xy² + ln x + R at (1, 0, R) is z = x + y + (R - 1).

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