ORRELATION
Please complete the following quiz. Use the data set attached. Please upload your Word doc for your submission. Include your SPSS output in this document as part of Step 3.
Test for the significance of the correlation coefficient at the .05 level using a two-tailed test between hours of studying and grade.
Hours of Study Grade
0 80
5 93
8 97
6 100
5 75
3 83
4 98
8 100
6 90
2 78
Sheet 1, Sheet 2, Sheet 3

The simplified expression after simplification is 3x² - 3x - 2.

To simplify the expression, we need to distribute the negative sign to the second polynomial and then combine like terms.

So,

(2xx² - 4) - (-x²+ 3x - 6)

= 2x² - 4 +x²- 3x + 6 (distributing the negative sign)

= 3x²- 3x + 2 (combining like terms)

To simplify the given expression, we first need to distribute the negative sign to the terms inside the second parentheses:

(2x² - 4) - (-x² + 3x - 6)

= 2x² - 4 + ² - 3x + 6 (distributing the negative sign changes the signs of all terms inside the second parentheses)

= 3x² - 3x + 2

Therefore, the simplified expression is 3x² - 3x - 2.

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Complete Question:

Simplify (2x²−4)−(−x²+3x−6)(² −4)−(−x² +3x−6).

Answer:

2(1/2)(4)(5.5) + 2(1/2)(5)(6) + 4(6) =

22 + 30 + 24 = 76 square centimeters

B is correct.

Answer:

.2250

Step-by-step explanation:

The cosine function is positive in the 1st quadrant. The value of cos 77 is given as .2250

Hope this helps

Answer:

$5,920.44

Step-by-step explanation:

The most general form to compute the amount accrued when interest is compounded with periodic contributions is given by the formula

[tex]A = P \dfrac{\left(1 + \dfrac{r}{n}\right)^{nt}-1}{\dfrac{r}{n}}[/tex]

where
A = Accrued amount (principal + interest)

P = Periodic contribution to the sinking fund,

r = Annual nominal interest rate as a decimal

R = Annual nominal interest rate as a percent

r = R/100

n = number of compounding periods per unit of time

We are given A as 412,000 (amount at the end of 4 years) and asked to compute P(monthly contribution)

We have R = 18%, so r = 18/100 = 0.18

t = 4 years

n = 12 because we are compounding monthly so in 1 year we compound 12 times

Plugging these values into the equation we get

[tex]412000 = P \dfrac{\left(1 + \dfrac{0.18}{12}\right)^{12 \cdot 4}-1}{\dfrac{0.18}{12}}\\\\[/tex]

We have

r/n = 0.18/12 = 0.015

1 + r/n = 1.015

nt = 12 x 4 = 48

[tex]412000 = P\dfrac{ (1.015)^{48} -1 } {0.015}\\\\[/tex]

[tex]412000 = P \dfrac{1.043478}{0.015}\\\\412000 = P \cdot 69.5652\\\\\P = \dfrac{412000}{69.5652}\\\\[/tex]

[tex]P = 5,922.4998[/tex]

There may be differences in the given answer choices because of round off errors. The amount computed comes closest to the first answer choice

$5,920.44

Answer:

  (a)  $5920.44

Step-by-step explanation:

You want the monthly payment required to a sinking fund that is expected to have a value of $412,000 in 4 years if the account pays 18% interest.

A table of sinking fund payment values will tell you that the monthly payment required at an 18% interest rate for 4 years is $14.37 per thousand of account value.

We want the account value to be 412 thousand, so the monthly payment will need to be ...

  412 × $14.37 = $5,920.44

__

Additional comment

The actual payment required is $5922.50. Using a multiplier rounded to cents understates the payment because of rounding error.

If the more precise multiplier $14.375 per thousand is used, then the payment value would be correctly computed.

If you simply divide the desired $412000 into 48 equal payments, each would be $8,583.33. Since interest is earned, you know the payment is less than this amount. $5,920.44 is the only reasonable answer choice.

Answer:

Step-by-step explanation:

Let's start with the wording of the question. Since the sphere is tangential to the faces of the cube, if we draw our radius perpendicular (forming a right angle with the face), we can see it makes a direct connection to the cube face. This means that the diameter of the sphere is equal to the length of the cube.

Next, the question is asking for the volume outside the sphere and inside the cube. To find this, we need to take the volume of the sphere and subtract it from the volume of the cube.

The volume of a sphere is given as: 4/3*pi*(r)^3

The volume of a cube (or any rectangle) is given as: l*w*h

Now all that's left is to plug in the radius and sides of the cube (which we know is double the radius) and subtract.

(6)*(6)*(6)  -  4/3*pi*(3)^3

216 - 113.1 = 102.9

The question asks for standard units, but we aren't given any units so I'm a bit unclear about this. Either way, volumes are measured in cubics (m^3, ft^3, etc.) so it would be the unit of the radius cubed.

Hope I could help!

The volume contained outside the sphere and inside the cube is 216 - 36π cubic units, which is approximately 99.425 cubic units when rounded to three decimal places.

To find the volume contained outside the sphere and inside the cube, we need to calculate the volume of the cube and subtract the volume of the sphere.

The cube's side length is equal to twice the radius of the inscribed sphere. Therefore, the cube's side length is 2 * 3 = 6 units.

The volume of a cube is calculated by raising the side length to the power of 3. So, the volume of the cube is [tex]6^3 = 216[/tex] cubic units.

The volume of a sphere is given by the formula where r is the radius. Substituting the value, we have [tex](4/3) * π * 3^3 = (4/3) * π * 27[/tex]= 36π cubic units.

Now, to find the volume contained outside the sphere and inside the cube, we subtract the volume of the sphere from the  [tex](4/3) * π * r^3[/tex],volume of the cube: 216 - 36π.

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If in a survey, 200 college students were asked whether they live on campus and if they own a car, 55% of college students in the survey don't own a car.

To find the percent of college students who don't own a car, we need to add up the number of students who don't own a car and divide it by the total number of students in the survey. In this case, the total number of students in the survey is 200.

From the table, we can see that there are 88 students who live on campus and don't own a car, and 22 students who don't live on campus and don't own a car. So the total number of students who don't own a car is 88 + 22 = 110.

To find the percentage, we divide the number of students who don't own a car by the total number of students in the survey and then multiply by 100 to get the percentage:

Percentage of students who don't own a car = (110/200) x 100% = 55%

When working with percentages, we need to divide the number we are interested in by the total and then multiply by 100 to get the percentage.

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The number of possible outcomes for each game is 3 (win, lose, or tie). Since there are 2 games left, the total number of possible outcomes is 3 x 3 = 9. Therefore, none of the given answers (a, b, or c) is correct. The correct answer is d.

To determine the number of possible outcomes for your favorite football team's remaining 2 games, we'll consider each game independently. Each game can have 3 possible outcomes: win, lose, or tie. For 2 games, you can use the multiplication principle:

Number of possible outcomes = Outcomes for Game 1 × Outcomes for Game 2

So, the number of possible outcomes is:

3 (win, lose, or tie in Game 1) × 3 (win, lose, or tie in Game 2) = 9

Since 9 is not among the given options, the correct answer is:

d. None of the other answers is correct.

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We can expect that in a random draw of 4 peas from this cross, the number of yellow peas is likely to fall within the range of 2 to 4, with a typical range of 2.134 to 3.866.

The ratio of 3 yellow to 1 green suggests that the cross is between two heterozygous pea plants, each carrying one dominant (yellow) and one recessive (green) allele. This type of cross is called a monohybrid cross.

We can use the binomial distribution to calculate the probability of obtaining a certain number of yellow pea plants in a sample of four. Let p be the probability of obtaining a yellow pea plant, and q be the probability of obtaining a green pea plant, where p + q = 1. Since the ratio is 3 yellow to 1 green, we have p = 3/4 and q = 1/4.

The probability of obtaining exactly k yellow pea plants out of n trials is given by the binomial probability formula:

P(k) = (n choose k) * p^k * q^(n-k)

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items out of n without regard to order. It can be calculated as:

(n choose k) = n! / (k! * (n-k)!)

where n! is the factorial of n.

To find the typical range of the number of yellow peas drawn from a random draw of 4 peas expressed as the 1st SD window, we need to calculate the mean and standard deviation of the binomial distribution. The mean is given by:

μ = n * p

and the standard deviation is given by:

σ = sqrt(n * p * q)

where sqrt represents the square root function.

Substituting n = 4, p = 3/4, and q = 1/4, we have:

μ = 4 * 3/4 = 3

and

σ = sqrt(4 * 3/4 * 1/4) = sqrt(3/4) = 0.866

The typical range of the number of yellow peas drawn from a random draw of 4 peas expressed as the 1st SD window is given by:

[μ - σ, μ + σ] = [3 - 0.866, 3 + 0.866] = [2.134, 3.866]

Therefore, we can expect that in a random draw of 4 peas from this cross, the number of yellow peas is likely to fall within the range of 2 to 4, with a typical range of 2.134 to 3.866.

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Answer:

45 2/3, or 45.666666..., rounded to the nearest tenth is 45.7.

On the basis of Reggie thinking of a secret number, which is divisible by 9 and 12, the smallest secret number that Reggie could be thinking is equals to the thirty-six.

We have Reggie is thinking of a secret number. Let the secret number be equal to x. According to scenario, x is divisible by 12. Also, it is divisible by 9. Consider that Reggie is telling the truth to both of them that is x is divisible by 9 and 12. We have to determine the smallest value of x. The smallest number divisible by both 9 and 12 is the smallest common multiple of 9 and 12. Now, Multiples of 9: 9, 18, 27, 36, 45, 54, 63...

Multiples of 12: 12, 24, 36, 48, 60...

The least common multiple of 9 and 12 from above list of multiples is 36. Hence, required value is 36.

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Answer:

Step-by-step explanation:

(x - 2)²/16 - (y - 1)²/4 = 1

(x - 2)² - 4(y - 1)² = 16

x² + 4 - 4x - 4(y² + 1 - 2y) = 16

x² + 4 - 4x - 4y² - 4 + 8y = 16

x² - 4x + 8y - 4y² = 16

x² - 4x = 16  ,  -4y² + 8y = 16

x(x - 4) = 16  ,  4y(-y + 2) = 16

x = 16, x = 20,  y = 4, y = -14

The coordinates of the point (-7,-3) after a 90 degree clockwise rotation about the origin are (-3,-7).

To rotate a point 90 degrees clockwise about the origin, we need to swap its x and y coordinates and negate the new x coordinate.

So, starting with point (-7,-3):

Swap the x and y coordinates to get (3,-7)

Negate the new x coordinate to get (-3,-7)

Therefore, the coordinates of the point (-7,-3) after a 90 degree clockwise rotation about the origin are (-3,-7).

In mathematics, coordinates are used to specify the position of a point or an object in a particular space. The number of coordinates needed depends on the dimension of the space in which the point or object exists.

In two-dimensional space (also called the Cartesian plane), a point is located by two coordinates, usually denoted as (x, y), where x represents the horizontal distance from a fixed reference point called the origin, and y represents the vertical distance from the origin.

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When the popularity of grapefruits increases, it leads to a higher demand for them in the market. Consequently, the price of grapefruits rises due to this increased demand. In response to the price increase, several changes occur in the market for grapefruits.

Firstly, as the price of grapefruits increases, the quantity demanded by consumers will likely decrease, as some individuals might be deterred by the higher cost. This is in accordance with the law of demand, which states that as the price of a good increases, the quantity demanded decreases, and vice versa.

Secondly, the higher price of grapefruits may encourage producers to increase their supply to take advantage of the increased revenue potential. As a result, the quantity supplied in the market will likely rise, following the law of supply, which states that as the price of a good increases, the quantity supplied increases, and vice versa.

In the long run, the market will seek to achieve equilibrium, where the quantity demanded equals the quantity supplied. This process will involve adjustments in both supply and demand until a new equilibrium price and quantity are established. The ultimate outcome will depend on the elasticity of both supply and demand for grapefruits, which determine how responsive they are to price changes.

In conclusion, an increase in the popularity of grapefruits resulting in a higher price leads to changes in the market, including decreased quantity demanded, increased quantity supplied, and eventually, a new market equilibrium.

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Answer:

129

Step-by-step explanation:

UR WELCOM

JUST TIMES AND USE PEDMAS AND ADDITION SUB AND MORE

Answer:

Step-by-step explanation:

33

The weighted mean score is:

90.7 / 100% = 0.907 or 90.7%

Rounding to two decimal places, the student's weighted mean score is 90.70%.

To calculate the weighted mean score, we need to multiply each score by its corresponding percent of the final grade, then sum these products, and finally divide by the total percent of the final grade.

In this case, we have:

Homework score: 85, percent of final grade: 10%

First quiz score: 87, percent of final grade: 10%

Second quiz score: 97, percent of final grade: 10%

Project score: 95, percent of final grade: 40%

Final exam score: 86, percent of final grade: 30%

To calculate the weighted mean score, we first need to calculate the products of the score and the percent of the final grade for each component:

Homework contribution to the final grade: 85 x 0.1 = 8.5

First quiz contribution to the final grade: 87 x 0.1 = 8.7

Second quiz contribution to the final grade: 97 x 0.1 = 9.7

Project contribution to the final grade: 95 x 0.4 = 38

Final exam contribution to the final grade: 86 x 0.3 = 25.8

Next, we sum these products:

8.5 + 8.7 + 9.7 + 38 + 25.8 = 90.7

Finally, we divide by the total percent of the final grade:

10% + 10% + 10% + 40% + 30% = 100%

So, the weighted mean score is:

90.7 / 100% = 0.907 or 90.7%

Rounding to two decimal places, the student's weighted mean score is 90.70%.

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In a comprehensive headache treatment program, people who were low users of analgesic medications achieved at least a 50 percent reduction in headache pain.

A comprehensive headache treatment program is a multi-disciplinary approach to managing and treating headaches. It typically involves a team of healthcare professionals, such as neurologists, pain specialists, psychologists, physical therapists, and nutritionists, who work together to develop a personalized treatment plan for the patient.

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The linear approximation of ln(1.09) is approximately equal to 0.09 (rounded to two decimal places).

To use a linear approximation to estimate ln(1.09) and produce a small error, we will follow these steps:
Step 1: Choose a value of 'a' close to 1.09 for which the natural logarithm is easy to calculate. In this case, we can choose a = 1.
Step 2: Find the derivative of the natural logarithm function, which is f'(x) = 1/x.
Step 3: Evaluate the derivative at the chosen value of 'a'. In our case, f'(1) = 1/1 = 1.
Step 4: Use the linear approximation formula to estimate ln(1.09):
ln(1.09) ≈ ln(a) + f'(a) * (1.09 - a)
Step 5: Plug in the values of 'a' and f'(a) into the formula:
ln(1.09) ≈ ln(1) + 1 * (1.09 - 1)
Since ln(1) = 0, we have:
ln(1.09) ≈ 1 * (0.09) = 0.09

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Answer:

Step-by-step explanation:

Here A be n×n matrix with real enties

y={x=n×n matrix with real enties | Ax=xA} is vector space.

Let m be set of all n*n matrix with real enties then m is vector space over IR.

we show y is vector subspace of m.

Here [tex]I_{n*n\\}[/tex] identity matrix

IA=AI

∴ I ∈ y

∴ y is non empty subset of m.

Also if [tex]x_{1}[/tex],[tex]x_{2}[/tex] ∈ y ⇒ A[tex]x_{1}[/tex]=[tex]x_{1}[/tex]A ,A[tex]x_{2}[/tex]=[tex]x_{2}[/tex]A

for [tex]\alpha[/tex] ∈ IR arbitrary

[tex](\alpha x_{1} +x_{2} )A=\alpha (x_{1}A)+x_{2} A\\=\alpha (Ax_{1})+Ax_{2}\\ =A(\alpha x_{1} +x_{2})\\[/tex]

Hence [tex]\alpha x_{1}+x_{2}[/tex] ∈ y ∀ [tex]x_{1},x_{2}[/tex] ∈ y

∴ y is subspace of m.

∴ y is vector space.

Are all the open sets that contain c, and hence all the neighborhoods of c in (X,T).

To show that (X,T) is not a normal space, we need to find two disjoint closed subsets of X that cannot be separated by open neighborhoods. Let A = {a,b,c,d} and B = {a,b,e} be two disjoint closed subsets of X. We can see that A and B cannot be separated by open neighborhoods as follows:

Suppose there exist open sets U and V in X such that A ⊆ U, B ⊆ V, U ∩ V = ∅. Then, since {a,b} is in both A and B, we must have a and b both in either U or V, say a and b are both in U. But then, U cannot be a subset of any open set containing {a,c,d}, since U also contains b, which is not in any such set. Therefore, there is no way to separate A and B by open neighborhoods, and (X,T) is not a normal space.

To find the collection of all neighborhoods of c, we need to find all open sets containing c. Since {a,c,d} is the smallest open set containing c, we have:

N(c) = {X, {a}, {a,b,c,d}, {a,c,d}, {a,b,c,d,e}, {a,b,c,e}, {a,c,d,e}}

These are all the open sets that contain c, and hence all the neighborhoods of c in (X,T).

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We can prove the congruency of ΔABE and ΔDBC by Side- Side- Angle rule , Side -Side- Side rule and Hypotenuse- Leg rule.

Hence option a, b and c are the correct options.

In the given figure we have,

Line segment, CD ≅ Line segment,,EA ____(1)

Line segment, AD is the perpendicular bisector of line segment of CE.

That is,

Line segment CE is bisected at point B so,

Line segment CB = Line segment EB ____(2)

And, angle ABE = 90°= angle CBD _____(3)

From equation (1), (2) and (3) we can apply Pythagoras theorem and conclude,

Line segment, AB = line segment, DB_____(4)

From equation (1), (2) and (3) we can apply Side - Side - Angle rule to say that ΔABE ≅ ΔDBC.

From equation (1), (3) and (4) we can apply Side - Side - Angle rule to say that ΔABE ≅ ΔDBC.

From equation (1), (2) and (4) we can apply Side - Side - Side rule to say that ΔABE ≅ ΔDBC.

From equation (1), and (2) we can apply Hypotenuse- Leg rule to say that the right triangles ΔABE ≅ ΔDBC where angle ABE = 90°= angle CBD .

And from equation (1), and (4) we can apply Hypotenuse- Leg rule to say that the right triangles ΔABE ≅ ΔDBC where angle ABE = 90°= angle CBD .

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The sum of squares of the real roots of p(x) = x³ - x² - 18x +18 is 37 for k= 18.

The cubic equation is given as,

p(x) = x³ - x² - 18x +k

To find the real roots of the cubic equation p(x) we can equate p(x) =0 , we get,

x³ - x² - 18x +k = 0

⇒ x² (x -1) - 18(x - k/18) = 0

For factoring the equation we can equate  (x -1) = (x - k/18) by comparing it with solving of general equations.

That is by arranging the cubic equation after equating (x -1) = (x - k/18)  we will get,

(x-1)(x² -18) =0

Thus we get,

(x -1) = (x - k/18)

⇒ k/18 =1

⇒ k =18

The cubic equation which will give us real roots will become,

p(x) = x³ - x² - 18x +18

By factoring we can find the real roots as,

x³ - x² - 18x +18 =0

⇒ (x² -18)(x -1) =0

⇒x= 1 , x = 3√2 and x= -3√2

Let us say, a = 1 , b = 3√2 and c = -3√2 are the required real roots.

The square of real roots are as follows,

a² = 1

b² = 18

c² = 18

Thus, the sum of squares of the real roots of p(x) = x³ - x² - 18x +18 is

= a² + b²+ c²

= 1 + 18 + 18

= 37

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There is 95% confident that the true difference in proportions of people who identify as a "dog person" in the city and rural areas is between -0.3008 and -0.0260

To find the 95% confidence interval for the difference in proportions of people who identify as a "dog person" in the city and rural areas, we can use the formula:

[tex](p1 - p2)± \frac{zα}{2} \sqrt{\frac{p1(1-p1)}{n1} } + \frac{p2(1-p2)}{n2}[/tex]

where:

p1 is the proportion of people in the city sample who identify as a "dog person"

p2 is the proportion of people in the rural sample who identify as a "dog person"

n1 is the size of the city sample

n2 is the size of the rural sample

[tex]\frac{za}{2}[/tex] is the critical value from the standard normal distribution for a 95% confidence level (which is 1.96)

Plugging in the values given in the problem, we get:

[tex]p1 = \frac{75}{231} = 0.3247[/tex]

[tex]p2 = \frac{51}{113} = 0.4513[/tex]

n1 = 231

n2 = 113

[tex]\frac{za}{2} = 1.96[/tex]

So the confidence interval for the difference in proportions is:

[tex](0.3247 - 0.4513)±1.96 \sqrt{0.3247(\frac{1-0.3247}{231} )} + (0.4513(\frac{1-0.4513}{113)} )[/tex]

= -0.1634 ± 0.1374

= (-0.3008, -0.0260)

Therefore, we are 95% confident that the true difference in proportions of people who identify as a "dog person" in the city and rural areas is between -0.3008 and -0.0260.

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The radius of the circle with tangent length of 12 cm is equal to √122 cm

The tangent to a circle theorem states that a line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency

The radius of the circle will form a right triangle with the tangent length 12 cm and the length 16 cm, thus the length of the radius can be derived using the Pythagoras rule as follows:

(16 cm)² = (12 cm)² + r² {r = radius}

r = √(16² - 12²) cm

r = √(256 - 144) cm

r = √112 cm.

Therefore, the radius of the circle with tangent length of 12 cm is equal to √122 cm

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The economic order quantity for Zhou Bicycle Company's Airwing bicycle is approximately 68. The answer is (b) 68.

To calculate the economic order quantity (EOQ) for Zhou Bicycle Company's Airwing bicycle, we need to use the following formula:

EOQ = √((2DS)/H)

where:

D = annual demand

S = cost of placing one order

H = holding cost per unit per year

First, we need to calculate the annual demand for Airwing bicycles. The table provided shows the sales data for the past two years:

Year 1: 300 Airwing bicycles sold

Year 2: 350 Airwing bicycles sold

Average annual demand = (300 + 350) / 2 = 325

Next, we need to calculate the cost of placing one order. The question states that each time an order is placed, ZBC incurs a cost of $65. Therefore, S = $65.

Finally, we must compute the annual holding cost per unit. According to the question, ZBC's inventory carrying cost is 1% per month (12% per year) of the purchase price. ZBC paid 60% of the suggested retail price of $170 for the purchase. Therefore, the purchase price paid by ZBC is 0.6 x $170 = $102.

Holding cost per unit per year = 12% x $102 = $12.24

Now we can plug these values into the EOQ formula:

EOQ = √((2 x 325 x $65)/$12.24) ≈ 68

Therefore, the economic order quantity for Zhou Bicycle Company's Airwing bicycle is approximately 68.

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Complete question:

Zhou Bicycle Company​ (ZBC), located in​ Seattle, is a wholesale distributor of bicycles and bicycle parts. Formed in 1981 by University of Washington Professor​ Yong-Pin Zhou, the​ firm’s primary retail outlets are located within a​ 400-mile radius of the distribution center. These retail outlets receive the order from ZBC within 2 days after notifying the distribution​ center, provided that the stock is available.​ However, if an order is not fulfilled by the​ company, no backorder is​ placed; the retailers arrange to get their shipment from other​ distributors, and ZBC loses that amount of business.

The company distributes a wide variety of bicycles. The most popular​ model, and the major source of revenue to the​ company, is the Airwing. ZBC receives all the models from a single manufacturer in​ China, and shipment takes as long as 4 weeks from the time an order is placed. With the cost of​ communication, paperwork, and customs clearance​ included, ZBC estimates that each time an order is​ placed, it incurs a cost of​ $65. The purchase price paid by​ ZBC, per​ bicycle, is roughly​ 60% of the suggested retail price for all the styles​ available, and the inventory carrying cost is​ 1% per month​ (12% per​ year) of the purchase price paid by ZBC. The retail price​ (paid by the​ customers) for the Airwing is​ $170 per bicycle.

ZBC is interested in making an inventory plan for 2019. The firm wants to maintain a​ 95% service level with its customers to minimize the losses on the lost orders. The data collected for the past 2 years are summarized in the following table. A forecast for Airwing model sales in 2019 has been developed and will be used to make an inventory plan for ZBC.

The only equilibrium of the replicator equation is the neutrally stable equilibrium at x1 = x2 = x3 = 1/3.

The replicator equation is a dynamical system that models the evolution of a population of players in a game based on their payoffs. For a two-player game with payoffs given by the matrix R, the replicator equation for the frequency-dependent selection is given by:

dx/dt = x(Rx - x'R x)

where x is a vector of population frequencies, and x'R x is the average payoff of the population.

For the given payoff matrix:

1 1 3

2 1 3

2 -1 0

The replicator equation is given by:

dx1/dt = x1(1x1 + 2x2 + 2x3 - x1 - x2)

dx2/dt = x2(1x1 + 1x2 - 1x3 - x1 - x2)

dx3/dt = x3(3x1 + 3x2 + 0x3 - 2x1 - 3x2)

To find the equilibria of the replicator equation, we need to solve for dx/dt = 0. One possible equilibrium is when all players play the same strategy, i.e., x1 = x2 = x3 = 1/3. To check if this is a stable equilibrium, we need to compute the Jacobian matrix of the replicator equation evaluated at this equilibrium:

J = [2/3 -1/3 -1/3;

-1/3 2/3 -1/3;

1/3 -1/3 0 ]

The eigenvalues of this matrix are λ1 = 1, λ2 = 1/3, and λ3 = -1/3, which means that the equilibrium is neutrally stable (i.e., stable in some directions and unstable in others).

Another possible equilibrium is when x1 = 1 and x2 = x3 = 0 (i.e., all players play the first strategy). To check if this is a stable equilibrium, we need to evaluate the replicator equation at this point:

dx1/dt = 0

dx2/dt = x2(1 - x1 - x2)

dx3/dt = x3(3 - 2x1 - 3x2)

From the second equation, we see that x2 = 0 or x1 + x2 = 1. If x2 = 0, then x1 = 1 and dx3/dt = 3x3 > 0, which means that the equilibrium is unstable. If x1 + x2 = 1, then x1 = 1 - x2 and dx3/dt = 3x3 - 2x1 > 0 for x2 < 3/5, which means that the equilibrium is also unstable in this case.

Therefore, the only equilibrium of the replicator equation is the neutrally stable equilibrium at x1 = x2 = x3 = 1/3. This means that there is no dominant strategy in this game, and the population frequencies of the strategies will oscillate around the equilibrium in the long run.

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(1) The two triangles are similar because they have equal angles.

(2)  Triangle QRS is similar to triangle QLM because they have equal angles.

(3) Both triangles are similar and the value of x is 21.

Two triangles are said to be similar if they have equal sides, equal angles or both.

The missing angles of the triangles for the question is calculated as;

Bigger triangle; missing angle = 180 - (44 + 46) = 90

Smaller triangle; missing angle = 90 - 46 = 44⁰

Both triangles are similar.

For the second question; triangle QRS is similar to triangle QLM  because angle R is equal to angle L, and also they have common angle Q, which implies that angle S must be equal to angle L.

For third question, the triangles are similar because their corresponding angles are equal.

The value of x is calculated as;

48 + 4x + (180 - (56 + 76)) = 180 (sum of angles on a straight line)

48 + 4x + 48 = 180

4x = 84

x = 84/4

x = 21

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The value of x in the parallelogram is 1/3.

PQRT is a parallelogram

PT = 8x, TR = 6y, QT = 2x + 2, TS = 2y

We have to find the value of x

In a parallelogram the opposite sides are equal

8x=2x+2

Subtract 2x from both sides

6x=2

Divide both sides by 6

x=2/6

x=1/3

Hence, the value of x in the parallelogram is 1/3.

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The probability of getting exactly 417 girls in 811 births is approximately 0.0668.

The probability of getting 417 or more girls in 811 births is approximately 0.1349.

a. The probability of getting exactly 417 girls in 811 births is not 1. The correct probability can be calculated using the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

where n is the total number of births, k is the number of girls, p is the probability of a girl birth (assumed to be 0.5 for simplicity), and "n choose k" denotes the binomial coefficient, which is calculated as:

(n choose k) = n! / (k! * (n-k)!)

Plugging in the values, we have:

P(X = 417) = (811 choose 417) * 0.5^417 * 0.5^(811-417)

Using a calculator or software, we can simplify and evaluate this expression to find:

P(X = 417) ≈ 0.0668

So the probability of getting exactly 417 girls in 811 births is approximately 0.0668.

b. To find the probability of getting 417 or more girls in 811 births, we need to calculate the cumulative probability from 417 to 811:

P(X ≥ 417) = P(X = 417) + P(X = 418) + ... + P(X = 811)

We can use software or a calculator to calculate this sum, or we can use the complement rule:

P(X ≥ 417) = 1 - P(X < 417)

To calculate P(X < 417), we can use the cumulative distribution function (CDF) of the binomial distribution:

P(X < 417) = sum(i=0 to 416) (811 choose i) * 0.5^i * 0.5^(811-i)

Again, using software or a calculator, we can evaluate this expression to find:

P(X < 417) ≈ 0.8651

So the probability of getting 417 or more girls in 811 births is:

P(X ≥ 417) = 1 - P(X < 417) ≈ 1 - 0.8651 ≈ 0.1349

Rounding to four decimal places, the probability of getting 417 or more girls in 811 births is approximately 0.1349.

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The value of r² + s² is 17.25 if r and s represents the solution of the given quadratic equation 2x² + 9x + 3 = 0.

To find the sum of squares of the solutions of the given quadratic equation, we can use the formula

r² + s² = (r + s)² - 2rs

where r and s are the roots of the quadratic equation.

In this case, we have the equation

2x² + 9x + 3 = 0

We can use the quadratic formula to find the roots:

x = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = 9, and c = 3.

Plugging in these values, we get

x = (-9 ± √(9² - 4(2)(3))) / 4

Simplifying

x = (-9 ± √69) / 4

So the roots are

r = (-9 + √69) / 4

s = (-9 - √69) / 4

To find r² + s², we need to compute (r + s)² - 2rs

(r + s)² - 2rs = ((-9 + √69)/4 + (-9 - √69)/4)² - 2((-9 + √69)/4)((-9 - √69)/4)

Simplifying

= ((-18)/4)² - 2((-9 + √69)/4)((-9 - √69)/4)

= (9/2)² - 2((81 - 69)/16)

= 81/4 - 3/2

= 69/4

= 17.25

Hence, the sum of squares of the solutions of the given quadratic equation 2x² + 9x + 3 = 0 is 17.25.

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