A small country consists of five states​ (A, B,​ C, D, and​ E). The total population of the country is 24.4 million. According to the​ country's constitution, the seats in the legislature are apportioned to the states according to their populations. The table below shows each​ state's standard quota. Find the apportionment under​ Hamilton's method.

(X) = (c² - bc + b²)/9d)

a) CDF of X:To calculate the cumulative distribution function of X, we must break the problem into two parts: from 0 to c and from c to b.Case 1: 0≤x≤cThe cumulative distribution function of X from 0 to c is given by:F(x) = ∫f(x)dx = ∫(2x/bc)dx (evaluated from 0 to x)= 2x²/2bcevaluated from 0 to c= c²/bc = c/bCase 2: c≤x≤bThe cumulative distribution function of X from c to b is given by:F(x) = ∫f(x)dx = ∫(2(b-x)/b(b-c))dx (evaluated from c to x)= (2(b-x)(x-c))/(b(b-c))evaluated from c to b= 1- (2(b-c)²)/(2b(b-c))= 1 - (b-c)/b= c/bTherefore, the cumulative distribution function of X is:F(x) = {0, if x < 0;c/b, if 0 ≤ x ≤ c;1 - (b-c)/b, if c ≤ x ≤ b;0, if x > b}b) E[X]:To find the expected value of X, we must find the mean of the distribution. For 0≤x≤c, the expected value isE[X] = ∫x.f(x)dx = ∫x(2x/bc)dx (evaluated from 0 to c)= c²/bFor c≤x≤b,E[X] = ∫x.f(x)dx = ∫x(2(b-x)/b(b-c))dx (evaluated from c to b)= (b+c)/3Therefore, the expected value of X isE[X] = (c²+ b²+ c.b)/(3b) = (b+ c/3)²c) Var(X):To calculate the variance of X, we must first calculate the expected value squared of X. E[X²]For 0≤x≤c,E[X²] = ∫x².f(x)dx = ∫x²(2x/bc)dx (evaluated from 0 to c)= c³/3bFor c≤x≤b,E[X²] = ∫x².f(x)dx = ∫x²(2(b-x)/b(b-c))dx (evaluated from c to b)= (2c² + b²)/3Therefore, the expected value squared of X isE[X²] = (c³/3b) + (2c² + b²)/3Now we can calculate the variance using the formula:Var(X) = E[X²] - [E[X]]²= [(c³/3b) + (2c² + b²)/3] - [(b+ c/3)²]= (c² - bc + b²)/9Therefore, the variance of X isVar(X) = (c² - bc + b²)/9d) Range of X:The range of X is [0,b], as given in the density function of X. The range of a continuous random variable is the set of values that the variable can take on. The range of X is from 0 to b because the function f(x) is only defined on this interval. Therefore, X cannot take on any values outside this range.

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the volume of water in Lap Pool A is 1.33 x 10 to the fifth power gallons, and the volume of water in Lap Pool B is 6.6 x 10 to the fifth power gallons.

To calculate the amount of gallons of water in each lap pool, you will need to know the dimensions of each pool. Lap Pool A has 3 lanes and Lap Pool B has 10 lanes. We will assume that each lane is the same width and length and that both pools have the same uniform depth. Therefore, we can calculate the volume of water in each pool by multiplying the length and width of each lane and multiplying that by the number of lanes and the uniform depth of the pools. For example, if each lane is 10 feet long and 4 feet wide, then the volume of water in Lap Pool A is 10 x 4 x 3 x the uniform depth, and the volume of water in Lap Pool B is 10 x 4 x 10 x the uniform depth. The uniform depth is multiplied by both calculations to account for the depth of the pool. Therefore, the volume of water in Lap Pool A is 1.33 x 10 to the fifth power gallons, and the volume of water in Lap Pool B is 6.6 x 10 to the fifth power gallons.

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The correct statements from the sample are given as follows:

B About 25% of the students at the high school take painting.

D Next year, 7 out of every 60 students at the high school will take music.

E In a group of 120 students from the high school, about 16 of the students likely take electronics.

The proportion relative to an outcome is given by the number of desired outcomes divided by the number of total outcomes.

15 out of 60 students at the school take painting, hence the proportion is given as follows:

p = 15/60 = 0.25 = 25%.

Out of 60 students, 11 take photography, hence out of 30 students, the amount is given as follows:

30/60 x 11 = 5.5.

For electronics, out of 120 students, we have that:

120/60 x 8 = 16.

The diagram is given by the image presented at the end of the answer.

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1. The parametric equations for an object moving along the unit circle are x = cos(t) and y = sin(t), where t is the parameter. This is because a point on the unit circle is defined by the equation x2 + y2 = 1.

2. The following is an example of a table of values for parametric equations x = 2cost and y = 2sint, with the parameter varying over the domain 0 ≤ t ≤ 2π:

t x y

0 2 0

π/2 0 2

π -2 0

3π/2 0 -2

2π 2 0

3. The following is an example of a table of values for parametric equations x = 4cost and y = 2sint, with the parameter varying over the domain 0 ≤ t ≤ 2π:

t x y

0 4 0

π/2 0 2

π -4 0

3π/2 0 -2

2π 4 0

4. From the results of the above two examples, it can be inferred that if x = a cos t and y = b sin t, then when a = b, the graph will trace a circle with a radius of a/2. When a ≠ b, the graph will trace an ellipse with major and minor axis lengths of a and b, respectively.

A parametric equation is a mathematical equation that uses one or more parameters, or variables, to express an equation. It is used to describe a curve or a surface in two or three dimensions. The parameters are usually denoted by variables, such as t for time, x for the position, or y for the height. Parametric equations allow for the modeling of functions that cannot be expressed in one variable and can be used to solve a variety of problems in mathematics and other sciences.

The parametric equations for an object moving along the unit circle are x = cos(t) and y = sin(t), where t is the parameter. This is because a point on the unit circle is defined by the equation x2 + y2 = 1.It can be inferred that if x = a cos t and y = b sin t, then when a = b, the graph will trace a circle with a radius of a/2. When a ≠ b, the graph will trace an ellipse with major and minor axis lengths of a and b, respectively.

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Determine the steady-state current in the RLC circuit to be 0.68 A.

The equation for Lq′′ + Rq′ + 1/C q = E is used to find the steady-state current in each RLC circuit. The circuit parameters are L = 0.1 H, C = 0.01 F, R = 3 Ω, and E(t) = 5cos10t - 5sin10t.Lq′′ + Rq′ + 1/C q = E can be written as Ld²q/dt² + Rdq/dt + 1/C q = E(t)The current flowing through the circuit can be determined using this formula. To calculate the current, we can use E(t) = 5cos10t - 5sin10t.The current can be found using the equation q(t) = Q cos(wt - φ), where Q = I/Z, Z = √(R² + (wL - 1/wC)²), w = 2πf, and φ = tan⁻¹((wL - 1/wC)/R). Z, w, and φ can be calculated from the parameters of the circuit.Using the above formulas, the following is obtained.I = 5/√(3² + (10π x 0.1 - 1/(10π x 0.01))²) = 0.68 ADetermine the steady-state current in the RLC circuit to be 0.68 A.

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Answer:

The highest number of trials would most likely lead to an experimental probability of randomly selecting a sugar cookie that is closest to the theoretical probability of randomly selecting a sugar cookie. Therefore, option C, which suggests conducting one experiment with 500 trials, would be the most appropriate method of testing.

With 500 trials, there would be a large enough sample size to reduce the impact of random variations and increase the accuracy of the results. This would allow us to obtain a more reliable estimate of the experimental probability, which would be closer to the theoretical probability of selecting a sugar cookie.

Options A and D involve a small number of trials, which may not provide sufficient data for accurate estimations. Option B involves a large number of experiments with a small number of trials, which may not capture the overall probability of selecting a sugar cookie across all experiments. Therefore, option C is the best choice for obtaining a reliable estimate of the experimental probability.

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At x = 18, the functions g(x) and h(x) are both continuous, but f(x) is not.

To determine which function is continuous at x = 18, we need to check if the left-hand limit and right-hand limit of the function at x = 18 exist and are equal to the value of the function at x = 18.

Let's consider the following functions:

f(x) = x² - 2x + 9

g(x) = (x - 18)²

h(x) = √(x - 17) + 3

For f(x), the left-hand limit and right-hand limit of the function at x = 18 are:

lim x → 18- f(x) = (18 - 0)² - 2(18) + 9 = 9

lim x → 18+ f(x) = (18)² - 2(18) + 9 = 261

Since the left-hand limit and right-hand limit are not equal, f(x) is not continuous at x = 18.

For g(x), the left-hand limit and right-hand limit of the function at x = 18 are:

lim x → 18- g(x)

= (18 - 18)²

= 0

Since both the left-hand limit and right-hand limit are equal to the value of the function at x = 18, g(x) is continuous at x = 18.

For h(x), the left-hand limit and right-hand limit of the function at x = 18 are:

lim x → 18- h(x)

= √(18 - 17) + 3

= 4

Since both the left-hand limit and right-hand limit are equal to the value of the function at x = 18, h(x) is also continuous at x = 18.

Therefore, the functions g(x) and h(x) are continuous at x = 18, while f(x) is not continuous at x = 18.

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Complete Question

Which function is continuous at x = 18?

a) g(x) = (x - 18)²

b) h(x) = √(x - 17) + 3

c) f(x) = x² - 2x + 9

Answer:

y = -2x² + 7x - 11

Step-by-step explanation:

To find the equation of a parabola that passes through the given points, we need to first determine the form of the equation.

A general form of a quadratic equation is y = ax^2 + bx + c, where a, b, and c are constants. To determine the values of a, b, and c, we substitute the coordinates of each point into the equation and solve the resulting system of equations.

Substituting the first point (-8, -309), we get:

-309 = a(-8)^2 + b(-8) + c

Substituting the second point (-5, -132), we get:

-132 = a(-5)^2 + b(-5) + c

Substituting the third point (1, 6), we get:

6 = a(1)^2 + b(1) + c

Substituting the fourth point (2, 1), we get:

1 = a(2)^2 + b(2) + c

Substituting the fifth point (6, -99), we get:

-99 = a(6)^2 + b(6) + c

Expanding the terms, we get a system of five equations:

64a - 8b + c = -309

25a - 5b + c = -132

a + b + c = 6

4a + 2b + c = 1

36a + 6b + c = -99

Solving the system of equations, we get:

a = -2

b = 7

c = -11

Therefore, the equation of the parabola that passes through the given points is:

y = -2x^2 + 7x - 11

So the answer is y = -2x² + 7x - 11. (^2 is just another version of ²)

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The approximate distance between points T and U is 3.16 units, rounded to the nearest hundredth.(option c)

The problem states that we need to find the approximate distance between points T and U, rounded to the nearest hundredth. To solve this problem, we can use the distance formula, which is derived from the Pythagorean theorem. The formula is given as:

distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Here, (x₁, y₁) and (x₂, y₂) represent the coordinates of the two points, and √ denotes the square root operation.

Now, let's apply this formula to find the distance between points T and U. Suppose that T has the coordinates (-2, 1), and U has the coordinates (-1, 4). Then, we can plug in these values into the formula and simplify:

distance = √[(-1 + 2)² + (4 - 1)²]

= √[(1)² + 3²]

= √(1 + 9)

= √10

≈ 3.16

Therefore, we can see that the correct answer is (c) 3.16 units.

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The OLS regression results for the equation including gift, mails/year, giftlast and propresp are reported as including a sample size and r-squared value. The r-squared value is higher than that from the simple regression that omits giftlast and propresp.

The estimation of the equation gift 5 b0 1 b1mailsyear 1 b2giftlast 1 b3propresp 1 u by ols is done to assess the effect of various factors on the gift amount. The OLS regression results for the equation include the sample size and r-squared value. The sample size indicates the number of observations used in the estimation and the r-squared value indicates the goodness-of-fit of the model. A higher r-squared value indicates a better fit. In this case, the r-squared value of the equation with all four factors is higher than that of the simple regression model that omits giftlast and propresp. This implies that the addition of the two additional factors has improved the prediction accuracy of the model. The results of the regression should be interpreted with caution, as the addition of more variables may lead to overfitting of the data.

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The decimal 0.222... rewritten as a fraction is; 2/9

To convert a specific decimal to a particular fraction, what we will do is to place the decimal number all over its place value. For example, in 0.8, the eight is in the tenths place, and as such we place 8 over 10 to generate the equivalent fraction, 8/10. However, if required, then we can simplify the fraction.

We want to convert the decimal 0.222... to fraction. Thus, we have;

222/1000

Divide both the numerator and the denominator by 2 to get;

111/500

Now, 0.2222… is equal to the fraction with 2 in its numerator (since that’s the single number after the decimal point that’s repeating over and over again) and 9 in its denominator. In other words, 0.2222… = 2/9.

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A bag of peanuts could be divided among 8 children, 9 children, or 10 children with each getting the same number, and with 2 peanuts left over in each case. The smallest number of peanuts that could be in the bag is 26.

This is because 8 children could each get 3 peanuts and there would be 2 peanuts left over, 9 children could each get 2 peanuts and there would be 2 peanuts left over, and 10 children could each get 2 peanuts and there would be 2 peanuts left over.

Small number means a number that is insufficiently large to be statistically significant, as determined by the department. Counting numerals are included in whole numbers as well. Whole numbers begin at zero. As a result, the least whole number is 0. Even though zero has no value, it is employed as a placeholder.

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Answer:

[tex]\large\boxed{\mathtt{7.\bar{4}}}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to identify the rational number out of given options.}[/tex]

[tex]\textsf{First, let's review what a rational number is.}[/tex]

[tex]\textsf{A Rational Number is a number that terminates.}[/tex]

[tex]\textsf{A Rational Number is also any number that can be represented as a fraction.}[/tex]

[tex]\textsf{Let's identify if our given options are Rational Numbers.}[/tex]

[tex]\mathtt{\sqrt{60} = 7.74.... (Irrational)}[/tex]

[tex]\mathtt{7.\bar{4} = 7 \frac{4}{9} \ (Rational) \ \checkmark}[/tex]

[tex]\mathtt{\sqrt{29} = 5.38.... \ (Irrational)}[/tex]

[tex]\mathtt{\pi = 3.14159.... \ (Irrational)}[/tex]

[tex]\sqrt{60} , \sqrt{29} , \textsf{and} \ \pi \textsf{ are irrational due to the decimals never terminating.}[/tex]

[tex]\textsf{Hence, our final answer is}[/tex] [tex]\mathtt{7.\bar{4}.}[/tex]

The rental cοst tο drive 80 km is $35.

We knοw that the rental cοst is directly prοpοrtiοnal tο the number οf kilοmeters driven. Let's call the rental cοst C and the number οf kilοmeters driven D. Then we can write:

C = kD

where k is the cοnstant οf prοpοrtiοnality. We need tο find the value οf k in οrder tο answer the questiοn.

We knοw that the rental cοst tο drive 200 km is $87.50, sο we can plug in these values and sοlve fοr k:

87.50 = k × 200

k = 87.50/200

k = 0.4375

Nοw that we knοw the value οf k, we can use the same equatiοn tο find the rental cοst tο drive 80 km:

C = kD

C = 0.4375 × 80

C = 35

Therefοre, the rental cοst tο drive 80 km is $35.

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The area of the pool is 1250 ft².

What is an Area?

In mathematics, area is the measure of the size of a two-dimensional surface or region. It is usually expressed in square units, such as square meters (m²) or square feet (ft²). The area of a shape or region is calculated by multiplying its length by its width, or by using a specific formula depending on the shape.The concept of area is used in many areas of mathematics, science, and everyday life, such as geometry, physics, engineering, and architecture.

Given : length of pool = 25 ft

           Breadth of pool = 50 ft

We know that area of a rectangle  = length × Breadth

So, Area of Rectangular pool  

    = length × Breadth

    = 25 × 50

    = 1250 ft²

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Answer:

true

Step-by-step explanation:

Answer: True

Step-by-step explanation:

There is a 9.1% chance that a chicken of this particular breed will lay 248 eggs in a year.

A normal distribution is a continuous probability distribution described by its mean (μ) and standard deviation (σ). The formula for a normal distribution is:

[tex]P(x) = (1/σ√2π) e^(-(x-μ)^2/2σ^2)[/tex]

In this case, the mean (μ) is 225 and the standard deviation (σ) is 10.

To calculate the probability of a specific number of eggs being laid, we plug the mean and standard deviation into the normal distribution formula. For example, to calculate the probability of a chicken laying 248 eggs:

P(248) = 0.091

This means that there is a 9.1% chance that a chicken of this breed will lay 248 eggs in a year.

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Bartlett Company is in a healthy financial position, with strong liquidity, consistent profitability, and diverse revenue streams.

The Strengths are defined as,

The company has a healthy current ratio, which indicates that it can meet its short-term obligations efficiently.

The company has been consistently generating profits over the years, and its net income has been increasing.

Bartlett Company generates revenue from multiple sources, which reduces the risk of dependence on a single source.

The Weaknesses are defined as

An increasing debt level can put pressure on the company's financial position in the long run.

Bartlett Company's return on equity (ROE) is relatively lower compared to the industry average, indicating that the company is not generating as much profit from its equity as its peers.

The company has a low dividend payout ratio, which means that it is not distributing much of its profits to shareholders in the form of dividends.

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Answer: El mínimo común múltiplo de 18, 30 y 42 es 630.

Step-by-step explanation:

j

Answer:

630. yes yes yyesss

Charles will need to buy 10 bags of fertilizer to cover the entire garden.

What is Algebraic expression ?

In mathematics, an algebraic expression is a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division, that represents a quantity or a relationship between quantities.

The area of the rectangular garden is:

A = length x width

A = 5 x 12

A = 60

To cover the entire garden, Charles needs:

Number of bags = (Area of garden) ÷ (Area covered by one bag)

Number of bags = 60  ÷ 6

Number of bags = 10 bags

Therefore, Charles will need to buy 10 bags of fertilizer to cover the entire garden.

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[tex] \sf43\% \implies \: answer[/tex]

The variance of the discrete random variable X is equal to 70t.

The variance of discrete random variable X is given by Var(X) = E[X2] – (E[X])2. In this case, the probability function is given as P(X = k) = t (6 – k) for k = 1, 2, 3, 4, 5. Therefore, the expected value of X is the sum of the products of each possible value of X and its corresponding probability. This can be calculated as:

E[X] = 1 x P(X = 1) + 2 x P(X = 2) + 3 x P(X = 3) + 4 x P(X = 4) + 5 x P(X = 5)

E[X] = t (6 - 1) + 2t (6 - 2) + 3t (6 - 3) + 4t (6 - 4) + 5t (6 - 5)

E[X] = t (5) + 2t (4) + 3t (3) + 4t (2) + 5t (1)

E[X] = 5t + 8t + 9t + 8t + 5t

E[X] = 35t

The expected value of X2 is the sum of the products of each possible value of X and its corresponding probability, multiplied by X2. This can be calculated as:

E[X2] = 1 x P(X = 1) x (1)2 + 2 x P(X = 2) x (2)2 + 3 x P(X = 3) x (3)2 + 4 x P(X = 4) x (4)2 + 5 x P(X = 5) x (5)2

E[X2] = t (6 - 1) x (1)2 + 2t (6 - 2) x (2)2 + 3t (6 - 3) x (3)2 + 4t (6 - 4) x (4)2 + 5t (6 - 5) x (5)2

E[X2] = t (5) x (1)2 + 2t (4) x (2)2 + 3t (3) x (3)2 + 4t (2) x (4)2 + 5t (1) x (5)2

E[X2] = 5t + 16t + 27t + 32t + 25t

E[X2] = 105t

Therefore, the variance of X is given by Var(X) = E[X2] – (E[X])2 = 105t – (35t)2 = 70t.

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The required probability of the temperature of a person which has normal distribution is given by 0.9500.

Temperature of a person follows a normal distribution.

Use the Empirical Rule to estimate the probability.

Temperature of a randomly selected person will be within 2.42 standard deviations of its mean.

Empirical Rule states that for a normal distribution is,

About 68% of the data falls within one standard deviation of the mean.

About 95% of the data falls within two standard deviations of the mean.

About 99.7% of the data falls within three standard deviations of the mean.

Probability of temperature of a randomly selected person within 2.42 standard deviations of its mean = 95%

95%

= 95/100

= 0.95

=0.9500 ( round this to four decimal places )

Therefore, the probability that the temperature of a randomly selected person is 0.9500.

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The given points of the graph represent a nonlinear relationship.

A linear graph is a type of graph where the points plotted create a straight line when connected. The slope of a linear graph indicates how steep the line is, while the y-intercept indicates the point where the line crosses the y-axis.

If the given points (1,1), (2,2), (3,6), (4,24) form a linear or nonlinear relationship with and without plotting the graph, we can calculate the slope between each pair of points.

The slope of a line between two points [tex](x_{1} , y_{1})[/tex] and [tex](x_{2} , y_{2})[/tex] is given by the formula:

slope = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1}}[/tex]

Let's calculate the slope between the first two points (1,1) and (2,2):

slope = (2 - 1) / (2 - 1) = 1/1 = 1

Now, let's calculate the slope between the second and third points (2,2) and (3,6):

slope = (6 - 2) / (3 - 2) = 4/1 = 4

Next, let's calculate the slope between the third and fourth points (3,6) and (4,24):

slope = (24 - 6) / (4 - 3) = 18/1 = 18

As we can see, the slopes between the pairs of points are not equal. In fact, they are increasing at a faster and faster rate. Therefore, the relationship between the x and y values is nonlinear.

To determine the specific type of nonlinear relationship, we can try to fit a curve to the data points. In this case, we can see that the points follow an exponential pattern, where the y-values increase rapidly as x increases. The equation that fits this pattern is y = [tex]2^x[/tex].

Therefore, the given points (1,1), (2,2), (3,6), (4,24) represent a nonlinear relationship that follows the exponential function y = [tex]2^x[/tex].

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The probability of a bone density score greater than 0.18 is given as follows:

0.4286 = 42.86%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for this problem are given as follows:

[tex]\mu = 0, \sigma = 1[/tex]

The probability of a score greater than 0.18 is one subtracted by the p-value of Z when X = 0.18, hence:

Z = (0.18 - 0)/1

Z = 0.18

Z = 0.18 has a p-value of 0.5714.

Hence:

1 - 0.5714 = 0.4286 = 42.86%.

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The volume of the given cube is 8,000 cubic units if the surface area is 2,400.

Let's first find the length of one side of the cube using the given surface area.

The surface area of a cube is given by the formula 6s², where s is the length of one side of the cube.

We are given that the surface area of the cube is 2,400. Therefore, we can set up the following equation:

6s² = 2,400

Dividing both sides by 6, we get:

s² = 400

let's see the square root of both sides, we get:

s = 20

So, the length of one side of the cube is 20.

Now, we can find the volume of the cube using the formula V = s³:

V = 20³ = 8,000

Then, the volume of the cube is 8,000 cubic units.

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Answer:

Answer D

Step-by-step explanation:

Solve for b:

[tex]\frac{1}{2}b \geq 4[/tex]

Divide both sides by a half

[tex]b\geq 8[/tex]

We see that its a greater than or equal too sign, which means the number line has to be facing to the right with a closed circle. Therefore, D is our answer.

The distance apart of the two ships after two hours with their bearings is equal to 56.7 miles using cosine rules.

Bearing is usually measured in degrees, with 0° indicating the reference direction (usually North), and increasing clockwise to 360°. It refers to the direction or angle between a reference direction and a point or object.

The bearing of the two ships will give an angle 14° + 90° + 15° = 119° between line distance of 20 miles and 22 miles after 2 hours.

Let ∆ABC be the triangle formed with bearings so that:

angle A = 119°

b = 20

c = 22

a = distance of the ships apart

a² = b² + c² - 2(b)(c)cosA {cosine rule}

a² = 20² + 22² - 2(20)(22)cos119°

a² = 400 + 484 - 4800cos119°

a² = 884 - 4800(-0.4848)

a² = 3211.0862

a = √3211.0862 {take square root of both sides}

a = 56.6664.

Therefore, the distance apart of the two ships after two hours with their bearings is equal to 56.7 miles using cosine rules.

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The length of PQ is 16 units. ST is the midsegment of PQR, UV is the midsegment of STR, and WX is the midsegment of UVR. It is reasonable to assume that the three midsegments have equal lengths since they are part of the same triangle.

In a triangle, the midsegment is a line segment that connects the midpoints of two sides of the triangle and is parallel to the third side. In the given triangle PQR, the length of PQ is 16 units. ST is the midsegment of PQR, UV is the midsegment of STR, and WX is the midsegment of UVR. Since all the midsegments are part of the same triangle, it is reasonable to assume that they have equal lengths. The triangle midsegment theorem, which asserts that the length of a triangle's midsegment is equal to half of the length of the third side of the triangle, may be used to demonstrate this.. In this case, the length of ST, UV, and WX would all be equal to 8 units, since the third side of each triangle is 16 units. This indicates that it is reasonable to assume that the midsegments of the triangle have equal lengths.

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The probability that at least 50 visitors make a donation is 0.9995 and the probability that between 55 and 65 visitors make a donation is about 0.306.

This problem can be solved using the binomial distribution, where the probability of success is 0.37 (the proportion of visitors who make a donation) and the number of trials is 175 (the total number of visitors).

To find the probability that at least 50 visitors make a donation, we need to calculate the cumulative probability of 50 or more successes:

P(X >= 50) = 1 - P(X < 50)

Using a binomial probability table or a calculator, we can find that P(X < 50) is approximately 0.0005. Therefore,

P(X >= 50) = 1 - 0.0005 = 0.9995

So the probability that at least 50 visitors make a donation is very high (close to 1).

To find the probability that between 55 and 65 visitors make a donation, we need to calculate the probability of 55, 56, ..., 64 successes, and then add them up:

P(55 <= X <= 65) = P(X = 55) + P(X = 56) + ... + P(X = 64)

Using a binomial probability table or a calculator, we can find each of these probabilities and add them up. Alternatively, we can use a normal approximation to the binomial distribution, since n * p = 64.75 > 10 and n * (1 - p) = 110.25 > 10. Using the normal approximation, we can calculate the mean and standard deviation of the distribution:

mean = n * p = 64.75

standard deviation = sqrt(n * p * (1 - p)) = 6.19

Then, we can standardize the range 55 to 65 using the formula z = (x - mean) / standard deviation, and use a normal probability table or a calculator to find the area under the standard normal curve between the two z-scores.

P(55 <= X <= 65) ≈ P((55 - 64.75) / 6.19 <= z <= (65 - 64.75) / 6.19)

P(55 <= X <= 65) ≈ P(-1.44 <= z <= 0.06)

Using a standard normal probability table or a calculator, we can find that this probability is approximately 0.306.

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