PLEASE HELP! The image is below:

Steady-state probability for each system state π_1 = 3π_0 ≈ 0.495.

What is probability?

Probability is a measure of the likelihood of an event occurring.

To analyze the system, we can use the Markov chain approach. We can define the state of the system as the number of busy servers, ranging from 0 to 3. Let's denote the state of the system at time t as X(t). The transition rates between states depend on the arrival and service rates, as follows:

For X(t) = 0, the arrival rate is λ = 8 per hour, and the departure rate is μ = 1/20 per minute per server. Therefore, the transition rate from state 0 to state 1 is λ, and the transition rate from state 1 to state 0 is 3μ.

For X(t) = 1, the arrival rate is λ = 8 per hour, and the departure rate is μ = 1/20 per minute per server. Therefore, the transition rate from state 1 to state 2 is λ, and the transition rates from state 2 to state 1 and from state 1 to state 0 are both 2μ.

For X(t) = 2, the arrival rate is λ = 8 per hour, and the departure rate is μ = 1/20 per minute per server. Therefore, the transition rate from state 2 to state 3 is λ, and the transition rates from state 3 to state 2 and from state 2 to state 1 are both μ.

For X(t) = 3, the arrival rate is λ = 0 (since there is no waiting space), and the departure rate is μ = 1/15 per minute per server. Therefore, the transition rate from state 3 to state 2 is 3μ.

To find the steady-state probabilities for each system state, we can use the balance equations:

π_i * q_i,j = π_j * q_j,i

where π_i is the steady-state probability of being in state i, and q_i,j is the transition rate from state i to state j.

We can set up a system of four equations (one for each state) and solve for the unknown probabilities. The equations are:

λπ_0 = 3μπ_1

λπ_1 = 2μπ_2 + 2μπ_0

λπ_2 = μπ_3 + 2μπ_1

3μπ_3 = μπ_2

We also have the normalization condition:

π_0 + π_1 + π_2 + π_3 = 1

Solving the system of equations, we get:

π_0 = (1 - ρ) * (1 - ρ²) * (1 + 3ρ + 9ρ²) / (1 + 3ρ + 9ρ² + 9ρ³)

π_1 = 3π_0

π_2 = 3ρπ_0

π_3 = ρ³π_0

where ρ = λ/3μ is the traffic intensity.

Substituting the given values, we get:

ρ = (8/3) / (3 * (1/20)) = 32/3

π_0 = (1 - 32/3) * (1 - (32/3)^2) * (1 + 3*(32/3) + 9*(32/3)²) / (1 + 3*(32/3) + 9*(32/3)² + 9*(32/3)³) ≈ 0.165

π_1 = 3π_0 ≈ 0.495

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Answer:

5 in.

Step-by-step explanation:

For a square:

perimeter = 4 × side

20 in. = 4 × side

side = 20 in. / 4

side = 5 in.

Answer and Explanation:

1. The left column should be labeled coin toss because we can see that there are two outcomes from a coin toss: heads or tails.

We know that the right column should be labeled spinner because it has three outcomes (red, yellow, or blue) for every previous outcome.

2. We can label the top box of the left column as H (for Heads) and the lower box as T (for Tails). Then, we can label the top, middle, and bottom boxes in the right column as R, Y, and B, respectively (for Red, Yellow, Blue).

3. Next to each box on the right, label the two events that lead to that outcome. For example, "HR" means that you tossed a heads, then spun a red.

4. There is only 1 outcome in the event "toss a tails, spin a yellow" because there is only one way (out of two ways) to toss a tails, and from there, there is only one way to spin a yellow (out of three ways).

5. The probability of tossing tails, then spinning a yellow is:

[tex]\dfrac{1}{2} \cdot \dfrac{1}{3} = \boxed{\dfrac{1}{6}}[/tex]

The  B-matrix for the transformation X-_Ax is:

[17b1 - 3b2 - 54b3]
[22b1 + b2 + b3]

The B-matrix for the transformation X-_Ax is a matrix that represents the images of each basis vector in B under the linear transformation represented by the matrix A. To find the B-matrix, we first need to compute the product A*B, where A is the transformation matrix and B is the basis matrix.

In this case, we are given B = {b1, b2, b3} and A = [[17, -3, -54], [22, b2, b3]]. We multiply A by the column vector [b1, b2, b3] to get the image of each basis vector under the transformation. The resulting matrix has two columns, where each column represents the image of one of the basis vectors.

The B-matrix is then constructed by arranging the images of the basis vectors as columns of a matrix. So the B-matrix for the transformation X-_Ax is:

[17b1 - 3b2 - 54b3]
[22b1 + b2 + b3]

This matrix can be used to find the coordinates of any vector in terms of the basis B after it has been transformed by the linear transformation represented by A.

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On Monday the total number of sold hamburgers are 118.

Let's call the number of hamburgers sold "h" and the number of cheeseburgers sold "c".

From the problem, we know two things:

The total number of burgers sold is 472:

h + c = 472

The number of cheeseburgers sold is three times the number of hamburgers sold:

c = 3h

We can use substitution to solve for h:

h + 3h = 472

4h = 472

h = 118

Therefore, 118 hamburgers were sold on Monday.

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Answer: There were 180 dog owners in this survey that owned more than one dog.

Step-by-step explanation:

Since this word problem is a percentage problem, you will need to convert 45% of 400 to a hundredths decimal.

45% -> 45/100 -> 0.45

Then, to find the solution, multiply 0.45 by the total number of dog owners in the survey. (400)

0.45

x400

---------

 180!

The answer you'll get will be 180, so therefore, 180 dog owners in this survey owned more than one dog. Hope this helps!:)

Jack has a 7.5 meter (750 cm) rope, gives his sister a 150 cm piece, and cuts the remaining 600 cm into 10 equal sections, with each section being 60 cm long.

Jack's rope is 7.5 meters long, which is equal to 750 centimetres. He gives his sister a piece of 150 centimetres, which leaves him with 600 centimetres of rope.
Jack then cuts the remaining piece into 10 equal sections. To find the length of each section, we can divide the total length of the rope (600 cm) by the number of sections (10):
600 cm ÷ 10 sections = 60 cm per section
Therefore, each section of rope that Jack cuts will be 60 centimetres long.

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When determining whether or not a measured effect can be distinguished from zero, we are interested in statistical significance.

Statistical significance refers to the likelihood that the results of a study are not due to chance. In other words, it assesses whether the effect observed in a sample is likely to be a true effect in the population, or whether it could have occurred by chance. Statistical significance is typically assessed using hypothesis testing and a significance level (usually set at 0.05), which represents the probability of obtaining the observed results or more extreme results under the assumption that the null hypothesis (i.e., no effect) is true. If the probability is less than the significance level, the result is said to be statistically significant, indicating that the null hypothesis can be rejected and the observed effect is likely a true effect. Practical significance, on the other hand, refers to the importance or relevance of the observed effect in the context of the research question or real-world application.

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Answer:

-7+7=2x

Step-by-step explanation:

0=2x

0/2=2x/2

0/2=x

To answer this question, we would need to perform a hypothesis test using the given sample data and a significance level of α = 0.05. The null hypothesis would be that the standard deviation of the resting heart rates for students in this class is equal to 12 bpm, while the alternative hypothesis would be that it is different from 12 bpm.

We would then need to calculate the sample standard deviation from the given data and use it to compute the test statistic (either a t-score or a z-score, depending on the sample size and whether or not the population standard deviation is known). We would compare this test statistic to the critical value from the appropriate distribution (either a t-distribution or a standard normal distribution) using the given significance level.

If the test statistic falls outside the critical value region, we would reject the null hypothesis and conclude that there is enough evidence to say that the standard deviation of the resting heart rates for students in this class is different from 12 bpm. However, if the test statistic falls inside the critical value region, we would fail to reject the null hypothesis and conclude that there is not enough evidence to make such a claim.

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if f(x, y, z) = 4[tex]x[/tex][tex]y^{2}[/tex][tex]z^{3}[/tex] arcsin (xz) , the value of fₓzᵧ is "24xyz²√(1-x²z²)".

To find fₓzᵧ, we differentiate f(x,y,z) partially with respect to x, then z, and finally y.

First, we take the partial derivative of f with respect to x:

fₓ = 4y²z³(arcsin(xz)) + 4xy²z³(1-x²z²)⁻ᵐ

where m = 1/2 * (1 - x²z²)⁻ⁿ, n = -1/2

Next, we take the partial derivative of fₓ with respect to z:

fₓz = 12xyz²(arcsin(xz)) + 4y²z²(1-x²z²)⁻ᵐ + 8xy²z²x(1-x²z²)⁻ⁿ

Finally, we take the partial derivative of fₓz with respect to y:

fₓzᵧ = 24xyz²√(1-x²z²)

Therefore, fₓzᵧ = 24xyz²√(1-x²z²) is the solution, where x, y, and z are the values of the given function f.

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The expected value of the smaller loss can be found using the properties of the exponential distribution.

Since the exponential distribution is memoryless, the probability of the first loss being the smaller loss is the same as the probability of the second loss being the smaller loss. Therefore, the expected value of the smaller loss is half of the expected value of the minimum of two exponential random variables.

The minimum of two independent exponential random variables with the same mean is known to follow an Erlang distribution with parameters k=2 and λ=1. Therefore, the expected value of the minimum of two exponential random variables with mean 1 is given by 2/λ = 2.

Thus, the expected value of the smaller loss is 1, which is half of the expected value of the minimum of two exponential random variables.

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The correct option is C, as the statement suggests that there is a positive correlation between the number of Target stores and Walmart stores in a city.

This means that as the number of Walmart stores in a city increases, there is a corresponding increase in the number of Target stores in the same city. However, this does not necessarily mean that there is an exact one-to-one relationship between the two stores, as stated in option A.

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If we find that the null hypothesis, H_0 : B_j = 0, cannot be rejected when testing the contribution of an individual regressor variable to the model, we usually should 1. remove the variable from the model.

This is because if the variable is not contributing significantly to the model, it is not useful in predicting the outcome. Therefore, removing the variable will simplify the model and potentially improve its accuracy.
Options 2 and 3 (doing nothing or adding a quadratic term in x) would not be appropriate if the variable is not significant, as they would not address the issue of the variable's lack of contribution.
Option 4 is also incorrect because we do need to take action if a variable is not contributing significantly to the model.

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To write y as the sum of a vector in w and a vector orthogonal to w, we first need to find a basis for w. Since w is spanned by bold u 1 and bold u 2, we can use these vectors as our basis for w:
B = {bold u 1, bold u 2}

Now, we can use the orthogonal complement of w, denoted by w⊥, to find a vector that is orthogonal to w. By definition, w⊥ is the set of all vectors that are orthogonal to every vector in w. We can find w⊥ by taking the null space of the matrix whose rows are the basis vectors of w:

A = [bold u 1; bold u 2]

w⊥ = null(A)

Once we have a basis for w⊥, we can find a vector that is orthogonal to w by taking a linear combination of the basis vectors of w⊥. Let's call this vector z:

z = c_1*bold v_1 + c_2*bold v_2 + ... + c_k*bold v_k

where c_1, c_2, ..., c_k are constants, and bold v_1, bold v_2, ..., bold v_k are the basis vectors of w⊥.

Finally, we can express y as the sum of a vector in w and a vector orthogonal to w:

y = a*bold u 1 + b*bold u 2 + z

where a and b are constants that we can find by projecting y onto the basis vectors of w:

a = (y · bold u 1) / (bold u 1 · bold u 1)
b = (y · bold u 2) / (bold u 2 · bold u 2)

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After 3 years, Ana's account balance would be approximately $1221.96, assuming a constant withdrawal rate of $200 per year.

Ana opened a bank account with an initial balance of $1000 and an annual interest rate of 0.02, compounded continuously. The function P(t) models the balance of the account in dollars at time t.
The function P models the balance of Ana's account, given the information provided.

1. Ana opens a bank account with $1000. This is the initial amount in the account.
2. The account earns interest compounded continuously at an annual rate of r = 0.02.
3. Money can be withdrawn, but no additions can be made.

Since money can be withdrawn from the account at regular intervals and no additions can be made, we can assume that the withdrawals are made at a constant rate. Let's call this rate "w" (in dollars per year). Therefore, the function P(t) can be modeled as:
The function P models the balance of the account, in dollars, at time t. Since the interest is compounded continuously, we will use the continuous compound interest formula:
P(t) = P₀ * e^(rt)

Where:
- P(t) is the balance at time t
- P₀ is the initial balance ($1000)
- e is the base of the natural logarithm (approximately 2.71828)
- r is the annual interest rate (0.02)
- t is the time in years

We get: P(t) = 1000*e^(0.02t) - wt

where e is the mathematical constant, approximately equal to 2.71828. The first term represents the balance of the account with continuous compounding, while the second term represents the withdrawals made from the account over time.

To calculate the balance of the account at a specific time t, we can substitute that value into the function P(t). For example, if we want to find the balance of the account after 3 years and assume that the withdrawal rate is $200 per year, we can write:

P(3) = 1000*e^(0.02*3) - 200*3
P(3) = 1000*e^(0.06) - 600
P(3) ≈ $1221.96
This function P(t) represents the balance of Ana's account, in dollars, at any given time t in years, considering continuous compounding and the possibility of withdrawals.

Therefore, after 3 years, Ana's account balance would be approximately $1221.96, assuming a constant withdrawal rate of $200 per year.

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The correct answer is option B, which is 1/(x+4). To see why, first factor the denominator of the given expression:

x^2 - 16x + 64 = (x - 8)(x - 8) = (x - 8)^2

Now, we can rewrite the original expression as:

(x - 8)^2 / [(x - 8)(x + 4)]

Canceling the common factor (x - 8), we get:

(x - 8) / (x + 4)

This is equivalent to 1/(x+4) since (x - 8) / (x + 4) = (x + 4 - 12) / (x + 4) = 1 - 12 / (x + 4) = 1 - 3 / (x + 4/3). As x approaches infinity, 3/(x+4/3) approaches 0, so 1 - 3 / (x + 4/3) approaches 1. Thus, the expression is equivalent to 1/(x+4) for any value of x except x = -4.

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If dakota was hurt about 8 miles into her ride and her whole trip took 4 hours total, Neena's walking speed is 2 miles per hour.

Let's assume that Neena's walking speed is "x" miles per hour. As per the problem, Dakota walks at twice the speed of Neena, which means Dakota's speed is "2x" miles per hour.

Now, we know that the total distance traveled by Neena and Dakota is 8 miles, and their total travel time is 4 hours. We can set up the following equation using the distance formula:

distance = speed x time

For Neena:

distance = x * t₁

For Dakota:

distance = 2x * t₂

Total distance = 8 miles

Total time = t₁ + t₂ = 4 hours

Substituting the distance and time values, we get:

x * t₁ + 2x * t₂ = 8

t₁ + t₂ = 4

Solving for t₁, we get:

t₁ = 4 - t₂

Substituting this in the first equation and simplifying, we get:

x * (4 - t₂) + 2x * t₂ = 8

4x = 8

x = 2

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The sampling method described in the scenario is d. stratified sampling.

In stratified sampling, the population is divided into distinct subgroups or strata based on certain characteristics or variables. The researcher then randomly selects samples from each stratum in proportion to their representation in the population. This approach ensures that the sample is representative of the population's diversity.

In this case, the researcher has divided the population of teachers into two strata: male teachers and female teachers. By randomly selecting 50 male teachers and 50 female teachers, the researcher is ensuring that both genders are represented in the sample.

The researcher's intention is to have a sample that reflects the gender distribution of teachers in the population accurately. Therefore, stratified sampling is the appropriate method in this scenario.

Other sampling methods, such as systematic sampling, convenience sampling, random sampling, and cluster sampling, are not applicable because they do not specifically address the need to ensure proportional representation of genders in the sample.

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Answer:

A = 24.15cm2 ~ 24.2cm2

Step-by-step explanation:

Area is calculated by

[tex]a = l \times w[/tex]

a = area

l = length = 6.9cm

w = width = 3.5cm

A= 3.5cm × 6.9cm

A = 24.15 cm2

so the area is 24.15cm2 ~ 24.2cm2

The original number of friends planning the party was 25.

Let's assume the total number of friends originally planning the party is 'x'.

Initially, each friend would contribute an equal amount to raise $200. So the initial contribution per friend would be $200/x.

When five friends decided not to attend, the number of friends remaining is (x - 5). Now, each friend has to contribute $2.00 more than before.

So, the new contribution per friend is $200/(x - 5) + $2.

According to the given information, the new contribution is $2.00 more than the initial contribution:

$200/(x - 5) + $2 = $200/x

To solve this equation, we can eliminate the dollar signs and simplify:

200/(x - 5) + 2 = 200/x

Multiplying both sides of the equation by x(x - 5) to eliminate the denominators:

200x + 2x(x - 5) = 200(x - 5)

200x + 2x^2 - 10x = 200x - 1000

Rearranging the equation and simplifying:

2x^2 - 10x - 1000 = 0

Dividing the equation by 2:

x^2 - 5x - 500 = 0

Using the quadratic formula, we can find the values of x:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -5, and c = -500.

x = (-(-5) ± √((-5)^2 - 4(1)(-500))) / (2(1))

x = (5 ± √(25 + 2000)) / 2

x = (5 ± √2025) / 2

x = (5 ± 45) / 2

Simplifying further:

x1 = (5 + 45) / 2 = 50 / 2 = 25

x2 = (5 - 45) / 2 = -40 / 2 = -20

Since the number of friends cannot be negative, we discard x2 = -20 as an extraneous solution.

Therefore, the original number of friends planning the party was 25.

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Answer:

a≥3

Step-by-step explanation:

(i) (CUD-E) specified using set builder notation:

(CUD-E) = {x ∈ R | (x > 0 ∧ x ≤ 10) ∨ (x > 9 ∧ x ≤ 15) ∧ x ∉ {1, 2, 3}}

(ii) (CE) specified using interval notation and set operations concisely:

(CE) = (0, 10] ∩ {1, 2, 3} = {1, 2, 3}

(iii) (CDF) specified using the most concise notation:

(CDF) = (C ∩ D) ∩ F

(b) Proof using the element argument method:

Given: A and B are sets such that P(A) ≤ P(B).

To prove: A ≤ B.

Proof:

1. Let x be an arbitrary element in A.

2. Since x is in A, by definition, x is a subset of A. Hence, x ⊆ A.

3. Since x ⊆ A and A ≤ B, by the definition of ≤, x ⊆ B.

4. Therefore, x is a subset of B. Hence, x ∈ P(B), where P(B) is the power set of B.

5. Since x ∈ P(B), by definition, x is a subset of B. Hence, x ⊆ B.

6. Since x is an arbitrary element in A and x ⊆ B, by definition, A ≤ B.

7. Therefore, if P(A) ≤ P(B), then A ≤ B.

In this proof, we used the fact that if x is an element of A, then x is a subset of A. Also, if x is a subset of A and A ≤ B, then x is a subset of B. These properties are based on the definitions of subsets and the order relation between sets.

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Answer:

Step-by-step explanation:

Let x - be the number of sweets in small packs

y - be the number of sweets in big packs

Therefore, we have:

4x + 3y = 175 (1)

5x + 2y = 154 (2)

Now, we find the difference between (1) & (2) is:

y-x = 21. Thus, y = 21+x

Now we substitute the value of y = 21+x to any of the two statements, we have 4x + 3(21+x) = 175 => 4x + 63 + 3x = 175.

Hence, 7x = 175 - 63 = 112 or simply, x=16.

Now, finding the value of y:

5(16) + 2y = 154

80 + 2y = 154

2y = 154-80

2y = 74

y = 37.

Therefore, there are 16 sweets in each small pack and 37 sweets in each big pack.

The expression gave us the desired value of T(-1-4x) is 9 + 10x.

Linear transformations are a fundamental concept in mathematics that play a crucial role in various fields such as physics, engineering, and computer science.

Now, let's consider the given problem. We are given that T is a linear transformation on the vector space P1P1, which is the space of polynomials of degree at most one. Specifically, we are given two evaluations of T, namely T(1+5x) = 1+2x and T(5+24x) = -2-3x.

Using the linearity of T, we can express any polynomial in P1P1 as a linear combination of 1 and x, that is, p(x) = a + bx for some scalars a and b. Then, we can use the evaluations of T to determine its action on any such polynomial. For instance, let's consider the polynomial -1-4x. We can write this as -1-4x = -1(1+0x) - 4(x+0x), which is a linear combination of 1 and x.

Using the linearity of T, we can apply T to each term separately, obtaining:

T(-1-4x) = T(-1(1+0x) - 4(x+0x))

= T(-1(1+0x)) - T(4(x+0x))

= -1T(1+0x) - 4T(x+0x)

= -1(1+2x) - 4(-2-3x)

= 1-2x+8+12x

= 9+10x.

Therefore, T(-1-4x) = 9+10x.

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Answer:

do you choose the best

Step-by-step explanation:

9 1/2 + 1 1/2

Converting to improper fractions: 9 1/2 = (9 * 2 + 1) / 2 = 19/2 and 1 1/2 = (1 * 2 + 1) / 2 = 3/2

Adding the fractions: 19/2 + 3/2 = (19 + 3) / 2 = 22/2 = 11/1 = 11

5 + 3 1/3

Converting to improper fractions: 5 = 5/1 and 3 1/3 = (3 * 3 + 1) / 3 = 10/3

Adding the fractions: 5/1 + 10/3 = (5 * 3 + 10) / 3 = 25/3

2 4/5 + 16 2/5

Converting to improper fractions: 2 4/5 = (2 * 5 + 4) / 5 = 14/5 and 16 2/5 = (16 * 5 + 2) / 5 = 82/5

Adding the fractions: 14/5 + 82/5 = (14 + 82) / 5 = 96/5

7 1/2 + 2 1/4

Converting to improper fractions: 7 1/2 = (7 * 2 + 1) / 2 = 15/2 and 2 1/4 = (2 * 4 + 1) / 4 = 9/4

Adding the fractions: 15/2 + 9/4 = (15 * 2 + 9) / 2 = 39/4

8 5/6 + 3 2/3

Converting to improper fractions: 8 5/6 = (8 * 6 + 5) / 6 = 53/6 and 3 2/3 = (3 * 3 + 2) / 3 = 11/3

Adding the fractions: 53/6 + 11/3 = (53 * 3 + 11 * 6) / 6 = 257/6

2 7/8 + 3 1/2 + 5 + 3 1/4

Converting to improper fractions: 2 7/8 = (2 * 8 + 7) / 8 = 23/8, 3 1/2 = (3 * 2 + 1) / 2 = 7/2, 3 1/4 = (3 * 4 + 1) / 4 = 13/4

Adding the fractions: 23/8 + 7/2 + 5 + 13/4 = (23 * 2 + 7 * 8 + 5 * 8 + 13 * 2) / 8 = 101/8

5 1/2 + 2 1/3 + 9 1/6 + 2

Converting to improper fractions: 5 1/2 = (5 * 2 + 1) / 2 = 11/2

Answer:

3392.9

Step-by-step explanation:

A little Pythagoras first, just to show how it works:

In a right-angled triangle, a ² + b ² = c ²

41² = radius² + 40²

radius = 9

Volume of cone = (1/3) X vertical height X π r ²

= (1/3) (40) π (9) ²

= 1080π

= 3392.9 to nearest tenth

For the statement Q R, the Inverse is ~R→~Q, the Converse is R→Q, the Contrapositive is ~Q→~R, and the original statement is Q→R. The original statement is Q→R, which means that if Q is true, then R must also be true.

The Inverse is formed by negating both the hypothesis and the conclusion of the original statement. In this case, the hypothesis is Q and the conclusion is R, so the negation of both would be ~Q and ~R, respectively. The resulting statement is ~R→~Q. The Converse is formed by switching the hypothesis and the conclusion of the original statement. In this case, the hypothesis is Q and the conclusion is R, so the Converse is R→Q. The Contrapositive is formed by negating both the hypothesis and the conclusion of the Converse statement. In this case, the hypothesis is R and the conclusion is Q, so the negation of both would be ~R and ~Q, respectively. The resulting statement is ~Q→~R.

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