suppose we roll two dice. what is the probability that the sum is 7 given that neither die showed a 6?

Excluding the boundary lines shown on the graph, (4 , 6) is the required ordered pair inside the region of the graph satisfies the given system of linear equation.

The budget of a website to advertise in local newspaper is $600 , and they plan to run no more than 10 advertisements. An ad in the weekend edition is $75, and an ad in the weekday paper is $50.

Let  x be equal to the number of weekend ads, and y be equal to the number of weekday ads.

We can compute the point inside the region of the graph that satisfies the system of linear equations that can be formed as,

75x + 50y = 600 ___(1)

x + y = 10 ___(2)

Solving the system of linear equations in (1) and (2) as,

From equation (2), x= 10- y

Substituting x in equation (1) with x= 10- y, we get,

75( 10- y )+ 50y = 600

⇒ 750 - 75y + 50y = 600

⇒ 25y = 150

⇒ y = 6

Thus, x = 10 - 6 = 4

Hence (x, y) = (4 , 6) inside the region of the graph satisfies the given system of equation.

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The criteria prove that the quadrilateral is not a square

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

For parallel opposite sides, we have

A = (-12, 5), B = (-7, 12), C = (5, 7) and D = (0, 0)

Calculate the slopes of opposite sides

So, we have

AD = (0 - 5)/(0 + 12) = -5/12

BC = (12 - 7)/(-7 - 5) = -5/12

AC = (7 - 5)/(5 + 12) = 2/17

BD = (12 - 0)/(-7 - 0) = -12/7

The sides AC and BD are not parallel

For the congruent sides, we have

AD = √[(0 - 5)² + (0 + 12)²] = 13

BC = √[(12 - 7)² + (-7 - 5)²] = 13

AC = √[(7 - 5)² + (5 + 12)²] = 17.11

BD = √[(12 - 0)² + (-7 + 0)²] = 13.89

The four sides are not parallel

For the right angles

The slopes calculated do not show that opposite sides have opposite reciprocals as their slopes

So, the four sides are not right angled

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The theoretical probability that the coin will land tails up is 1/2 = 0.5 or 50%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

In this problem, we have a fair coin, meaning that in any throw, the coin is equally as likely to land in one of the two outcomes, which are heads up or tails up.

Hence the probability is given as follows:

p = 1/2 = 0.5 = 50%.

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The area of the brick border is: 7.678 m²

Using Pythagoras theorem, we can find the diameter of the circle as:

d = √(10² - 6²)

d = √64

d = 8 m

Since the width of the brick border is 40 cm, then converting to meters gives 0.4 m.

Area of semi circle with brick border = ¹/₂(π * (8.8/2)²) = 30.411 m²

Area of semi circle without brick border = ¹/₂(π * (8/2)²) = 25.133 m²

Area of circular part of border = 30.411 m² - 25.133 m²

= 5.278 m²

Area of rectangular part of border = 6 * 0.4 = 2.4 m²

Total area of border = 5.278 m² + 2.4 m²

= 7.678 m²

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The false statement is (a) A chi-square distribution with k degrees of freedom is more right-skewed than a chi-square distribution with k + 1 degrees of freedom.

In fact, as the degrees of freedom increase, the chi-square distribution becomes more symmetric and approaches a normal distribution.

The other statements are true: (b) A chi-square distribution never takes negative values, (c) The degrees of freedom for a chi-square test are determined by the number of categories being compared minus one, (d) P(X'>10) is greater when the degrees of freedom are k + 1 than when they are k, and (e) The area under a chi-square density curve is always equal to 1 and is determined by the sample size and degrees of freedom.

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The simplified expression is (8x² - 7x - 4) / [(4x + 1)(x - 1)(4x - 1)].

We have,

To simplify the expression

2(x - 1) / (4x² - 3x - 1) + (x + 2) / (4x² + 7x - 2)

we need to find the least common denominator (LCD) of the two denominators:

(4x² - 3x - 1) and (4x² + 7x - 2).

To find the LCD, we need to factor in both denominators.

We can factor the first denominator as:

4x² - 3x - 1 = (4x + 1)(x - 1)

We can factor the second denominator by using the quadratic formula or by factoring by grouping:

4x² + 7x - 2 = (4x - 1)(x + 2)

Therefore, the LCD is the product of the factors of both denominators, with each factor appearing once at most:

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

To get each fraction to have the same denominator, we need to multiply the numerator and denominator of the first fraction by (4x - 1) and the numerator and denominator of the second fraction by (x - 1):

2(x - 1)(4x - 1) / [(4x + 1)(x - 1)(4x - 1)] + (x + 2)(x - 1) / [(4x + 1)(x - 1)(4x - 1)]

Now that both fractions have the same denominator, we can add the numerators and simplify:

[2(x - 1)(4x - 1) + (x + 2)(x - 1)] / [(4x + 1)(x - 1)(4x - 1)]

Multiplying out the numerator and simplifying, we get:

[8x^2 - 7x - 4] / [(4x + 1)(x - 1)(4x - 1)]

Therefore,

The simplified expression is (8x² - 7x - 4) / [(4x + 1)(x - 1)(4x - 1)]

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If we consider the two vectors a and b with known magnitudes of 1 m and 10 m respectively, but unknown directions, we can still perform some calculations with these vectors. However, without knowing their directions, we cannot determine the resultant vector of their addition or subtraction.

The direction of a vector is a crucial component in determining the overall effect of the vector. Therefore, to fully understand the impact of vectors a and b, we need to know their directions as well.
Based on the information given, you have two vectors: vector a with a magnitude of 1 m and vector b with a magnitude of 10 m. However, since the directions of these vectors are unknown, you cannot determine their specific position, direction, or any further information about their relationship to each other. To analyze or perform operations on these vectors, you would need additional information regarding their directions.

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To prove that arc AC is congruent to arc BD, we will follow these steps:
1. Draw a circle with center O, and draw chords AB and CD parallel to each other. Since, angles AOB and COD are congruent, their intercepted arcs must also have the same measure.

To prove that arc AC is congruent to arc BD, we need to use the fact that AB is parallel to CD.

First, we can draw a diagram of the circle with the chords AB and CD intersecting at a point E. Since AB is parallel to CD, we know that angle AEB is congruent to angle CED (corresponding angles).

Next, we can draw radii from the center of the circle to the endpoints of the chords, creating right triangles AOE and COF. Since the radii of a circle are congruent, we know that AO is congruent to CO and OE is congruent to OF.

Using these congruences and the fact that angle AOE is congruent to angle COF (both are right angles), we can apply the Side-Angle-Side (SAS) congruence theorem to triangle AOE and triangle COF. Therefore, we can conclude that triangle AOE is congruent to triangle COF.

Now, we can use the congruence of triangle AOE and triangle COF to show that arc AC is congruent to arc BD. Angle AOE is congruent to angle COF (by the congruence of the triangles) and arc AC is the measure of twice angle AOE while arc BD is the measure of twice angle COF. Therefore, we can conclude that arc AC is congruent to arc BD.

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To assign 2-digit numbers to these classes such that these numbers considered as 2-dimensional vectors will be in a partial order relation determined by the component-wise s between these vectors, we can follow the given steps.

1. Identify the classes and their prerequisites:
- Class A is a prerequisite for classes B and C
- Class D is a prerequisite for classes B and E
- Class C is a prerequisite for classes E and F

2. Draw a directed graph representing the prerequisites:
```
A -> B -> E -> F
 \-> C -> E
D -----^
```

3. Assign numbers to the classes in such a way that the numbers assigned to prerequisite classes are smaller than those assigned to dependent classes. We can use the following numbering scheme:
- Class A: 10
- Class B: 20
- Class C: 30
- Class D: 40
- Class E: 50
- Class F: 60

4. Represent these numbers as 2-dimensional vectors with the first digit representing the horizontal component and the second digit representing the vertical component:
- Class A: (1,0)
- Class B: (2,0)
- Class C: (3,0)
- Class D: (4,0)
- Class E: (5,0)
- Class F: (6,0)

5. Check if these vectors are in a partial order relation determined by the component-wise ≤ between these vectors:
- (1,0) ≤ (2,0) since 1 ≤ 2
- (1,0) ≤ (3,0) since 1 ≤ 3
- (4,0) ≤ (2,0) since 4 ≤ 2
- (4,0) ≤ (5,0) since 4 ≤ 5
- (3,0) ≤ (5,0) since 3 ≤ 5
- (5,0) ≤ (6,0) since 5 ≤ 6

Therefore, the assignment of numbers to these classes and their representation as 2-dimensional vectors satisfy the partial order relation determined by the component-wise ≤ between these vectors.

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The length of BD is 53 units.

We are given here that BD is (8x - 27) units and EC is (2x + 33) units. BD and EC are the diagonals of a trapezoid. As we know that diagonals of a trapezoid are equal. Therefore, we will equate the given diagonals of a trapezoid.

BD  =  EC

Substituting the given values of BD and EC

8x - 27  =  2x + 33

combining the like terms

8x - 2x  =  33 + 27

6x  =  60

x = 60/6

x = 10

Now, as we have to find BD, we will substitute the value of x in the given equation for BD.

BD = 8x - 27

BD = 8(10) - 27

BD = 80 - 27

BD = 53

Therefore, BD is 53 units.

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The complete question is " If BD = 8x - 27 and EC-2x + 33, find BD​ with respect to the image shown."

Based on the inscribed angle theorem and the tangent theorem, the angles that must be right angles are: ∠XLS, ∠LEG, ∠LTG, and ∠XLQ.

Since, The theorem states that an inscribed angle of a semicircle equals 90°.

And, According to the tangent theorem, a tangent is at right angle at the point of tangency with the radius.

Hence, Based on these theorems, the angles that must be right angles are:

∠XLS, ∠LEG, ∠LTG, and ∠XLQ.

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The correct statement is "Find half the difference of 14 and 6, multiply by 6, then add 4".

The solution of the fraction expression is calculated as follows;

1/[2 x (14 - 6) x 6 + 4]

To solve the above expression, we will apply the rule of BODMAS as shown below;

The difference of 14 and 6 = 14 - 6 = 8

The next is to divide 8 by 2

= 8/2

= 4

The next step is to multily the result by 6

= 4 x 6

= 24

The final step is to add 4 to it;

= 24 + 4

= 28

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y(x) = a_0 (1 - x^2/3 + 2x^4/45 - 8x^6/315 + ...)

that the solution is only valid for |x| < 1, since the differential equation is singular at x = ±1.

We can solve the given differential equation using the Frobenius method, by assuming that the solution can be represented as a power series:

y(x) = ∑(n=0)^(∞) a_n x^n

Differentiating the series twice, we get:

y'(x) = ∑(n=1)^(∞) n a_n x^(n-1)

y''(x) = ∑(n=2)^(∞) n(n-1) a_n x^(n-2)

Substituting these into the differential equation, we get:

(1-x^2) ∑(n=2)^(∞) n(n-1) a_n x^(n-2) - 2x ∑(n=1)^(∞) n a_n x^(n-1) + 2 ∑(n=0)^(∞) a_n x^n = 0

Simplifying and shifting the indices, we get:

∑(n=0)^(∞) [(n+2)(n+1) a_{n+2} - 2n a_n + 2a_n] x^n = 0

This gives us the following recurrence relation for the coefficients:

(n+2)(n+1) a_{n+2} = 2n a_n - 2a_n

Simplifying further, we get:

a_{n+2} = - (2n/(n+2)(n+1)) a_n

Starting with n = 0, we can compute the coefficients a_n in terms of a_0:

a_2 = - 2/3 a_0

a_4 = 2/15 a_2 = - 4/45 a_0

a_6 = - 2/21 a_4 = 8/315 a_0

a_8 = 2/99 a_6 = - 16/3465 a_0

...

The general form of the solution is then:

y(x) = a_0 (1 - x^2/3 + 2x^4/45 - 8x^6/315 + ...)

that the solution is only valid for |x| < 1, since the differential equation is singular at x = ±1.

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The multiple choice question is "Ο Α True", which means "Option A is true".

Recall that Newton's method involves the iterative formula:

x_(n+1) = x_n - f(x_n)/f'(x_n)

where f(x) is the function we want to solve for, and f'(x) is its derivative.

In this case, we have f(x) = 2x^3 - 83x^2 - 80x - 2.02 and f'(x) = 6x^2 - 166x - 80. So, using the initial guess of x_0 = 3, we have:

x_1 = x_0 - f(x_0)/f'(x_0)

= 3 - (2(3)^3 - 83(3)^2 - 80(3) - 2.02)/(6(3)^2 - 166(3) - 80)

= 2.838

Therefore, the next approximation to the solution using Newton's method with an initial guess of x_0 = 3 is x_1 = 2.838.

The answer to the multiple choice question is "Ο Α True", which means "Option A is true".

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Answer:

m+10

Step-by-step explanation:

We know that yesterday, eric had m baseball cards. Thus, we can denote that the total number of baseball cards he had yesterday is m.

We know that he got 10 more today. Since he is receiving more, he is adding to his collection. Since he is getting more, we have to add 10 to how many baseball cards he used to have. He used to have m baseball cards, so today he has m+10 baseball cards.

This is the answer as we cannot combine 10 and m. Since m is a variable with no set value as of now, and 10 is a constant number that has no variable, they are not like terms and cannot be added. So the answer is m+10 baseball cards.

I hope this helped.

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A 95% confidence interval for the true mean hours of exercise per week is (2.08, 8.40) for males and (2.95, 5.49) for females

To find a 95% confidence interval for the true mean hours of exercise per week for each group, we will use the following formula:

[tex]Confidence interval = Sample mean± (t \frac{Sample standard deviation}{\sqrt{n} } )[/tex]
where t is the t-score based on the degrees of freedom and the desired confidence level (95%).

(a) Males:

1. Determine the t-score: For a 95% confidence interval and 6 - 1 = 5 degrees of freedom, the t-score is approximately 2.571.

2. Calculate the margin of error: [tex]2.571 (\frac{3.01}{\sqrt{6} }) = 3.16[/tex]

3. Find the confidence interval: 5.24 ± 3.16 = (2.08, 8.40)

Males: (2.08, 8.40)

(b) Females:

1. Determine the t-score: For a 95% confidence interval and 14 - 1 = 13 degrees of freedom, the t-score is approximately 2.160.

2. Calculate the margin of error: [tex]2.160 (\frac{2.33}{\sqrt{14} })  = 1.27[/tex]

3. Find the confidence interval: 4.22 ± 1.27 = (2.95, 5.49)

Females: (2.95, 5.49)

In conclusion, a 95% confidence interval for the true mean hours of exercise per week is (2.08, 8.40) for males and (2.95, 5.49) for females.

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The design described in the first scenario uses a related sample with matched subjects.

In the first scenario, Suzanne Thomas is interested in comparing alcoholics with social phobia to alcoholics without social phobia. She matches participants from both groups on several variables, such as age and gender, in order to create comparable groups. This indicates the use of matched subjects design, where participants in one group are matched with participants in another group based on certain criteria to create comparable groups for comparison. Suzanne Thomas then collects data on the severity of alcohol dependence from each group. Therefore, the design described in the first scenario uses a related sample with matched subjects.

In the second scenario, the design described does not involve the use of a related sample. It is mentioned that the researcher wants to compare kleptomaniacs who are taking antidepressants to the population average, without matching or pairing participants.

Therefore, the design in the second scenario does not involve the use of a related sample.

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The correct answer is (c) multiplication law. This law helps us calculate the probability of two independent events occurring together by simply multiplying their individual probabilities.

The probability of an intersection of two events is computed using the multiplication law. The multiplication law states that the probability of two independent events occurring together is the product of their individual probabilities. For example, if event A has a probability of 0.4 and event B has a probability of 0.3, then the probability of both events A and B occurring together is 0.4 x 0.3 = 0.12.

The probability of an intersection of two events is computed using the multiplication law. This law states that the probability of two independent events occurring simultaneously is equal to the product of their individual probabilities. Mathematically, it can be represented as:

P(A ∩ B) = P(A) × P(B)

Where P(A ∩ B) is the probability of the intersection of events A and B, P(A) is the probability of event A occurring, and P(B) is the probability of event B occurring. Remember that this law is only applicable if the events are independent, meaning that the occurrence of one event does not affect the probability of the other event. If the events are not independent, you would need to use conditional probability.

It is important to note that the multiplication law applies only when the two events are independent, meaning that the occurrence of one event does not affect the probability of the other event occurring. If the events are dependent, then the multiplication law cannot be used and the calculation becomes more complex.

In contrast, the addition law is used to compute the probability of the union of two events, meaning either one or the other event occurs or both events occur. The subtraction law and division law are not typically used for computing probabilities of intersections or unions, but instead are used in other probability calculations such as conditional probability or Bayes' theorem.

In summary, the correct answer is (c) multiplication law. This law helps us calculate the probability of two independent events occurring together by simply multiplying their individual probabilities.

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The equation of the plane tangent to the surface z = x²y + xy² + ln x + R at (1, 0, R) is z = x + y + (R - 1).

To find the equation of the plane tangent to the surface z = x²y + xy^2 + ln x + R at the point (1, 0, R), we'll need to find the partial derivatives with respect to x and y, and then use the point-slope form of the tangent plane equation. Here are the steps:

1. Find the partial derivatives of the surface function with respect to x and y:
∂z/∂x = 2xy + y² + (1/x)
∂z/∂y = x² + 2xy

2. Evaluate the partial derivatives at the given point (1, 0, R):
∂z/∂x(1, 0) = 2(1)(0) + (0)² + (1/1) = 1
∂z/∂y(1, 0) = (1)² + 2(1)(0) = 1

3. Use the point-slope form of the tangent plane equation:
z - z₀ = a(x - x₀) + b(y - y₀)

4. Substitute the point (x₀, y₀, z₀) = (1, 0, R) and the partial derivative values a = ∂z/∂x = 1, b = ∂z/∂y = 1:
z - R = 1(x - 1) + 1(y - 0)

5. Simplify the equation:
z - R = x - 1 + y

6. Rearrange the equation to the standard form:
z = x + y + (R - 1)

The equation of the plane tangent to the surface z = x²y + xy² + ln x + R at (1, 0, R) is z = x + y + (R - 1).

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The car will travel approximately 11.829 if its wheels revolve 10,000 times without slipping.

First, we need to find the circumference of the wheel, which is given by the formula:

circumference = pi x diameter

where pi is approximately equal to 3.14.

So, the circumference of the wheel = 3.14 x 24 = 75.36 inches.

Next, we need to find the distance traveled by the car in one revolution of the wheel, which is equal to the circumference of the wheel.

Distance traveled by the car in one revolution of the wheel = 75.36 inches = 0.0011829 miles (1 inch = 0.000015783 miles).

Therefore, the distance traveled by the car in 10,000 revolutions of the wheel = 0.0011829 x 10,000 = 11.829 miles.

Rounding this answer to two decimal places, we get the final answer as approximately 11.829.

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a)  The value of f(x) is 1/51, for 193 ≤ x ≤ 244

b) The mean is  218.5

c) σ =  = 17.65

d) P(x > 227) = 0.333

e) P(202 ≤ x ≤ 219) = 0.333

f)  P(x ≤ 222) = 0.569

We have,

a)

Since the distribution is uniform, the probability density function (PDF) is constant within the range [193, 244] and zero elsewhere.

The PDF is given by:

f(x) = 1 / (244 - 193) = 1/51, for 193 ≤ x ≤ 244

b)

The mean of a uniform distribution is the average of the minimum and maximum values, so we have:

mean = (193 + 244) / 2 = 218.5

c)

The standard deviation of a uniform distribution is given by:

σ = (b - a) / √12

where a and b are the minimum and maximum values, respectively. Substituting in the values given, we get:

σ = (244 - 193) / √12 = 17.65

d)

The probability of x > 227 is the area under the PDF to the right of x = 227. Since the PDF is constant, this is simply the proportion of the total range that lies to the right of 227:

P(x > 227) = (244 - 227) / (244 - 193) = 17 / 51 ≈ 0.333

e)

The probability of 202 ≤ x ≤ 219 is the area under the PDF between x = 202 and x = 219. This is:

P(202 ≤ x ≤ 219) = (219 - 202) / (244 - 193) = 17 / 51 ≈ 0.333

f)

The probability of x ≤ 222 is the area under the PDF to the left of x = 222. This is:

P(x ≤ 222) = (222 - 193) / (244 - 193) = 29 / 51 ≈ 0.569

Thus,

a) f(x) = 1/51, for 193 ≤ x ≤ 244

b)  mean = 218.5

c) σ =  = 17.65

d) P(x > 227) = 0.333

e) P(202 ≤ x ≤ 219) = 0.333

f)  P(x ≤ 222) = 0.569

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Answer: B. 6.500

Step-by-step explanation: I just took the quiz.

To show that x ∈ iso(X) if and only if {x} is open in (X, d)

We need to prove both directions: (1) if x ∈ iso(X), then {x} is open in (X, d) and (2) if {x} is open in (X, d), then x ∈ iso(X).

(1) If x ∈ iso(X), then x is an isolated point of X. By definition, this means there exists an open ball B(x, r) centered at x with radius r > 0 such that B(x, r) ∩ X = {x}. Now, consider the set {x}. To show that {x} is open in (X, d), we need to show that for each point x in {x}, there exists an open ball centered at x that is entirely contained in {x}. Since B(x, r) ∩ X = {x}, it follows that B(x, r) ⊆ {x}. Thus, {x} is open in (X, d).

(2) If {x} is open in (X, d), then for each point x in {x}, there exists an open ball B(x, r) centered at x with radius r > 0 such that B(x, r) ⊆ {x}. In other words, B(x, r) ∩ X = {x}. This means that no other points of X are in B(x, r), which is the definition of an isolated point. Therefore, x ∈ iso(X).

In conclusion, x ∈ iso(X) if and only if {x} is open in (X, d).

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False.

The Cp and Cpk statistics are both used to assess the capability of a process to meet specifications, but they have different purposes.

The Cp statistic is a measure of how well the process spread (variation) fits within the specification limits. It assumes that the process mean is centered on the target value. It is calculated as the ratio of the specification width to the process spread (six times the process standard deviation).

The Cpk statistic, on the other hand, takes into account both the process spread and the deviation of the process mean from the target value. It is calculated as the minimum of two values: the difference between the process mean and the closest specification limit divided by three times the process standard deviation (assuming the process is centered within the specification limits), or the Cp value adjusted for the deviation of the process mean from the target value.

Both statistics have their uses and limitations depending on the situation. If the process mean is not centered on the target value, the Cpk statistic may be more useful than the Cp statistic since it takes into account both the spread and centering of the process.

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Answer: -12

Step-by-step explanation:

As per the given values, the area of the hemisphere is 45065.41 m³

Volume = 450000 m³

Calculating the volume of the hemisphere -

Volume = 2/3 πr³

Substituting the values -

450,000 =2/3 x 3.14 x r³

Solving for r³

r³ = 450000 x 3/(2 x 3.14)

r³ = 214968.1

r = √214968.1

r = 59.9

Calculating the area of the hemisphere -

Area = 4πr²

Substituting the values -

Area = 4 x 3.14 x (59.9)²

= 12.56 x (59.9)²

= 45065.41

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Answer: 62

Step-by-step explanation:

top row has 12

the next row has 1 more =13

and it alternates

12+13+12+13+12=62

The solution to the IVP is Y(t) = [ -5e^(4t) - 5e^(-t); 3e^(4t) - e^(-t)] with initial condition Y(0) = [5; 3].

To solve the IVP dY/dt - [3 10; -1 5] Y = [8; 0] with initial condition Y(0) = [5; 3], we can use the matrix exponential method.

First, we need to find the eigenvalues and eigenvectors of the matrix A = [3 10; -1 5]. We can do this by solving the characteristic equation det(A - λI) = 0, where I is the identity matrix and λ is the eigenvalue.

det(A - λI) = (3-λ)(5-λ) + 10 = λ^2 - 8λ + 25 = (λ-4)^2

So, the eigenvalue is λ = 4 with multiplicity 2. To find the eigenvectors, we need to solve (A - λI)x = 0 for each eigenvalue.

For λ = 4, we have

(A - λI)x = [3 10; -1 5 - 4] [x1; x2] = [0; 0]

which gives us the equation 3x1 + 10x2 = 0 and -x1 + x2 = 0. Solving these equations, we get x1 = -10/3 and x2 = 1. So, the eigenvector corresponding to λ = 4 is [ -10/3; 1].

Since we have repeated eigenvalues, we need to find the generalized eigenvector. We can do this by solving (A - λI)x = v, where v is any vector that is not an eigenvector.

Let v = [1; 0], then (A - 4I)x = [1; 0] gives us 3x1 + 10x2 = 1 and -x1 + x2 = 0. Solving these equations, we get x1 = -2/3 and x2 = 1/3. So, the generalized eigenvector corresponding to λ = 4 is [ -2/3; 1/3].

Now, we can form the matrix P = [ -10/3 -2/3; 1 1/3] and the diagonal matrix D = [4 1; 0 4], where the diagonal entries are the eigenvalues.

Using the formula Y(t) = e^(At) Y(0), we can write Y(t) as

Y(t) = P e^(Dt) P^(-1) Y(0)

= [ -10/3 -2/3; 1 1/3] [ e^(4t) 0; 0 e^(4t)] [ -1/2 1/2; 2 1] [5; 3]

= [ -5e^(4t) - 5e^(-t); 3e^(4t) - e^(-t)]

Therefore, the solution to the IVP is Y(t) = [ -5e^(4t) - 5e^(-t); 3e^(4t) - e^(-t)] with initial condition Y(0) = [5; 3].

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The probability that a srs of 20 students will spend an average of between 600 and 700 dollars is 0.92081.

We need to find the probability that a simple random sample of 20 students will spend an average of between 600 and 700 dollars.

To solve this problem, we will use the central limit theorem, which states that the sampling distribution of the sample means will be approximately normally distributed with a mean of μ and a standard deviation of σ/√(n), where n is the sample size.

Thus, the mean of the sampling distribution is μ = 650 and the standard deviation is σ/sqrt(n) = 120/√(20) = 26.83.

We need to find the probability that the sample mean falls between 600 and 700 dollars. Let x be the sample mean. Then:

Z1 = (600 - μ) / (σ / √(n)) = (600 - 650) / (120 / √t(20)) = -1.77

Z2 = (700 - μ) / (σ / √(n)) = (700 - 650) / (120 / √(20)) = 1.77

Using a standard normal distribution table or calculator, we can find the area under the standard normal distribution curve between these two Z-scores as:

P(-1.77 < Z < 1.77) = 0.9208

Therefore, the probability that a simple random sample of 20 students will spend an average of between 600 and 700 dollars is 0.9208, or approximately 0.92081 when rounded to five decimal places.

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