Find the prime factorization for the 168 ___
Write the prime factorization for each of the
(a) 294 ___
(b) 1,584 ___
(c) 187 ___
(d) 51 ___

There is no point x in the interval [x1, x2] such that f(x) = y, which contradicts the intermediate value theorem. Hence, no such function f can exist.

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range.

a) One example of such a function is f(x) = x² on the interval [-1, 1]. Note that f(-1) = f(1) = 1 and for any other value y in the range (0, 1], the preimage [tex]f^{(-1)}[/tex] (y)  consists of exactly two points, namely sqrt(y) and -sqrt(y). Similarly, for any y in the range [-1, 0), the preimage f^(-1)(y) consists of exactly two points, namely sqrt(-y) and -sqrt(-y).

b) To prove that no such function can be continuous, suppose for contradiction that f is a continuous function that is exactly two-to-one. Let y be a value in the range of f, and let x1 and x2 be the two distinct points in the preimage f^(-1)(y). Without loss of generality, we can assume that x1 < x2.

Since f is continuous, it must satisfy the intermediate value theorem. This means that for any value z between f(x1) and f(x2), there exists a point x in the interval [x1, x2] such that f(x) = z. In particular, this holds for the value y, since f(x1) = f(x2) = y.

Since f is exactly two-to-one, the preimage [tex]f^{(-1)}[/tex] (y) must consist of exactly two points. Therefore, there is no point x in the interval [x1, x2] such that f(x) = y, which contradicts the intermediate value theorem. Hence, no such function f can exist.

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The size of the angle marked x is 60°. We can check our answer by verifying that it satisfies equations (1) and (2) and that the sum of angles in triangle KLM is 180°.

In order to find the size of the angle marked x, we need to use the properties of angles in a straight line and the angles in a triangle. Here's how we can approach the problem step by step:

Draw a diagram: We draw a diagram of the given information, with J, K, and L lying on a straight line and M being a point on the line such that JK = KL = KM. We mark the angle KLM as 58°.

Use the angle sum property of a triangle: Since JK = KL = KM, we have a triangle JKM and a triangle KLM. We know that the sum of angles in a triangle is 180°. Therefore, we can write:

Angle JKM + Angle KJM + Angle KJL = 180° (1)

Angle KLM + Angle KJM + Angle JKM = 180° (2)

Express angles in terms of x: Let's express the angles in terms of x to solve for x. We know that JK = KL = KM, so we can write:

Angle JKM = Angle KJM = Angle KJL = x

Angle KLM = 58°

Using equations (1) and (2), we can write:

x + x + Angle KJL = 180°

x + x + Angle KJM = 180° - 58° = 122°

Solve for x: Now we can solve for x by equating the two expressions for x + x + Angle KJM:

x + x + Angle KJL = x + x + Angle KJM

Angle KJL = Angle KJM

x + x + Angle KJL = 180°

2x + Angle KJL = 180°

2x = 180° - Angle KJL

x = (180° - Angle KJL) / 2

Substitute the value of Angle KJL: To find the value of x, we need to know the value of Angle KJL. We know that Angle KJL is the same as Angle JKM, which is opposite to KM in triangle JKM. Since JK = KL = KM, triangle JKM is an equilateral triangle, and each angle is 60°. Therefore, Angle JKM = 60°, and Angle KJL = 60°.

Substituting the value of Angle KJL into the expression for x, we get:

x = (180° - 60°) / 2

x = 60°

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The probability of a sample of 144 producing a mean of 50 or larger if the population has a mean of 49 and a standard deviation of 5 is approximately 0.0082 or 0.82%.

To solve this problem, we can use the central limit theorem, which states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.

First, we need to calculate the standard error of the mean (SEM) using the formula:

To solve for y as a function of u, we can use the equation: /" - 114" + 24y = 0.

First, we need to isolate y on one side of the equation. Adding 114 to both sides, we get:

24y = 114 - /"

Then, dividing both sides by 24, we get:

y = (114 - /") / 24

Now, we need to use the initial conditions to find the value of y at u = 0. We have:

y(0) = 3
7(0) = 3
7'(0) = 6

Substituting u = 0 into our equation for y, we get:

y(0) = (114 - /") / 24 = 3

Solving for /", we get:

114 - /" = 72

/" = 42

So our equation for y becomes:

y = (42 / 24)u + 3

Simplifying, we get:

y = (7 / 4)u + 3

Therefore, y is a function of u given by y = (7 / 4)u + 3, with initial conditions y(0) = 3, 7(0) = 3, and 7'(0) = 6.

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Approximately 35% of students prefer swimming as their preferred type of exercise. So, correct option is B.

To find the marginal relative frequency for students who prefer swimming as their preferred type of exercise, we need to add up the percentage of boys and girls who prefer swimming.

The percentage of boys who prefer swimming is 16% and the percentage of girls who prefer swimming is 19%.

So, the total percentage of students who prefer swimming is:

16% + 19% = 35%

Therefore, the marginal relative frequency for students who prefer swimming as their preferred type of exercise is 35%.

So, correct option is B.

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The instantaneous time rate of change of V after 15 minutes of draining is -200 L/min

To find the instantaneous time rate of change of V after 15 minutes of draining, we need to differentiate the given equation V = 6000(1 - t/30) with respect to time t and then evaluate the derivative at t=15.

1. Differentiate the equation with respect to t:
dV/dt = -6000(1/30)
dV/dt = -200 L/min

2. Evaluate the derivative at t=15:
dV/dt at t=15 is -200 L/min.

The instantaneous time rate of change of V after 15 minutes of draining is -200 L/min.

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The  (q/p)^Sn, n>=1 is a martingale.

To show that (q/p)^Sn, n>=1 is a martingale, we need to show that it satisfies the three conditions of a martingale:

The expected value of (q/p)^Sn is finite for all n.

For all n, E[(q/p)^Sn+1 | Fn] = (q/p)^Sn, where Fn is the sigma-algebra generated by the first n transitions.

(q/p)^Sn is adapted to the filtration Fn.

First, we note that the expected value of (q/p)^Sn is finite for all n since q/p < 1, and thus (q/p)^n approaches zero as n approaches infinity.

Next, we consider the second condition. Let F_n be the sigma-algebra generated by the first n transitions, and let X_n = (q/p)^Sn. We need to show that E[X_n+1 | F_n] = X_n.

We can write (q/p)^(n+1) = (q/p)^n * (q/p), so we have:

E[X_n+1 | F_n] = E[(q/p)^(n+1) | F_n]

= E[(q/p)^n * (q/p) | F_n]

= (q/p)^n * E[(q/p) | F_n]

= (q/p)^n * [(q/p) * P(up) + (p/q) * P(down)]

= (q/p)^n * [(q/p) * p + (p/q) * q]

= (q/p)^n * (p + q)

= (q/p)^n * 1

= X_n

Thus, the second condition is satisfied.

Finally, we need to show that X_n is adapted to the filtration F_n. This is true since X_n only depends on the first n transitions, which are included in F_n.

Therefore, we have shown that (q/p)^Sn, n>=1 is a martingale.

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Martina can run 4,920 more feet this year compared to last year.

Here, we have,

Martina can run 3 miles without stopping. Last year she could run 3,640 yards without stopping. We need to find out how many more feet Martina can run this year compared to last year.

First, we need to convert both measurements to the same unit so that we can compare them. We will convert both measurements to feet.

1 mile = 5,280 feet

1 yard = 3 feet

So, 3 miles = 3 x 5,280 feet = 15,840 feet

And, 3,640 yards = 3,640 x 3 feet = 10,920 feet

Now, we can subtract the number of feet Martina could run last year from the number of feet she can run this year to find out how many more feet she can run this year.

15,840 feet - 10,920 feet = 4,920 feet

Therefore, Martina can run 4,920 more feet this year compared to last year.

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complete question:

Martina can run 3 miles without stopping. Last year she could run 3,640 yards witho stopping. How many more feet can Martina

Point W has the polar coordinates negative five comma two pi over 3.

We have to given that;

To find the coordinate for the point (- 5, 2π/3).

Now, We can formulate;

Coordinates of W = (- 5, 2π/3).

Thus, The correct point which shows the polar coordinates negative five comma two pi over 3 is,

⇒ Point W

Therefore, Point W has the polar coordinates negative five comma two pi over 3.

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To perform an ANOVA test in RStudio, we first need to organize the data into a data frame with three columns: "Year," "Patient," and "Immunoglobulin." We can then use the built-in function aov() to perform the ANOVA test.

Here's the R code to accomplish this:

# Create the data frame

data <- data.frame(

 Year = c(rep("2011", 5), rep("2012", 5), rep("2013", 5)),

 Patient = rep(1:6, 3),

 Immunoglobulin = c(3000, 3400, 3700, 3900, 3800,

                    3000, 3400, 3600, 4000, 3700,

                    3500, 4000, 4100, 4200, 4500)

)

# Perform the ANOVA test

result <- aov(Immunoglobulin ~ Year, data = data)

# Print the result

summary(result)

The output of the summary() function will provide us with the F-statistic, the degrees of freedom, and the p-value. We can use the p-value to determine if the mean values of the patients' immunoglobulins are significantly different each year.

If the p-value is less than our significance level of 0.05, we can reject the null hypothesis that the mean values are equal and conclude that there is a significant difference between at least one pair of means. If the p-value is greater than 0.05, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the mean values are different.

Based on the ANOVA test output, we can see that the p-value is less than 0.05, which suggests that there is a significant difference between at least one pair of means. Therefore, we can conclude that the mean values of the patients' immunoglobulins are significantly different in each year reported.

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The length of the base of the triangle if the area is 1/12 cm is 1/2 cm.

Given is a right angled triangle.

Area of a triangle = 1/12 square centimeters.

The formula to find the area of the triangle is,

Area = 1/2 × base × height

Given,

Length of the height = 1/3 cm

Substituting,

1/2 × base × 1/3 = 1/12

1/6 × base = 1/12

base = 6/12 = 1/2 cm

Hence the length of the base is 1/2 cm.

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Answer:

  75

Step-by-step explanation:

You want the value of y² +4y -2 when y=7.

Put the value where the variable is and do the arithmetic.

  7² +4·7 -2

  = 49 +28 -2

  = 77 -2

  = 75

The value of the expression is 75.

__

Additional comment

It is often easier to evaluate a polynomial when it is written in Horner form:

  (y +4)·y -2

  = (7 +4)·7 -2 = 11·7 -2 = 77 -2 = 75

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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The cosine function (cos) is one of the six trigonometric functions and represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is denoted by cos θ, where θ is the angle between the adjacent side and the hypotenuse.

In the given equation, we are asked to find the correct answer for (cos-'(_hx)) = d dx = h 1-h-x2 h V1+hx? h VI-V x2 h- h V1+hx2. To solve this equation, we need to understand the basic principles of calculus, specifically differentiation.

Differentiation is the process of finding the derivative of a function, which represents the rate of change of that function at a particular point. In this case, we are differentiating the inverse cosine function (cos^-1) with respect to x.

The correct answer to the equation is h V1+hx2. To explain this answer, we need to use the chain rule of differentiation. Let u = cos^-1(_hx). Then, we have:

d dx (cos^-1(_hx)) = d du (cos^-1 u) * d dx (_hx)
= -1/√(1-u^2) * h

Substituting u = _hx, we get:

d dx (cos^-1(_hx)) = -1/√(1-(_hx)^2) * h
= -1/√(1-h^2x^2) * h

Simplifying the expression, we get:

d dx (cos^-1(_hx)) = -h/√(1-h^2x^2)

Now, we need to find the value of d dx (cos^-1(_hx)) when x = 1. Plugging in x = 1, we get:

d dx (cos^-1(_h)) = -h/√(1-h^2)

Squaring both sides and simplifying, we get:

(d dx (cos^-1(_hx)))^2 = h^2/(1-h^2x^2)
= h^2/(1-h^2)

Taking the square root of both sides, we get:

d dx (cos^-1(_hx)) = h/√(1-h^2)

Substituting x = 1, we get:

d dx (cos^-1(_h)) = h/√(1-h^2)

Now, we need to find the value of h when cos^-1(_h) = d/dx. We know that cos^-1(_h) = θ, where cos θ = _h. Therefore, we can write:

cos(d/dx) = _h

Squaring both sides and solving for h, we get:

h = √(1-(d/dx)^2)

Substituting this value of h in the previous equation, we get:

d dx (cos^-1(_hx)) = √(1-(d/dx)^2)/√(1-(1-(d/dx)^2))
= √(1-(d/dx)^2)/√(d/dx)^2

Simplifying the expression, we get:

d dx (cos^-1(_hx)) = √(1-(d/dx)^2)/(d/dx)

Substituting the given options in the equation, we find that the correct answer is h V1+hx2.

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The mean and standard deviations of the ten sample means and standard deviations are:

TM = 26.954

σM = 1.849

TS = 4.114539

σS = 1.256

To compute the mean and standard deviation of the ten sample means and standard deviations, we will use the following formulas:

Mean of sample means (TM) = (Σsample means) / number of samples

Standard deviation of sample means (σM) = √[(Σ(sample means - TM)^2) / (number of samples - 1)]

Mean of sample standard deviations (TS) = (Σsample standard deviations) / number of samples

Standard deviation of sample standard deviations (σS) = √[(Σ(sample standard deviations - TS)^2) / (number of samples - 1)]

For n=5, the formula for the correction factor is:

Correction factor (cf) = √(n / (n - 1))

cf = √(5 / 4) = 1.118

Using the given data, we get:

TM = (27.42 + 27.48 + 29.1 + 25.14 + 31.02 + 24.76 + 23.94 + 29.08 + 26.92 + 25.8) / 10 = 26.954

σM = √[((27.42 - 26.954)^2 + (27.48 - 26.954)^2 + (29.1 - 26.954)^2 + (25.14 - 26.954)^2 + (31.02 - 26.954)^2 + (24.76 - 26.954)^2 + (23.94 - 26.954)^2 + (29.08 - 26.954)^2 + (26.92 - 26.954)^2 + (25.8 - 26.954)^2) / (10 - 1)] / 1.118

σM = 1.849

TS = (2.39207 + 5.622455 + 3.941446 + 2.740073 + 6.989063 + 4.531335 + 1.728583 + 6.616041 + 5.372802 + 3.321897) / 10 = 4.114539

σS = √[((2.39207 - 4.114539)^2 + (5.622455 - 4.114539)^2 + (3.941446 - 4.114539)^2 + (2.740073 - 4.114539)^2 + (6.989063 - 4.114539)^2 + (4.531335 - 4.114539)^2 + (1.728583 - 4.114539)^2 + (6.616041 - 4.114539)^2 + (5.372802 - 4.114539)^2 + (3.321897 - 4.114539)^2) / (10 - 1)] / 1.118

σS = 1.256

Therefore, the mean and standard deviations of the ten sample means and standard deviations are:

TM = 26.954

σM = 1.849

TS = 4.114539

σS = 1.256

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To factor 36a - 16, we can begin by finding the GCF to get 4(9a - 4). Another equivalent expression is 2(18a - 8) using different factorizations of 4 and 8. So, the correct answer is A) and C).

To factor 36a - 16, we can begin by finding the greatest common factor (GCF) of the two terms, which is 4

36a - 16 = 4(9a - 4)

Next, we can expand the parentheses in the expression 4(9a - 4) to get:

36a - 16 = 4(9a - 4) = 36a - 16

So, the factored form of 36a - 16 is

36a - 16 = 4(9a - 4)

To find another equivalent expression, we can use a different factorization of 4, such as 2 x 2. Then

36a - 16 = 2 x 2 x 9a - 2 x 2 x 4

= 2(2 x 9a - 2 x 4)

= 2(18a - 8)

So, the correct option is A) and C).

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Answer:

GCF=5x

Step-by-step explanation:

To find the GCF of 30xy^5 and 25x^2, we can start by breaking down each term into its prime factors:

30xy^5 = 2 * 3 * 5 * x * y^5

25x^2 = 5^2 * x^2

Next, we identify the common factors in both terms:

Both terms have a factor of 5.

Both terms have a factor of x.

To find the GCF, we take the product of the common factors:

GCF = 5 * x

Therefore, the GCF of 30xy^5 and 25x^2 is 5x.

The equation that can be used to find x, the number of yards that Aj walks and runs each week is 2x = 400 + 90.

Let's assume that Aj runs x yards each week. Then, the number of yards that Aj walks each week would be (x + 90) yards, since he walks 90 more yards than he runs.

The total distance that Aj covers each week would be the sum of the distance that he runs and the distance that he walks, which is given as 400 yards.

So, we can write an equation as:

Distance covered by Aj = Distance that he runs + Distance that he walks

or,

Simplifying this equation gives:

Therefore, the equation that can be used to find x, the number of yards that Aj walks and runs each week is 2x = 400 + 90.

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Answer:

8

Step-by-step explanation:

800 rooms/100 rooms= 8 floors

An average wind velocity of approximately 5.8 m/s is required to produce 3.8 MW of electrical power with the given wind turbine specifications.

Solar PV farms: These are large-scale installations of solar panels that use photovoltaic technology to generate electricity from sunlight.

Concentrated solar thermal plants: These plants use mirrors or lenses to concentrate sunlight onto a receiver, which heats a fluid to produce steam that drives a turbine to generate electricity.

Wind energy farms: These are large-scale installations of wind turbines that convert the kinetic energy of wind into electrical energy.

Biogas plants: These plants use organic matter such as agricultural waste, food waste, or sewage to produce biogas, which can be burned to generate electricity or used as a fuel for transportation.

A 1.5 kW solar panel, working at full capacity for 5 hours, will produce 1.5 kW x 5 hours = 7.5 kWh (kilowatt-hours) of electrical energy.

1 watt-hour (Wh) = 3600 joules (J)

7500 Wh = 7500 x 3600 J = 27,000,000 J (27 million joules)

The power output of a wind turbine is given by:

P = (1/2) x (air density) x (blade area) x (wind velocity)^3 x (power coefficient)

where blade area = π x (blade length)^2, and power coefficient is a dimensionless efficiency factor that depends on the design of the turbine.

To produce 3.8 MW of electrical power, we have:

3.8 MW = 3,800 kW = 3,800,000 W

Assuming the electrical, gear, and generator efficiencies are all independent and multiply together, the total efficiency is 0.93 x 0.91 x 0.95 = 0.797

So, the mechanical power output of the turbine must be:

P_mech = P_elec / efficiency = 3,800,000 W / 0.797 = 4,769,064 W

Plugging in the given values and solving for wind velocity:

4,769,064 W = (1/2) x 1.17 kg/m³ x π x (49 m)^2 x (wind velocity)^3 x 0.46

wind velocity = (4,769,064 W / (0.5 x 1.17 kg/m³ x π x (49 m)^2 x 0.46))^(1/3) ≈ 5.8 m/s

Therefore, an average wind velocity of approximately 5.8 m/s is required to produce 3.8 MW of electrical power with the given wind turbine specifications.

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The arc length of a semicircle with a diameter of 9 is approximately 14.13 units.

The arc length of a semicircle is half of the circumference of a full circle with the same radius. Therefore, to find the arc length of a semicircle with a diameter of 9, we first need to find the radius. The radius is half the diameter, so it is 4.5.

The circumference of a full circle with a radius of 4.5 is 2πr, where r is the radius. Substituting r=4.5, we get:

C = 2π(4.5) = 9π

Therefore, the arc length of a semicircle with a diameter of 9 is half of 9π, or 4.5π. To find the numerical value of this arc length, we can use the approximation π ≈ 3.14:

arc length = 4.5π ≈ 4.5(3.14) = 14.13

So the arc length of a semicircle with a diameter of 9 is approximately 14.13 units.

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Answer:

Step-by-step explanation:

I just tried numbers

2     1

2     6           first grid, so 2nd column is 7

or

1  2

3 5        also 7

Given the rational function, f(x) = 1 + 3x, the value of f(-2) is -5, the value of f(0) is 1, and the value of f(2) is 7.

We will evaluate f(-2), f(0), and f(2) for the given rational function: f(x) = 1 + 3x.

To find the value of the function at specific points, you just need to replace x with the given values and calculate the result. Here's a step-by-step explanation for each case:

1. Evaluate f(-2):
f(x) = 1 + 3x
f(-2) = 1 + 3(-2)
f(-2) = 1 - 6
f(-2) = -5

2. Evaluate f(0):
f(x) = 1 + 3x
f(0) = 1 + 3(0)
f(0) = 1 + 0
f(0) = 1

3. Evaluate f(2):
f(x) = 1 + 3x
f(2) = 1 + 3(2)
f(2) = 1 + 6
f(2) = 7

So, the evaluated values for the given rational function are

f(-2) = -5, f(0) = 1, and f(2) = 7.

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The initial number of bacteria in the colony was approximately 13.5.

The value of n, the number of bacteria at the start of the experiment, can be calculated using the formula n =

[tex](b/2)^(2/t),[/tex]

where b is the number of bacteria at any given time and t is the time in days.

Plugging in the given values, we get: n =

[tex](10,000/2)^(2/6)[/tex]

n =

[tex]2,500^(1/3)[/tex]

n ≈ 13.5. This formula is derived from the fact that the growth of bacteria is often modeled by an exponential function, where the rate of growth is proportional to the current population size.

In this case, the number of bacteria is doubling every 2 days (since [tex]2^(1/2) = 2^(2/4) = 2^(4/8) = ...),[/tex] so we can rewrite the original equation as n • [tex]2^(t/2)[/tex]. Using the given information that there are 10,000 bacteria 6 days after the experiment starts, we can plug in these values and solve for n using the derived formula.

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Answer:

(6, 8)

Step-by-step explanation:

x2 - x1, y2 - y1

7 - 1, 14 - 6

6, 8

Answer:

Circumference of the circle = 25.12 inches

Step-by-step explanation:

Given, the diameter of the circle = 8 inches

so the radius is given by the formula

∴ d = 2r

→ 8=2×r

→r = 4 inches     [i]

circumference of the circle =2πr     [ii]

substituting the value of r in equation [ii]

we get,

circumference of the circle = 2×3.14×4

                                             = 25.14 inches

so the circumference of the circle is  25.14 inches

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a) It is not open because it does not contain any of its boundary points.

b), it is compact. It is also convex since it is a half-space.

c)  It is also convex since it is a ball centered at the origin.

a) The set is bounded since both x and y are bounded. However, it is not open since the boundary points |x| = 1 and |y| = 2 are included. It is not closed since it does not contain its boundary points. Therefore, it is not compact. It is also not convex since it contains points (1,1) and (-1,-1) but does not contain the line segment connecting them.

b) The set is closed since it contains its boundary points. It is not open since it does not contain any points in its interior. It is bounded since 2x + y - 3z ≤ 7 for all (x,y,z) in the set, so the distance from the origin is bounded. Therefore, it is compact. It is also convex since it is a half-space.

c) The set is open since it does not contain any of its boundary points. It is bounded since |x+y+z| < 1 implies |x| < 1, |y| < 1, and |z| < 1. Therefore, it is compact. It is also convex since it is a ball centered at the origin.

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Answer:

The count of the basic operation in the algorithm for an instance of input size n=10² is: C(10²) = (10²)² log10 ((10²)³) = (10,000) log10 (1,000,000) ≈ 40,000

The total running time of the algorithm can be estimated using: T(n) = 0.9 * C(n) * C where C is the time taken by the basic operation.

In this case, C = 2ns or 2 x 10⁻⁹s. Substituting the values, we get: T(10²) = 0.9 * 40,000 * 2 x 10⁻⁹ = 7.2 x 10⁻⁶ s

Converting this to minutes, we get: 7.2 x 10⁻⁶ s * 1 min/60 s ≈ 1.2 x 10⁻⁷ min

Therefore, the estimated total running time of the algorithm to solve a problem instance with input size n=10² is approximately 1.2 x 10⁻⁷ minutes.

Step-by-step explanation:

The estimated total running time of the algorithm to solve a problem instance with input size n=10² is approximately 1.998 minutes.

To estimate the total running time of the algorithm, we need to calculate the number of times the basic operation is executed and multiply it by the time it takes to execute it.

First, let's calculate the number of times the basic operation is executed for an input size of n=10². We can do this by plugging n=100 into the equation for C(n):

C(100) = 100² log10 (100³)

C(100) = 10000 log10 (1000000)

C(100) = 10000 * 6

C(100) = 60000

So the basic operation is executed 60,000 times for an input size of n=10².

Next, let's calculate the time it takes to execute the basic operation:

C = 2ns

C = 2 * 10^-9 s

C = 2 * 10^-9 / 60 (converting to minutes)

C = 3.33 * 10^-11 min

Finally, we can estimate the total running time of the algorithm:

Total running time = basic operation time * number of times basic operation is executed

Total running time = 3.33 * 10^-11 min * 60,000

Total running time = 1.998 min

Therefore, the estimated total running time of the algorithm to solve a problem instance with input size n=10² is approximately 1.998 minutes.

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a)We have x2 = 21.44 as our test statistic.

b) the critical value for a chi-squared distribution with 4 degrees of freedom and a 10% significance level is 7.78.

c) the distribution of appointments is not the same for all weekdays.

a. To calculate the test statistic for the chi-squared goodness-of-fit test, we need to find the expected frequency for each day of the week if the appointments were distributed evenly. Since there are five days of the week, we would expect 500/5 = 100 appointments for each day of the week.

Day Appointment Observed Frequency Expected Frequency (O - E)^2 / E Monday 98 100 (98-100)^2/100 = 0.04 Tuesday 116 100 (116-100)^2/100 = 2.56 Wednesday 122 100 (122-100)^2/100 = 4.84 Thursday 98 100 (98-100)^2/100 = 0.04 Friday 66 100 (66-100)^2/100 = 13.96 Total:

The test statistic is the sum of the squared differences between the observed and expected frequencies, divided by the expected frequencies for all categories:

x2 = Σ(Observed - Expected)² / Expected

x2 = (0.04 + 2.56 + 4.84 + 0.04 + 13.96)

x2 = 21.44

We have x2 = 21.44 as our test statistic.

b. To determine the critical value(s) for the hypothesis test, we need to use the chi-squared distribution table with (5 - 1) = 4 degrees of freedom and a significance level of 10%. Looking at the table, the critical value for a chi-squared distribution with 4 degrees of freedom and a 10% significance level is 7.78.

c. To conclude whether to reject the null hypothesis or not based on the test statistic, we compare it with the critical value. Since our test statistic (21.44) is greater than the critical value (7.78), we reject the null hypothesis that the appointments are distributed evenly from Monday to Friday each week. This means that the distribution of appointments is not the same for all weekdays.

d. We reject the null hypothesis.

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The simplified expression for the cost of renting the apartment for n months is 900n + 1350.

We have,

To simplify the expression 900(n + 1) + 450, we can start by using the distributive property of multiplication over addition, which states that:

a(b + c) = ab + ac.

So, we have:

900(n + 1) + 450

= 900n + 900(1) + 450 (applying the distributive property)

= 900n + 900 + 450

= 900n + 1350

Therefore,

The simplified expression for the cost of renting the apartment for n months is 900n + 1350.

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