Please Help!!

Kim's flower was 2 inches high when she got it. It grew 0.75 inches per month until it was 12 months old. She keeps track of her flower's growth on a coordinate grid by graphing its height every two months and connecting the points to show the growth between months.


Which statements are true? (Choose 3)


Responses


The function is increasing over time.



The function is discrete.



The functions are continuous.



The function decreases over time.



The function is Linear.



The function is Nonlinear.

The volume of the composite solid, which consists of a cylinder with a height of 4 feet and a cone with a height of 6 feet, both having a diameter of 16 feet, is 384π cubic feet.

The volume of a cylinder is given by the formula V_cylinder = πr²h, where r is the radius of the cylinder's base and h is the height of the cylinder.

Given that the diameter of the cylinder is 16 feet, we can find the radius by dividing the diameter by 2:

r = 16 ft / 2 = 8 ft

Substituting the values into the formula, we get:

V_cylinder = π(8 ft)²(4 ft)

V_cylinder = π(64 ft²)(4 ft)

V_cylinder = 256π ft³

The volume of a cone is given by the formula V_cone = (1/3)πr²h, where r is the radius of the cone's base and h is the height of the cone.

Since the cone has the same diameter as the cylinder, the radius of the cone is also 8 feet. Using the height of the cone, we have:

V_cone = (1/3)π(8 ft)²(6 ft)

V_cone = (1/3)π(64 ft²)(6 ft)

V_cone = 128π ft³

To find the total volume of the composite solid, we add the volumes of the cylinder and the cone together:

V_total = V_cylinder + V_cone

V_total = 256π ft³ + 128π ft³

V_total = 384π ft³

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Complete Question:

Find the volume of the composite solid. Round your answer to the nearest tenth

The Polynomial of least degree with roots 7 and -9 is x^2 + 2x - 63.

To polynomial with roots at 7 and -9, we can use the fact that if a number r is a root of a polynomial, then the corresponding factor is (x - r).

Let's begin by setting up the factors for the given roots:

Factor 1: (x - 7)

Factor 2: (x - (-9)) = (x + 9)

To find the polynomial of least degree, we multiply these factors together:

Polynomial = (x - 7)(x + 9)

To simplify further, we can use the distributive property:

Polynomial = x(x + 9) - 7(x + 9)

Expanding the terms:

Polynomial = x^2 + 9x - 7x - 63

Combining like terms:

Polynomial = x^2 + 2x - 63

Therefore, the polynomial of least degree with roots 7 and -9 is x^2 + 2x - 63. This polynomial is in standard form with a leading coefficient of 1, which is the desired format.

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To sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions, we first need to understand the given conditions.

The polar coordinate system consists of two variables: r, which represents the distance from the origin, and θ, which represents the angle formed between the positive x-axis and a line connecting the point to the origin.

In this case, the conditions state that the distance from the origin (r) must be between 2 and 3, and the angle (θ) must be between 7/4 and 9/4.

To visualize this region, we can start by drawing a circle centered at the origin with a radius 2 and another circle centered at the origin with a radius 3. Then, we can shade the region between these two circles.

Next, we need to consider the angle conditions. To do this, we can draw two lines radiating from the origin at angles 7/4 and 9/4. Then, we can shade the region between these two lines within the shaded region between the circles.

Overall, the region in the plane consisting of points whose polar coordinates satisfy the given conditions is the shaded region between the circles with radii 2 and 3, and between the lines radiating from the origin at angles 7/4 and 9/4.

In summary, the region in the plane with the given conditions is a shaded region between two circles and two lines radiating from the origin at certain angles.

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To determine whether the series is absolutely convergent, conditionally convergent, or divergent, we can use the Ratio Test. Answer : the series is divergent.

Let's analyze the given series:

4, 7, 4 · 10, 7 · 9, 4 · 10 · 16, 7 · 9 · 11, 4 · 10 · 16 · 22, 7 · 9 · 11 · 13, ...

We will calculate the ratio of consecutive terms:

(7/4), (40/7), (63/40), (352/63), (1386/352), (7722/1386), ...

Now, we will calculate the limit of the absolute value of the ratios:

lim(n->∞) |a(n+1)/a(n)| = lim(n->∞) |(7722/1386) / (1386/352)| = lim(n->∞) |(7722/1386) * (352/1386)| = lim(n->∞) |7722/1386 * 352/1386| = |2039328/1933156| = 1.055...

The limit of the absolute value of the ratios is greater than 1. According to the Ratio Test, if the limit is greater than 1, the series diverges. Therefore, the given series is divergent.

In conclusion, the series is divergent.

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The expression 6⁻¹⁰.41⁻⁴.11⁻¹³ in positive exponent is 1/(6¹⁰.41⁴.11¹³)

The expression  (-2)⁷.19⁻³/31⁻¹ in positive exponent is (-2)⁷.31¹/19³

The expression 15⁰.8⁻⁶.23⁵ in positive exponent is  15⁰.23⁵/8⁶

The expression  3²⁵.16⁰/5⁻⁹.52⁻³in positive exponent is 3²⁵.16⁰.5⁹.52³

The given expression is 6⁻¹⁰.41⁻⁴.11⁻¹³

We have to rewrite this expression using only positive exponents

6⁻¹⁰.41⁻⁴.11⁻¹³

1/(6¹⁰.41⁴.11¹³)

Now (-2)⁷.19⁻³/31⁻¹

Rewrite this expression using only positive exponents

(-2)⁷.31¹/19³

Now 15⁰.8⁻⁶.23⁵

15⁰.23⁵/8⁶

and 3²⁵.16⁰/5⁻⁹.52⁻³

3²⁵.16⁰.5⁹.52³

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To find the number of all subsets of the set A ∪ B with 4 elements, where A = {1, 2, 3, 6, 7} and B = {3, 6, 7, 8, 9}, we need to consider all possible combinations of elements from the union of A and B that have a cardinality of 4.

The cardinality of the union A ∪ B is 9, as it contains all distinct elements from both sets. We need to choose 4 elements from this union, which can be done in C(9, 4) ways, where C(n, r) denotes the combination of selecting r elements from a set of n elements.

Using the formula for combinations, C(n, r) = n! / (r! * (n - r)!), we can calculate the number of subsets.

C(9, 4) = 9! / (4! * (9 - 4)!) = 9! / (4! * 5!) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 126.

Therefore, there are 126 subsets of the set A ∪ B with 4 elements.

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The flux of the vector field f(x, y) = ⟨x - y, y - x⟩ along the square bounded by x = 0, x = 1, y = 0, and y = 1 is given by the double integral ∫[0,1]∫[0,1] (x - y) dx dy. Evaluating this integral will provide the final answer for the flux.

To calculate the flux, we need to evaluate the surface integral of the dot product between the vector field f(x, y) and the outward-pointing unit normal vector on the surface. In this case, the surface is the square bounded by x = 0, x = 1, y = 0, and y = 1.

We can parameterize the surface as r(x, y) = ⟨x, y⟩, where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. The outward-pointing unit normal vector is given by n = ⟨0, 0, 1⟩.

The dot product between f(x, y) and n is (x - y) × 0 + (y - x) × 0 + (x - y) × 1 = x - y.

Next, we compute the surface integral over the square by integrating x - y with respect to x and y. The limits of integration are 0 to 1 for both x and y.

∫∫(x - y) dA = ∫[0,1]∫[0,1] (x - y) dx dy.

Evaluating this double integral will give us the flux of the vector field along the square bounded by x = 0, x = 1, y = 0, and y = 1.

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Using trigonometric identity, cos(A-B) is:

[tex]cos (A-B) = \frac{2\sqrt{8}\ + \sqrt{5}}{9}[/tex]

Trigonometry deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

If sin A = 1/3 and A terminates in Quadrant 1. All trigonometric functions in Quadrant 1  are positive

sin A = 1/3 (sine = opposite/hypotenuse)

adjacent = √(3² - 1²)

               = √8 units

cosine = adjacent/hypotenuse. Thus,

[tex]cos A = \frac{\sqrt{8} }{3}[/tex]

If cos B = 2/3 and B terminates in Quadrant 4.

opposite = √(3² - 2²)

                = √5

In  Quadrant 4, sine is negative. Thus:

[tex]sin B = \frac{\sqrt{5} }{3}[/tex]

We have:

cos(A-B) = cosA CosB + sinA sinB

[tex]cos (A-B) = \frac{\sqrt{8} }{3} * \frac{2}{3} + \left \frac{1}{3} * \frac{\sqrt{5} }{3}[/tex]

[tex]cos (A-B) = \frac{2\sqrt{8} }{9} + \left\frac{\sqrt{5} }{9}[/tex]

[tex]cos (A-B) = \frac{2\sqrt{8}\ + \sqrt{5}}{9}[/tex]

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The first four terms of the Maclaurin series of f(x) can be determined using the provided values. The Maclaurin series is an expansion of a function around x = 0. In this case, the series can be expressed as f(x) = -10 + 4x - (1/2)x^2 + (11/6)x^3 + ...

To find the coefficients of the series, we can use the formula for the Maclaurin series coefficients. The coefficient of x^n is given by f^(n)(0) / n!, where f^(n)(0) represents the nth derivative of f(x) evaluated at x = 0.

Using the provided values, we have f(0) = -10, f'(0) = 4, f"(0) = -2, and f"'(0) = 11. Plugging these values into the formula, we can find the coefficients for each term in the series.

For the first four terms, the coefficients are as follows:

The coefficient of x^0 is f(0) = -10.

The coefficient of x^1 is f'(0) = 4.

The coefficient of x^2 is f"(0) / 2! = -2 / 2 = -1.

The coefficient of x^3 is f"'(0) / 3! = 11 / 6.

Therefore, the first four terms of the Maclaurin series for f(x) are -10 + 4x - (1/2)x^2 + (11/6)x^3.

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The equation for the exponential function in the table of ordered pairs (x, y) is; y = 3·6ˣ

An exponential function or equation can be presented as follows;

y = a·bˣ, where; x is the input  variable and y is the value of the function.

The values in the table of the ordered pair indicates;

(-1, 1/2), (0, 3), (1, 18)

1/2 = a·b^(-1)

3 = a·b^(0) = a

a = 3

1/2 = 3·b^(-1) = 3/b

b = 3/(1/2) = 6

The possible exponential function is; y = 3·6ˣ

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Answer:

(2,-1)

Step-by-step explanation:

It is indeed possible to find non-empty families f and g such that the intersection of f and g, denoted as (f ∩ g), is not equal to the intersection of f and the intersection of g, denoted as (f) ∩ (g).

Let's consider the following example to prove this statement. Assume we have two families of sets: f = {{1, 2, 3}, {2, 3, 4}} and g = {{3, 4, 5}, {4, 5, 6}}. In this case, the intersection of f and g is f ∩ g = {{3}}.

Now, let's find the intersection of f and the intersection of g. The intersection of g, denoted as (g), is {3, 4, 5, 6}. Therefore, (f) ∩ (g) = {{1, 2, 3}, {2, 3, 4}} ∩ {3, 4, 5, 6} = {}.

As we can see, f ∩ g = {{3}} is not equal to (f) ∩ (g) = {}, which confirms that there exist non-empty families f and g for which the intersection of f and g is not equal to the intersection of f and the intersection of g.

This example illustrates that the intersections of families of sets do not necessarily distribute over each other, leading to distinct results in different orderings of intersections.

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The number of weeks in which Sarah and Asher have saved the same amount of money is 8 weeks.

The number of weeks in which Sarah and Asher have saved the same amount of money is calculated by setting the two equations equal to each other as follows;

12x + 6 = 9x + 30

Simplify the equation by collecting similar terms as follows;

12x - 9x  = 30 - 6

3x = 24

x = 24/3

x = 8

Thus, the number of weeks in which Sarah and Asher have saved the same amount of money is 8 weeks.

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The best way to describe the center of the data represented in this line plot is; mean = 1.8 inches and median = 1.5 inches

Line plots, also known as dot plots, are a type of graphical representation used to display data. They are particularly useful for showing the distribution and frequency of values in a dataset.

Line plots consist of a number line where each data point is represented by a dot or symbol placed above the corresponding value on the line.

Considering the given line plot:

Mean = (0 * 3 + 1 * 3 + 2 * 1 + 3 * 1 + 4 * 1 + 6 * 1)/10

Mean = 1.8 inches

Median = (1 + 2)/2

Median = 1.5 inches

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If the probability of obtaining a score less than x = 70 is p = 0.0013, the score that corresponds to a probability of 0.0013 is x = 38.2.

We are referring to a normal distribution with a mean (µ) of 100 and a standard deviation (σ) of 20. You want to find the probability of obtaining a score less than x = 70, and you provided that the probability (p) is 0.0013. In a normal distribution with µ = 100 and σ = 20, the probability of obtaining a score less than x = 70 is p = 0.0013. Based on the information given, we know that the probability of obtaining a score less than x = 70 is p = 0.0013. This means that the z-score for x = 70 is -3.09 (found using a standard normal distribution table or calculator).

To find the z-score, we use the formula:

z = (x - µ) / σ

Plugging in the values we know:

-3.09 = (70 - 100) / 20

Solving for x:

-3.09 = (x - 100) / 20

-3.09 * 20 = x - 100

-61.8 + 100 = x

x = 38.2

Therefore, the score that corresponds to a probability of 0.0013 is x = 38.2.

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To determine the optimal solution to the given linear programming problem, we need to solve the problem and find the values of X1 and X2 that maximize the objective function while satisfying the constraints.

However, the problem formulation provided is incomplete and contains some errors. The objective function and constraints are not properly defined. It seems there are missing symbols and equations.

Without the correct formulation of the objective function and constraints, we cannot determine the optimal solution. Therefore, none of the options (a, b, c, d, e) can represent the optimal solution to the problem as presented.

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To find the vector u3 using the Gram-Schmidt process, we start with the given vectors u1 = (1, 2, 1) and u2 = (2, 1, -1). The Gram-Schmidt process involves orthogonalizing each vector with respect to the previous vectors in the set.

Step 1: Normalize u1 to obtain the first orthonormal vector v1.

v1 = u1 / ||u1|| = (1, 2, 1) / √(1^2 + 2^2 + 1^2) = (1/√6, 2/√6, 1/√6)

Step 2: Find the projection of u2 onto v1 and subtract it from u2 to obtain a new vector u2' that is orthogonal to v1.

projv1(u2) = (u2 · v1) * v1 = (2/√6, 4/√6, 2/√6)

u2' = u2 - projv1(u2) = (2, 1, -1) - (2/√6, 4/√6, 2/√6) = (2 - 2/√6, 1 - 4/√6, -1 - 2/√6)

Step 3: Normalize u2' to obtain the second orthonormal vector v2.

v2 = u2' / ||u2'|| = ((2 - 2/√6)/√(1 + (2 - 2/√6)^2 + (1 - 4/√6)^2 + (-1 - 2/√6)^2), (1 - 4/√6)/√(1 + (2 - 2/√6)^2 + (1 - 4/√6)^2 + (-1 - 2/√6)^2), (-1 - 2/√6)/√(1 + (2 - 2/√6)^2 + (1 - 4/√6)^2 + (-1 - 2/√6)^2))

Finally, u3 is the remaining vector after orthogonalizing u3' with respect to v1 and v2. Since u3' is orthogonal to v1 and v2, u3 will also be orthogonal to both v1 and v2. Therefore, u3 can be expressed as u3 = (a, b, c), where a, b, and c are constants to be determined.

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The value of the test statistic, z, is -7.5.

To find the value of the test statistic, z, we can use the following formula:

z = (x - μ) / (σ / √n)

Where:

x = sample mean (84)

μ = population mean under the null hypothesis (90)

σ = population standard deviation

n = sample size (100)

Given that the population standard deviation is not provided, we'll assume it is unknown and use the sample standard deviation as an estimate for the population standard deviation.

Therefore, we'll use the given sample standard deviation of 8 as the estimate for σ.

Substituting the values into the formula, we have:

z = (84 - 90) / (8 / √100)

 = -6 / (8 / 10)

 = -6 / 0.8

 = -7.5

Hence, the value of the test statistic, z, is -7.5.

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a) To solve the recurrence relation an = an−1 + 6an−2 with initial conditions a0 = 3 and a1 = 6, we can use the characteristic equation r^2 - r - 6 = 0.

Factoring the quadratic equation, we get (r - 3)(r + 2) = 0.

So, the roots are r = 3 and r = -2.

The general solution is an = c1(3^n) + c2((-2)^n), where c1 and c2 are constants to be determined from the initial conditions.

Using the initial conditions a0 = 3 and a1 = 6, we can substitute these values into the general solution:

a0 = c1(3^0) + c2((-2)^0) = c1 + c2 = 3a1 = c1(3^1) + c2((-2)^1) = 3c1 - 2c2 = 6

Solving these equations simultaneously, we find c1 = 2 and c2 = 1.

Therefore, the solution to the recurrence relation with the given initial conditions is:

an = 2(3^n) + (-2)^n

b) Similarly, for the recurrence relation an = 7an−1 − 10an−2 with initial conditions a0 = 2 and a1 = 1, we can find the roots of the characteristic equation r^2 - 7r + 10 = 0, which are r = 2 and r = 5.

The general solution is an = c1(2^n) + c2(5^n).

Using the initial conditions a0 = 2 and a1 = 1:

a0 = c1(2^0) + c2(5^0) = c1 + c2 = 2

a1 = c1(2^1) + c2(5^1) = 2c1 + 5c2 = 1

Solving these equations simultaneously, we find c1 = -3 and c2 = 5.

Therefore, the solution to the recurrence relation with the given initial conditions is:

an = -3(2^n) + 5(5^n)

c), d), e), f) and g) will be solved in the next response due to space limitations.

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Answer: (7,-6)

Step-by-step explanation:

9-x = 2  --> x=7

-8-y = -2  --> y = -6

B = (7,-6)

Given the function F0(x) = 1 - 1/(1+x) for x ≥ 0, we can find expressions for the requested terms:

(a) S0(x) is the survival function, which is the complement of the cumulative distribution function F0(x). Therefore, S0(x) = 1 - F0(x). Substituting F0(x) into the equation, we get:
S0(x) = 1 - (1 - 1/(1+x)) = 1/(1+x)
(b) f0(x) is the probability density function (pdf) and can be found by taking the derivative of the cumulative distribution function F0(x) with respect to x:
f0(x) = dF0(x)/dx = d(1 - 1/(1+x))/dx = 1/(1+x)^2
(c) To find Sx(t), we need to find the survival function for an individual aged x at time t. Since we know S0(x), we can find Sx(t) using the following relationship:
Sx(t) = S0(x+t)/S0(x)
By substituting S0(x) into the equation, we get:
Sx(t) = (1/(1+x+t))/(1/(1+x)) = (1+x)/(1+x+t)
Now we can calculate the requested values:
(d) p20 is the probability of surviving one more year for an individual aged 20. It is given by:
p20 = S20(1)/S20(0)
Substitute 20 for x and 1 for t in Sx(t):
p20 = (1+20)/(1+20+1) = 21/22
(e) The term 10|5q30 does not follow the standard notation used in survival analysis. Please provide more context or clarify the term to receive an appropriate answer.

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It is essential to use an appropriate sample size and confidence level when calculating the margin of error to ensure the accuracy of the estimate.

The computations for the margin of error rely on the mathematical properties of the sampling distribution of the statistic. When we take a random sample from a population, we assume that the sample is representative of the population, which means that it has the same characteristics as the population.

The sampling distribution of the statistic is the distribution of all the possible values of the statistic that could be obtained from all the possible samples of a certain size from the population. The margin of error is calculated based on this distribution and the desired level of confidence.

The margin of error is an important statistical concept because it quantifies the uncertainty associated with the sample estimate. It tells us how much we should expect the sample estimate to vary from the true population parameter. The margin of error depends on the sample size, the level of confidence, and the variability of the population.

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The computations for the margin of error rely on the mathematical properties of the sampling distribution of the statistic.

Specifically, the margin of error is a function of the sample size and the standard error of the statistic, which is determined by the population standard deviation and the sample size. The confidence level determines the critical value used to calculate the margin of error, which is based on the standard normal distribution or the t-distribution depending on the sample size and the assumptions about the population distribution. However, the margin of error itself is based on the properties of the sampling distribution of the statistic, which describes the distribution of the statistic over all possible samples of the same size from the population.

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The new pairs of polar coordinates are (3,2π) for r>0 and 2π≤θ<4π, and (-3,π) for r<0 and 0≤θ<2π.

(a) You are given the point (3,0) in polar coordinates.

(i) To find another pair of polar coordinates for this point such that r>0 and 2π≤θ<4π, follow these steps:

1. Start with the given coordinates (3,0).
2. Since we want to keep r>0, r remains 3.
3. To find a new angle θ that is between 2π and 4π, we can add 2π to the current angle (0 + 2π = 2π).
4. The new pair of polar coordinates is (3,2π).

(ii) To find another pair of polar coordinates for this point such that r<0 and 0≤θ<2π, follow these steps:

1. Start with the given coordinates (3,0).
2. To make r<0, we can multiply the current r by -1: (-3).
3. To find a new angle θ that is between 0 and 2π, we can add π to the current angle (0 + π = π).
4. The new pair of polar coordinates is (-3,π).

So, the new pairs of polar coordinates are (3,2π) for r>0 and 2π≤θ<4π, and (-3,π) for r<0 and 0≤θ<2π.

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The length of the intercepted arc in the given circle is approximately 0.97 units.

To find the length of the intercepted arc, we need to use the formula that relates the angle of the intercepted arc to the length of the arc and the radius of the circle. The formula is as follows:

Length of Arc = Radius x Angle

In our case, the radius of the circle is given as 9.7 units, and the angle of the intercepted arc is 0.1 radians. Therefore, substituting these values into the formula, we can calculate the length of the arc as follows:

Length of Arc = 9.7 units x 0.1 radians

To find the product of 9.7 and 0.1, we simply multiply these two numbers together:

Length of Arc = 0.97 units

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The arc of the satellite refers to the path that a satellite follows as it orbits around a celestial body such as the Earth, and is determined by the gravitational forces between the two objects.

We are given the orbital radius and angular separation between two adjacent communication satellites, and we need to find the arc length that separates them.

Here's a step-by-step explanation:

1. Given the orbital radius (r) is 8.46 * 10^7 m.
2. Given the angular separation (θ) is 4.00 degrees.
3. To find the arc length (s), we can use the formula s = r * θ, where θ should be in radians.
4. Convert the angular separation from degrees to radians: θ (radians) = θ (degrees) * (π / 180) = 4 * (π / 180) = 4π / 180 radians.
5. Calculate the arc length: s = r * θ = (8.46 * 10^7) * (4π / 180) ≈ 5.92 * 10^6 m.

So, the arc length that separates the satellites is approximately 5.92 * 10^6 m (option b).

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The standard deviation of x can be  0.725.

The table describing the probability distribution of x is as follows

x P(X=x)

0 10/120

1 48/120

2 42/120

3 20/120

To find the probabilities, we can use the hypergeometric distribution formula:

P(X=x) = (C(4,x) * C(6,3-x)) / C(10,3)

where C(n,r) represents the number of combinations of n things taken r at a time.

The mean of x can be found using the formula:

E(X) = Σ(x * P(X=x))

= 0*(10/120) + 1*(48/120) + 2*(42/120) + 3*(20/120)

= 1.4

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Answer:

72

Step-by-step explanation:

(2^3)=8

(3^2)=9

8*9=72

H does not satisfy all three properties required for a subspace, we can conclude that H is not a subspace of R2.

The set H is a subspace of R2, we need to check if it satisfies the three properties required for a subspace

1. The zero vector is in H.

2. H is closed under vector addition.

3. H is closed under scalar multiplication.

Now each property

1. The zero vector (0, 0) is in H since it lies on the unit circle.

2. To check closure under vector addition, suppose we have two vectors (x₁, y₁) and (x₂, y₂) in H. If we add them together, (x₁, y₁) + (x₂, y₂), the resulting vector will not necessarily lie on the unit circle. For example, if we add (1, 0) and (-1, 0), the result is (0, 0), which is not on the unit circle. Therefore, H is not closed under vector addition.

3. To check closure under scalar multiplication, suppose we have a scalar c and a vector (x, y) in H. If we multiply them, c × (x, y), the resulting vector will not necessarily lie on the unit circle. For example, if we multiply (1, 0) by 3, the result is (3, 0), which is not on the unit circle. Therefore, H is not closed under scalar multiplication.

Since H does not satisfy all three properties required for a subspace, we can conclude that H is not a subspace of R2.

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Janelle buys 1 pound of raspberries, she can buy a maximum of 4.625 pounds of blackberries.

Let's assume Janelle wants to buy blackberries and raspberries and has a maximum budget of $40. We need to find the maximum amount of fruit she can purchase while staying within her budget.

Let's denote the pounds of blackberries as "b" and the pounds of raspberries as "r." The cost of blackberries is $8 per pound, and the cost of raspberries is $9 per pound.

Based on this information, we can set up the following equations:

8b + 9r ≤ 40 (Total cost of blackberries and raspberries should be less than or equal to $40)

b, r ≥ 0 (Pounds of blackberries and raspberries should be non-negative)

To find the maximum amount of fruit Janelle can buy, we need to find the values of b and r that satisfy the given conditions.

There are various methods to solve this problem, such as graphing, substitution, or elimination. Let's use the substitution method:

We can rearrange the first equation as:

8b ≤ 40 - 9r

b ≤ (40 - 9r)/8

Since b and r should be non-negative, we can consider different values of r and substitute them into the equation to find the corresponding maximum values of b.

For example, if we assume r = 0, the equation becomes:

b ≤ (40 - 9(0))/8

b ≤ 5

So, if Janelle buys 0 pounds of raspberries, she can buy a maximum of 5 pounds of blackberries.

Similarly, for r = 1:

b ≤ (40 - 9(1))/8

b ≤ 4.625

Therefore, if Janelle buys 1 pound of raspberries, she can buy a maximum of 4.625 pounds of blackberries.

By exploring different values of r within the given constraints, we can determine various combinations of blackberries and raspberries that Janelle can purchase while staying within her $40 budget.

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