Suppose the mean height in inches of all 9th grade students at one high school is estimated. The population standard deviation is 3 inches. The heights of 7 randomly selected students are 60,62,65,72,70,61 and 69.
mean=
margin of error 90% confidence level=
90% confindence interval = [smaller value,larger value]

We know that one complete revolution in degrees is equal to 360 degrees, which is also equal to 2π radians. The measure of an angle in degrees is given and we are required to find its measure in radians.

Therefore, we can use the proportion:

frac{360^{\circ}}{2\pi \text{ radians}}=\frac{18^{\circ}}{x \text{ radians}}

Simplifying the above proportion,

we get:  x = \frac{18}{360} \cdot 2\pi = \frac{1}{20}\cdot \pi

Therefore, 18 degrees is equivalent to π/20 radians.

Thus, the correct option is D.π/20 radians.

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(2x^2 - xy + 1)(x - y) = K. This is the general solution to the given differential equation.

To solve the given differential equation:

y(2x^2 - xy + 1)dx + (x - y)dy = 0

We can start by rearranging the equation:

ydx(2x^2 - xy + 1) + (x - y)dy = 0

Next, we can divide both sides by (2x^2 - xy + 1)(x - y) to separate the variables:

ydx/(2x^2 - xy + 1) + dy/(x - y) = 0

Now, we can integrate both sides of the equation with respect to their respective variables.

∫(ydx/(2x^2 - xy + 1)) + ∫(dy/(x - y)) = 0

To integrate the first term, we can use the substitution u = 2x^2 - xy + 1:

∫(ydx/u) = ∫(dy/(x - y))

Differentiating u with respect to x, we get:

du/dx = 4x - y - xy'

Rearranging, we have:

dy/dx = 4x - xy - du/dx

Substituting this into the second term, we get:

∫(dy/(x - y)) = ∫(du/dx/(4x - xy - du/dx))

Simplifying the integral, we have:

∫(dy/(x - y)) = ∫(du/(4x - y - u))

Now, we can integrate both terms:

∫(ydx/u) + ∫(dy/(x - y)) = 0

ln|u| + ln|x - y| = C

ln|u(x - y)| = C

Taking the exponential of both sides:

u(x - y) = e^C

Since C is a constant, we can write it as e^C = K:

u(x - y) = K

Substituting back the expression for u, we have:

(2x^2 - xy + 1)(x - y) = K

This is the general solution to the given differential equation.

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The hypotenuse if the legs of a right triangle measure 7 cm and 24 cm is 25 cm.

To find the length of the hypotenuse in a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

In this case, the lengths of the legs are given as 7 cm and 24 cm.

Let's use the Pythagorean theorem to find the length of the hypotenuse:

c² = a² + b²

c² = 7² + 24²

c² = 49 + 576

c² = 625

Taking the square root of both sides, we get:

c = √625

c = 25

Therefore, the length of the hypotenuse is 25 cm.

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The value of ‘x’ rounded to the nearest hundredth is 84.97 feet.

Let the height of the totem pole be ‘h’ and the distance between the two people be ‘d’.Given: Height of the totem pole, h = 65 feetDistance between the two people, d = 200 feetAngle of elevation of the top of the totem pole from the person closest to it,

θ = 32°We need to find the value of ‘x’. From the given diagram, we can see that the distance between the person closest to the totem pole and the base of the totem pole can be given by:

Distance = h / tanθ = 65 / tan 32°= 115.03 Feel Now,

we can calculate the distance between the two people by adding this distance to ‘x’.

Therefore, d = 115.03 + x Solving for ‘x’,

we get : x = d - 115.03x = 200 - 115.03x = 84.97 feet (rounded to the nearest hundredth)

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The limit of the function is solved and lim(x → -∞) (x - 9) = -∞

Given data ,

To find the limits in the given problem for the function f(x) = x - 9, we need to evaluate the limits as x approaches certain values.

The properties of limit are the following:

Sum of limits

product rule

Difference rule

Constant multiply rule

Law of constant, etc .

a)

The limit of f(x) as x approaches -∞ (negative infinity) can be evaluated as:

lim(x → -∞) (x - 9)

As x approaches -∞, the value of (x - 9) will also approach -∞. Therefore, the correct choice is:

A. lim f(x) = -∞

Hence , the limit is lim(x → -∞) (x - 9) = -∞

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Statement A is false. Statement B cannot be evaluated as the number of classrooms is not provided. Statement C cannot be confirmed as it does not necessarily imply all students are waiting in the queue. Statement D is false.

Based on the information given, we can evaluate the statements to determine their truth:

A. The arrival rate at the building is 150 students per hour.

To calculate the arrival rate, we divide the total number of students (2000) by the total time the building is open (20 hours): 2000/20 = 100 students per hour. Therefore, statement A is false. The arrival rate is 100 students per hour, not 150.

B. The building must have at least 5 classrooms.

The information provided does not give any indication of the number of classrooms in the building. Therefore, we cannot determine the truth of statement B based on the given information.

C. The number of students waiting in the queue is 150.

Since the average number of students in the building is 150, it does not necessarily mean that all of them are waiting in the queue. Some students may be inside classrooms, while others may be in common areas or moving between rooms. Therefore, statement C cannot be confirmed based on the given information.

D. Students spend an average of 90 minutes in the building.

To calculate the average time spent by each student in the building, we divide the total time the building is open (20 hours) by the total number of students (2000): 20/2000 = 0.01 hours or 0.01 * 60 = 0.6 minutes. Therefore, statement D is false. On average, students spend 0.6 minutes (or 36 seconds) in the building, not 90 minutes.

In summary, based on the given information:

Statement A is false. The arrival rate is 100 students per hour, not 150.

Statement B cannot be evaluated as the number of classrooms is not provided.

Statement C cannot be confirmed as it does not necessarily imply all students are waiting in the queue.

Statement D is false. On average, students spend 0.6 minutes (or 36 seconds) in the building, not 90 minutes.

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Answer:

[tex]\textsf{a)}\quad y=-\dfrac{2}{3}x+4[/tex]

Step-by-step explanation:

A line of best fit is a straight line that represents the general trend in a set of data points. The line is determined by minimizing the overall distance between the line and the data points.

If we add a line of best fit to the given scatter plot (see attachment):

We can use these two points to calculate the slope of the line by substituting them into the slope formula:

[tex]\textsf{Slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{0-4}{6-0}=-\dfrac{4}{6}=-\dfrac{2}{3}[/tex]

The line intercepts the y-axis at y = 4, so the y-intercept is 4.

Substitute the found slope and the y-intercept into the slope-intercept formula to create the equation of the line of best fit:

[tex]\begin{aligned}y&=mx+b\\\implies y&=-\dfrac{2}{3}x+4\end{aligned}[/tex]

Therefore, the equation of the line of best fit is

[tex]\boxed{y=-\dfrac{2}{3}x+4}[/tex]

The unknown angles in triangle ABC are 0°, 37.3°, and 52.7°.

In this triangle, angle A is equal to 37.3°, angle B is equal to 90°, and angle C is equal to 52.7°. To find the missing angles, we must use the Triangle Sum Theorem, which states that the sum of the three angles of a triangle must equal 180°. Therefore, we can calculate the missing angles by subtracting the known angles from 180°.

Angle A = 180° - (37.3° + 90° + 52.7°) = 180° - 180.0° = 0°
Angle B = 180° - (0° + 90° + 52.7°) = 180° - 142.7° = 37.3°
Angle C = 180° - (0° + 90° + 37.3°) = 180° - 127.3° = 52.7°

Therefore, the unknown angles in triangle ABC are 0°, 37.3°, and 52.7°.

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The volume of the cylindrical container in this problem is given as follows:

V = 603.2 cm³.

The volume of a cylinder of radius r and height h is given by the equation presented as follows:

V = πr²h.

The parameters for this problem are given as follows:

Hence the volume of the cylindrical container is given as follows:

V = π x 4² x 12

V = 603.2 cm³.

The problem asks for the volume of the cylinder.

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a. The sampling distribution of p is normal distribution. b. The probability of obtaining x = 790 or more individuals with the characteristic is 0.0.

a. The sampling distribution of p is approximately normal distribution with a mean of p = 0.76 and a standard deviation of sqrt((p(1-p))/n) = sqrt((0.76(1-0.76))/1000) = 0.0184.

b. To find the probability of obtaining x = 790 or more individuals with the characteristic, we need to calculate the z-score and look up the corresponding probability in the standard normal distribution table.

z = (790 - np) / sqrt(np(1-p)) = (790 - 1000(0.76)) / sqrt(1000(0.76)(1-0.76)) = -6.52

Looking up the z-score of -6.52 in the standard normal distribution table, we find that the probability is extremely low, approximately 0.0.

Therefore, the exact probability of obtaining x = 790 or more individuals with the characteristic is essentially 0.0.

Note: The question is incomplete. The complete question probably is: Suppose a simple random sample of size n = 1000 is obtained from a population whose size is N = 1,000,000 and whose population proportion with a specified characteristic is p = 0.76. a. Describe the sampling distribution of p. b. What is the probability of obtaining x = 790 or more individuals with the characteristic?

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The amount of gallons of water the Jug can hold is  8.66 gallons.

The water jug is in the shape of a prism the area of the base is 100 square inches and the height is 20 inches.

Therefore, the number of gallons of water the jug can hold can be calculated as follows:

volume of the prism = Bh

where

Therefore,

volume of the prism = 100 × 20

volume of the prism = 2000 inches³

Therefore,

231 inches³ = 1 gallon

2000 inches³  = ?

cross multiply

amount of water the jug can hold = 2000 / 231

amount of water the jug can hold = 8.66 gallons

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The problem states that 12 less than a variable, represented by 'm,' is no less than 132. The solution to the inequality is that 'm' is greater than or equal to 144.

To translate the given statement into an inequality, we can express "12 less than m" as "m - 12" and "no less than 132" as "≥ 132". Combining these expressions, we have the inequality: m - 12 ≥ 132. To solve for 'm,' we can add 12 to both sides of the inequality: m - 12 + 12 ≥ 132 + 12, which simplifies to m ≥ 144. Thus, the solution to the inequality is that 'm' is greater than or equal to 144. In interval notation, this can be written as [144, +∞), indicating that 'm' lies between 144 (inclusive) and positive infinity.

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Main Answer: The value of f(n) = 16(f(k))^4 when n = 3k.

Supporting Question and Answer:

How can we determine the value of f(n) when n = 3k using the given recurrence relation and initial condition?

By analyzing the given recurrence relation f(n) = 2f(n/3)^4 and the initial condition f(1) = 1, we can recursively calculate the value of f(n) for n = 3k. Using the recurrence relation, we can express f(n) in terms of f(n/3) and apply it iteratively. The value of f(n) when n = 3k is given by f(n) = 16(f(k))^4, where f(1) = 1 is used as the base case.

Body of the Solution:To find the value of f(n) when n = 3k, where f satisfies the recurrence relation f(n) = (2f(n/3))^4 with f(1) = 1, we can use the recurrence relation to recursively calculate the values of f(n).

Given that f(1) = 1, we can calculate the values of f(n) for n = 3, 9, 27, and so on.

f(3) = (2f(3/3))^4

= ((2f(1)))^4

= 2^4(1)^4

= 16

f(9) = (2f(3))^4

= (2(16))^4

= 1048576

f(27) =(2f(9))^4

= (2(1048576))^4

=(2097152)^4

Therefore, f(n) when n = 3k is given by:

f(3K)  =16(f(k))^4

So, f(n) =16(f(k))^4  when n = 3k, where f satisfies the given recurrence relation and f(1) = 1.

Final Answer:Therefore, f(n) =16(f(k))^4 when n = 3k.

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If 0 is an eigenvalue of the matrix of coefficients in a homogeneous system of n linear equations in n unknowns, it indicates that the system has infinitely many solutions.

1. The given statement is always true. When 0 is an eigenvalue of the matrix, it means that the matrix is singular or non-invertible.

2. A singular matrix implies that the system of linear equations has dependent rows or columns, leading to linearly dependent equations.

3. Linearly dependent equations result in an infinite number of solutions because they do not provide enough independent information to uniquely determine the values of the unknowns.

4. Therefore, if the matrix of coefficients has 0 as an eigenvalue, the system of linear equations will have infinitely many solutions.

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A procedure used to compare more than two groups of scores, each of which is from an entirely separate group of people is called an analysis of variance.

The correct option is (A) analysis of variance (ANOVA).

ANOVA is a statistical method used to compare the means of two or more groups. It is a useful technique for analyzing data in experiments where multiple groups are being compared.

The purpose of ANOVA is to determine whether the means of the groups are significantly different from each other or not.

ANOVA works by comparing the variation between groups with the variation within groups. The ratio of these two variations is known as the F-ratio.

If the F-ratio is large enough, then it suggests that the variation between groups is significant and that the means are significantly different from each other.

ANOVA can be used in a wide variety of settings, including in clinical trials, psychology experiments, and business research. It is particularly useful in experimental designs where there are multiple treatment groups, such as in randomized controlled trials.

There are several types of ANOVA, including one-way ANOVA, two-way ANOVA, and repeated measures ANOVA. The choice of which ANOVA to use depends on the specific research question and design.

In conclusion, ANOVA is a powerful statistical method used to compare the means of two or more groups. It is a useful technique for analyzing data in a wide range of fields and can provide valuable insights into the differences between groups.

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The measure of the supplement angle is 54 degrees.

Step-by-step explanation:

Let's denote the measure of the obtuse angle as x.

One-half of the obtuse angle is 9 more than its supplement,

The supplement of an angle is 180 degrees minus the angle itself. Therefore, the supplement of the obtuse angle is 180 - x.

The equation based on the given information:

(1/2)x = (180 - x) + 9

Solve for x:

(1/2)x = 189 - x

Multiply both sides of the equation by 2 to eliminate the fraction:

x = 2(189 - x)

x = 378 - 2x

Add 2x to both sides of the equation:

x+2x = 378 -2x+2x

3x = 378

Divide both sides of the equation by 3:

x = 378 / 3

x = 126

Therefore, the measure of the obtuse angle is 126 degrees.

To find the measure of the supplement, subtract the obtuse angle from 180:

Supplement = 180 - x = 180 - 126 = 54 degrees

Hence, the measure of the supplement is 54 degrees.

The portfolio that cannot lie on the efficient frontier is Portfolio W with an expected return of 1500% and a standard deviation of 36%.

To determine which portfolio cannot lie on the efficient frontier, we need to compare the risk-return characteristics of each portfolio. The efficient frontier represents the set of portfolios that offer the highest expected return for a given level of risk.

Looking at the given portfolios:

Portfolio W has an expected return of 1500% and a standard deviation of 36%. This is an extreme outlier and unlikely to be achievable in a realistic investment scenario. Therefore, portfolio W cannot lie on the efficient frontier.

Portfolios X, Z, and Y have more reasonable risk-return profiles. Portfolio X has a higher expected return compared to portfolios Z and Y, but it also has a higher standard deviation. Portfolios Z and Y have lower expected returns but also lower standard deviations.

Therefore, the portfolio that cannot lie on the efficient frontier is Portfolio W with an expected return of 1500% and a standard deviation of 36%.

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The option that does not represent an e-commerce business model is d. community provider. The remaining options, a. portal, b. hub, and c. market creator, are valid e-commerce business models commonly employed by companies operating in the online marketplace.

The option that is not an e-commerce business model is d. community provider.

E-commerce business models refer to different approaches or strategies that companies adopt to conduct online business and generate revenue. Let's explore each option to identify the one that does not fit the description of an e-commerce business model:

a. Portal: A portal refers to a website or platform that serves as a gateway or entry point to various services, information, or resources. In the context of e-commerce, a portal acts as a central hub that connects users to multiple online stores or services. It typically offers a wide range of products or services from different vendors, allowing users to access various options within a single platform. Examples of e-commerce portals include Amazon and eBay.

b. Hub: A hub, in the e-commerce context, represents a centralized platform or marketplace where multiple sellers or vendors can showcase and sell their products or services. It acts as a hub that brings together buyers and sellers, facilitating transactions and providing a common platform for commerce. Examples of e-commerce hubs include Shopify and Etsy.

c. Market creator: A market creator is an e-commerce business model that involves establishing and creating a new market or category within the industry. This model focuses on introducing innovative products, services, or platforms that disrupt traditional markets or create entirely new markets. Market creators often bring unique value propositions, leveraging technology and innovative approaches to capture market share. Examples of market creators include companies like Uber and Airbnb.

d. Community provider: Unlike the other options, a community provider does not align with a distinct e-commerce business model. While communities and online forums can exist within e-commerce platforms to facilitate user interactions and discussions, "community provider" does not represent a specific e-commerce business model. Instead, it refers to a broader concept of building and nurturing online communities around specific interests, hobbies, or topics.

In summary, the option that does not represent an e-commerce business model is d. community provider. The remaining options, a. portal, b. hub, and c. market creator, are valid e-commerce business models commonly employed by companies operating in the online marketplace.

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(a) The mean of X is 228, and the standard deviation of X is 10.12.

(b) B. For every 400 adults, the mean is the number of them that would be expected to believe that the overall state of moral values is poor.

(c) No.

We have,

(a)

To compute the mean of X, we multiply the total number of adults (400) by the proportion of adults who believe that the overall state of moral values is poor (57%).

The mean of X is therefore 400 x 0.57 = 228.

To compute the standard deviation of X, we use the formula for the standard deviation of a binomial distribution, which is √(np (1 - p)).

Here, n is the sample size (400), p is the proportion of adults who believe the state of moral values is poor (0.57), and (1 - p) is the proportion of adults who do not believe the state of moral values is poor (1 - 0.57 = 0.43). Plugging in these values, we get √(400 x 0.57 x 0.43) = 10.12.

(b)

The mean represents the average number of adults out of the 400 randomly selected who would be expected to believe that the overall state of moral values is poor.

So, for every 400 adults, we can expect around 228 of them to believe that the state of moral values is poor.

(c)

No, it would not be unusual if 215 of the 400 adults surveyed believed that the overall state of moral values is poor.

The probability of a result as extreme or more extreme than this can be calculated using the binomial distribution. If this probability is low (usually below a certain threshold, like 5%), we would consider the result unusual. However, without knowing the exact probability, we cannot determine whether it is unusual or not.

Thus,

(a) The mean of X is 228, and the standard deviation of X is 10.12.

(b) B. For every 400 adults, the mean is the number of them that would be expected to believe that the overall state of moral values is poor.

(c) No.

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A 2x2x3 fully-crossed factorial design means that there are 3 independent variables, each with 2 levels, 2 levels, and 3 levels, respectively. Therefore, it is not true that the experiment has 2 independent variables or 3 dependent variables. It also does not have any extraneous variables because all variables are manipulated and measured.

The experiment has a total of 12 conditions, which is calculated by multiplying the levels of each independent variable together (2x2x3=12). This means that each participant in the experiment will be exposed to all 12 conditions, which can be time-consuming and may require a large sample size.

In summary, the only statement that is definitely true is that one of the variables in the experiment has 2 levels, another has 2 levels, and the third has 3 levels. The experiment has 3 independent variables, 12 conditions, and no extraneous variables.

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The probability of a customer ordering a Soft Drink is 0.80. The likelihood of a customer ordering a Soft Drink is high. The probability of a customer ordering a Daily Special is 0.25. The likelihood of a customer ordering Daily Special is low. The probability of a customer ordering Dessert is 0.48.

The likelihood of a customer ordering Dessert is moderate. The probability of a customer ordering appetizers is 0.06. The likelihood of a customer ordering appetizers is low. The words to describe the likelihood of a customer ordering each item are:

High

Low

Moderate

Therefore, the likelihood that a probability will order Soft Drink is high, the likelihood that a customer will order Daily Special is low, the likelihood that a customer will order a Dessert is moderate, and the likelihood that a customer will order Appetizer is low.

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The  object took 9.97 seconds to fall to the ground.

To determine the time it takes for an object to fall from a certain height, we can use the formula for the time of free fall:

t = √(2h/g)

where t is the time in seconds, h is the height in feet, and g is the acceleration due to gravity, which is 32.2 feet per second squared.

In this case, the height of the building is 1,600 feet and the average speed of the fall is 160 feet per second.

Plugging in these values into the formula, we have:

t = √(2 x 1600 / 32.2)

t = √(3200 / 32.2)

t = √(99.3795)

t ≈ 9.97 seconds

Therefore, the object took 9.97 seconds to fall to the ground.

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The company will be out of debt 3.1 years after opening its business. Option 3.

To determine approximately how many years after opening its business the company will be out of debt, we need to find the value of x when the debt amount represented by the function f(x) equals zero.

The given function is:

f(x) = [tex]-8x^2 + 8x + 50[/tex]

Setting f(x) equal to zero:

[tex]-8x^2 + 8x + 50 = 0[/tex]

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -8, b = 8, and c = 50.

Plugging in the values into the quadratic formula:

x = (-8 ± √(8^2 - 4(-8)(50))) / (2(-8))

x = (-8 ± √(64 + 1600)) / (-16)

x = (-8 ± √1664) / (-16)

x = (-8 ± 40.8) / (-16)

We get two solutions:

x1 = (-8 + 40.8) / (-16) ≈ -2.55

x2 = (-8 - 40.8) / (-16) ≈ 3.05

Since time cannot be negative, we can conclude that the company will be out of debt approximately 3.1 years after opening its business.

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The radii and the centers are explained below.

Given that are equations of circles we need to find the center and the radius of each circle.

1) x² + 6x + y² - 4y = -9

x² + 2·3·x + y² - 2·2·y = -9

Add 13 to both side,

x² + 2·3·x + y² - 2·2·y + 13 = -9 + 13

x² + 2·3·x + y² - 2·2·y + 9 + 4 = 4

(x+3)² + (y-2)² = 4

The center = (-3, 2) and the radius = 2

2) x² + 10x + y² - 8y = -31

x² + 2·5·x + y² - 2·4·y = -31

Add 41 to both sides,

x² + 2·5·x + y² - 2·4·y + 41 = -31 + 41

x² + 2·5·x + y² - 2·4·y + 25 + 16 = 10

(x+5)² + (y-4)² = 10

The center = (-5, 4) and the radius = √10

3) x² - 2x + y² + 4y -11 = 0

x² - 2x + y² + 4y = 11

x² - 2·1·x + y² - 2·2·y = 11

Add 5 to both sides,

x² - 2·1·x + y² - 2·2·y + 5 = 11 + 5

x² - 2·1·x + y² - 2·2·y + 4 + 1 = 16

(x-1)² + (y-2)² = 4²

The center = (1, 2) and the radius = 4

4) x² + 9x + y² = 0

[tex]\left(x-\left(-\frac{9}{2}\right)\right)^2+\left(y-0\right)^2=\left(\frac{9}{2}\right)^2[/tex]

The center = (-9/2, 0) and the radius = 9/2

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The required original number that June was thinking of is 36.

Let's assume the original number June was thinking of is represented by "x". According to the problem, June doubles the original number (2x) and adds 18 to get an answer of 90. We can write this as the equation:

[tex]2x + 18 = 90[/tex]

To find the value of x, we need to isolate it on one side of the equation. Let's subtract 18 from both sides:

[tex]2x = 90 - 18 \\ 2x = 72[/tex]

Now, we divide both sides of the equation by 2 to solve for x:

[tex]x = 72 / 2 \\ x = 36[/tex]

Therefore, the original number that June was thinking of is 36.

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Without the form of the recurrence relation, we cannot determine the power of its closed form from the given terms.  

To determine the power of the closed form of the recurrence relation, we can examine the pattern in the given terms.

Looking at the sequence 5, 2, 17, 30, 245, 590, 1217, we can observe that the terms seem to be increasing at an exponential rate. Taking a closer look, we can see that each term can be obtained by multiplying the previous term by a certain number and then adding another number.

If we calculate the ratios between consecutive terms, we get the following:

2/5 = 0.4

17/2 = 8.5

30/17 = 1.76

245/30 = 8.17

590/245 = 2.41

1217/590 = 2.06

From these ratios, we can see that the terms are not growing at a consistent exponential rate. Therefore, it is unlikely that the recurrence relation has a closed form expression that follows a simple power relationship.

In conclusion, the power of the closed form of the recurrence relation cannot be determined based on the given terms.

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a) For n = 4, P(x>x) = 0.05 holds true for x₀=3.

b) For n=8, P(x?>xê) = 0.025 holds true for x₀=3.

a) Given, P(x>x)=0.05 for n=4

We know that, P(x>x) = 1 - P(x≤x)

Now, P(x≤x) can be calculated by using the following formula:

P(x≤x) = [nCx . pˣ . q⁽ⁿ⁻ˣ⁾ ]

for x=0,1,2,....,n

where, n=4 and

p=q=0.5 for a fair coin

Now, P(x>x)=1-P(x≤x) = 0.05

⇒ P(x≤x) = 1 - 0.05

= 0.95

From binomial distribution table, for n=4

and p=q=0.5

the probability P(x≤x) = 0.6875

for x=0, 1, 2, 3, 4

So, we need to find x such that P(x≤x) = 0.95

⇒ P(x=3)

= 0.6875

(0.5)⁽⁴⁻³⁾] = 0.25

Hence, for n=4, P(x>x) = 0.05 holds true for x₀=3.

x₀=3

b) Given,

P(x?>xê)=0.10

for n=12

Also given, P(x?>xê) = 0.025

for n=8

Now, we know that P(x>xê)= P(x≥xê) =

1- P(xxê) = 0.10

for n=12

So, P(xxê)⇒ P(xxê) = 0.10

Similarly, for n=8 and

p=q=0.5, we get

P(x<4) = [8C1 . (0.5)¹ . (0.5)⁽⁸⁻¹⁾] + [8C2 . (0.5)² . (0.5)⁽⁸⁻²⁾] + [8C3 . (0.5)³ . (0.5)⁽⁸⁻³⁾] + [8C4 . (0.5)⁴ . (0.5)⁽⁸⁻⁴⁾] = 0.6367(approx.)

We can see that for x=3, the probability becomes 0.5439

So, we can take xê=3 as the required value which satisfies

P(x>xê) = 0.025

Hence, for n=12,

P(x?>xê) = 0.10 holds true for

xo=5 and

for n=8,

P(x?>xê) = 0.025 holds true

for x₀=3.

x₀=5 and

x₀=3.

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The number of weeks required to reach the population of the city 3,000,000 people is equal to 80 weeks.

Current population of the city = 2,800,000

Increase in population per week = 2500

The final population of a city = 3000000

To find the number of weeks until the population reaches 3,000,000,

we can set up an equation based on the growth rate of the population.

Let us denote the number of weeks as 'w' and the initial population as 2,800,000.

The growth rate is given as 2,500 people per week.

The equation can be written as,

2,800,000 + 2,500w = 3,000,000

To solve for 'w' rearrange the equation and isolate the variable,

⇒2,500w = 3,000,000 - 2,800,000

⇒ 2,500w = 200,000

Now, divide both sides of the equation by 2,500 to solve for 'w'.

⇒ w = 200,000 / 2,500

⇒ w = 80

Therefore, it will take approximately 80 weeks for the population to reach 3,000,000 people.

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The point of maximum growth for the logistical growth function is (a) (6.8, 15)

From the question, we have the following parameters that can be used in our computation:

[tex]f(x) = \frac{30}{1+2e^{-0.5x}}[/tex]

The above equation is a logistical growth function

Next, we plot the graph of the logistical growth function (see attachment)

From the attached graph, we have the maximum point on the graph to be (6.8, 15)

Hence, the point of maximum growth for the logistical growth function is (a) (6.8, 15)

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The solution to the equation is x = 0 or 3x² - 3 = 0 => x² = 1 => x = ±1 So the zeros of the function are x = 0, 1 and -1.

The method of solving for the zeros of function algebraically is incorrect. Let us see why.

The function f(x) is given as:f(x) = 3x³ – 3xWe factor x out of this equation: f(x) = x (3x² - 3).

This is correct up until here.

After this, the method is wrong.

The given method factors 3 out of (3x² - 3) and leaves it as (x² - 3). Instead of solving the equation directly from here, they add a 0 and set it equal to zero.

This is not necessary. Instead, the equation can be set as: f(x) = x (3x² - 3) = 0

The product is zero when one or both of the factors are zero.

So the solution to the equation is x = 0 or 3x² - 3 = 0 => x² = 1 => x = ±1 So the zeros of the function are x = 0, 1 and -1.

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