In
the context of mechanical vibrations, explain and differentiate
dynamic instability and resonance phenomena related to vibrations
of linear systems.

The distance between the points (1, 2) and (1, -10) is 12 square units

We have to find the distance between (1, 2) and (1, -10)

The length along a line or line segment between two points on the line or line segment.

Distance=√(x₂-x₁)²+(y₂-y₁)²

=√(1-1)²+(-10-2)²

=√-12²

=√144

=12 square units

Hence,  the distance between (1, 2) and (1, -10) is 12 square units

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Answer:

y = 3/2x-14

Step-by-step explanation:

      The given line is y=-2/3x. So, the slope of the given line is -2/3.

Now, we have to find the perpendicular line to y= -2/3x passing through the point (4,-8).

       The product of two perpendicular lines is -1.

m1.m2 = -1.

-2/3.m2= -1

m2 = 3/2

        Now, we need to find the equation of the line passing through the point (4,-8) with slope 3/2.

The equation of slope-point form is (y-y1) = m(x-x1)

        y-(-8) = 3/2 (x-4)

        y+8 = 3/2x -6

Now, we have to add 6 on both sides.

        y + 8 + 6 = 3/2x - 6 + 6.

        y + 14 = 3/2x

        y = 3/2x - 14.

The sphere is tangent to all 12 edges, meaning that it just touches each edge at one point without intersecting it.
First, we need to find the radius of the sphere. Since the sphere is tangent to each edge, it can be thought of as inscribed within the cube.

Drawing a diagonal of the cube creates a right triangle with legs of length 6. Using the Pythagorean theorem, we find that the length of the diagonal is $6\sqrt{3}$.
Since the sphere is inscribed within the cube, its diameter is equal to the diagonal of the cube. Therefore, the radius of the sphere is half of the diagonal, which is $\frac{1}{2}(6\sqrt{3}) = 3\sqrt{3}$.

Now that we have the radius of the sphere, we can use the formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3$. Substituting in the value for the radius, we get:

$V = \frac{4}{3}\pi (3\sqrt{3})^3 \approx 113.10$

So the volume of the sphere is approximately 113.10 cubic units.
To find the volume of the sphere tangent to all 12 edges of a cube, we'll first need to determine the sphere's radius.

1. Consider the cube with edge length 6. Let's focus on one of its vertices.
2. At this vertex, there are 3 edges, each tangent to sphere S.
3. Since the sphere is tangent to all these edges, they form a right-angled triangle inside the sphere, with the edges being its legs and a diameter of the sphere being its hypotenuse.
4. Let r be the radius of sphere S.
5. Using the Pythagorean theorem, we have: (2r)^2 = 6^2 + 6^2 + 6^2
6. Simplifying, we get: 4r^2 = 108
7. Solving for r, we have: r^2 = 27, so r = √27

Now, we can find the volume of the sphere using the formula:

Volume = (4/3)πr^3

8. Substitute the value of r into the formula: Volume = (4/3)π(√27)^3
9. Simplifying, we get: Volume ≈ 36π(√27)

Thus, the volume of sphere S tangent to all 12 edges of the cube with edge length 6 is approximately 36π(√27) cubic units.

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The magnitude and direction of the vector that represents the straight path from Aaron's home to the park are approximately 8.5 km and N 34° E, respectively.

We can solve this problem by using vector addition. Let's break down Aaron's path into three vectors:

1. The first vector is 3 km at a bearing of N 30° E, which we can represent as a vector with components <2.598, 1.5>.

2. The second vector is 6 km due east, which we can represent as a vector with components <6, 0>.

3. The third vector is 4 km at a bearing of N 50° E, which we can represent as a vector with components <2.828, 3.053>.

To find the vector that represents the straight path from Aaron's home to the park, we need to add these three vectors together. We can do this by adding their components:

<2.598, 1.5> + <6, 0> + <2.828, 3.053> = <11.426, 4.553>

So the vector that represents the straight path from Aaron's home to the park has a magnitude of √(11.426² + 4.553²) = 12.3 km (rounded to the nearest tenth) and a direction of tan⁻¹(4.553/11.426) = 21° (rounded to the nearest degree) north of east.

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The perimeter of the square based on the dimensions of the diagonal is 145.27 millimeters.

We will begin with calculating the side of square from the diagonal of square. It will form right angled triangle and hence the formula will be represented as -

diagonal² = 2× side²

Keep the value of diagonal

(37✓2)² = 2× side²

Side² = 2638/2

Side² = 1319

Side = ✓1319

Side = 36.32 millimetres

Perimeter of the square = 4 × side

Perimeter = 145.27 millimeters

Thus, the perimeter of the square is 145.27 millimeters.

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Answer:

cos A = 45

Step-by-step explanation:

90 - sin A = 45 = cos A

90 Minus whatever the cos value for the angle is = the sin and vice versa.

The area of the composite figure is

The area is calculated by dividing the figure into simpler shapes.

The simple shapes used here include

Area of rectangle = length x width

= 12 x 7

= 84 square ft

Area of triangle = 1/2 base x height

= 1/2 x 12 x 6

= 36 square ft

Total area

= 84 square ft + 36 square ft

=  120 square ft

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The depth of the ocean is 650 feets at the bottom of the sunlight zone.

The distance travelled by echo sound is given by the formula -

Speed = 2×distance/time

So, calculating the speed of sound from the formula using distance and time

Speed = 2×25/(1/100)

Speed = 50×1000

Speed of sound = 5000 feet/second

Now, calculating the distance or depth of ocean at the bottom of the sunlight zone -

Distance = (speed×time)/2

Distance = (5000×26/100)/2

Distance = 1300/2

Distance = 650 feets

Hence, the depth of ocean is 650 feets.

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The complex fourth roots of 3−33i are: [tex]z_0[/tex] = 3.062[tex]e^{(-21.603)}[/tex], [tex]z_1[/tex] = 1.513[tex]e^{(22.247)}[/tex], [tex]z_2[/tex] = 0.3826[tex]e^{(22.247)}[/tex] and [tex]z_3[/tex] = 1.198[tex]e^{(76.247)}[/tex].

To find the complex fourth roots of 3-33i, we can use the polar form of the complex number:

3-33i = 33∠(-86.41)

Then, the nth roots of this complex number are given by:

[tex]z_k[/tex] = [tex]33^{(1/n)}[/tex] × ∠((-86.41 + 360k)/n) for k = 0, 1, 2, ..., n-1

For n = 4, we have:

[tex]z_0[/tex] = [tex]33^{(1/4)}[/tex] × ∠(-86.41/4) ≈ 3.062∠(-21.603°)

[tex]z_1[/tex] = [tex]33^{(1/4)}[/tex] × ∠(88.99/4) ≈ 1.513∠(22.247°)

[tex]z_2[/tex] = [tex]33^{(1/4)}[/tex] × ∠(196.99/4) ≈ 0.3826∠(49.247°)

[tex]z_3[/tex] = [tex]33^{(1/4)}[/tex] × ∠(304.99/4) ≈ 1.198∠(76.247)

So the complex fourth roots of 3-33i are approximate:

[tex]z_0[/tex] = 3.062[tex]e^{(-21.603)}[/tex]

[tex]z_1[/tex] = 1.513[tex]e^{(22.247)}[/tex]

[tex]z_2[/tex] = 0.3826[tex]e^{(22.247)}[/tex]

[tex]z_3[/tex] = 1.198[tex]e^{(76.247)}[/tex]

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An estimated regression equation that can be used to predict a residence's average monthly gas bill for last year given its age is [tex]$\hat{y} = 115.14 - 3.167x$[/tex]. The average monthly gas bill for last year increases by $0.2456 on average.

Using age as the predictor variable and average monthly gas bill as the response variable, we can use linear regression to develop an estimated regression equation:

[tex]$\hat{y} = b_0 + b_1 x$[/tex]

where [tex]\hat{y}[/tex] is the predicted average monthly gas bill, x is the age of the residence, b₀ is the intercept and b₁ is the slope.

Using the given data, we can find the values of b₀ and b₁:

[tex]$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 17.6$[/tex]

[tex]$\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 55.906$[/tex]

[tex]$s_x = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} = 8.564$[/tex]

[tex]$s_y = \sqrt{\frac{\sum_{i=1}^{n} (y_i - \bar{y})^2}{n-1}} = 24.193$[/tex]

[tex]$r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2} \sqrt{\sum_{i=1}^{n} (y_i - \bar{y})^2}} = -0.577$[/tex]

[tex]$b_1 = r \frac{s_y}{s_x} = -3.167$[/tex]

[tex]$b_0 = \bar{y} - b_1 \bar{x} = 115.14$[/tex]

Therefore, the estimated regression equation is:

[tex]$\hat{y} = 115.14 - 3.167x$[/tex]

where [tex]\hat{y}$[/tex] is the predicted average monthly gas bill and x is the age of the residence.

This equation suggests that as the age of the residence increases by one year, the average monthly gas bill for last year increases by $0.2456 on average.

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The probability that in the next 1,000 passengers

1 No mishandled bags: 0.0295

2 Four or fewer mishandled bags: 0.3449

3 At least one mishandled bag:  0.9705.

4 At least two mishandled bags:  0.8672

1. To discover the likelihood of no misused sacks within the next 1000 travelers, able to utilize the Poisson dispersion equation:

P(X = 0) =[tex]e^(-λ) * (λ^0)[/tex] / 0!

Where λ is the anticipated number of misused sacks per 1000 travelers, which is rise to 3.52.

P(X = 0) =[tex]e^(-3.52) * (3.52^0)[/tex] / 0!

P(X = 0) = 0.0295

Hence, the likelihood of no misused sacks within the other 1000 passengers is 0.0295.

2. To discover the likelihood of  fewer misused packs within another 1000 travelers, we will utilize the total Poisson dispersion:

P(X ≤ 4) = Σ k=0 to 4 [[tex]e^(-λ) * (λ^k)[/tex]/ k! ]

P(X ≤ 4) = [[tex]e^(-3.52) * (3.52^0) / 0! ] + [ e^(-3.52) * (3.52^1) / 1! ] + [ e^(-3.52) * (3.52^2) / 2! ] + [ e^(-3.52) * (3.52^3) / 3! ] + [ e^(-3.52) * (3.52^4)[/tex]/ 4! ]

P(X ≤ 4) = 0.3449

Subsequently, the likelihood of fewer misused sacks within another 1000 travelers is 0.3449.

3. To discover the likelihood of at least one misused sack within the following 1000 travelers, able to utilize the complementary likelihood:

P(X ≥ 1) = 1 - P(X = 0)

P(X ≥ 1) = 1 - 0.0295

P(X ≥ 1) = 0.9705

Subsequently, the likelihood of at slightest one misused pack within the following 1000 passengers is 0.9705.

4. To discover the likelihood of at slightest two misused sacks within the other 1000 travelers, we can utilize the complementary likelihood once more:

P(X ≥ 2) = 1 - P(X = 0) - P(X = 1)

P(X ≥ 2) = 1 - 0.0295 - [ [tex]e^(-3.52) * (3.52^1)[/tex] / 1! ]

P(X ≥ 2) = 1 - 0.0295 - 0.1033

P(X ≥ 2) = 0.8672

Subsequently, the likelihood of at slightest two misused packs within the following 1000 travelers is 0.8672. 

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The mean life of the 25 items will be less than 5.9 years (rounded to one decimal place) 8% of the time.

We'll use the concepts of normal distribution, mean, standard deviation, and the z-score.

Step 1: Calculate the standard error of the mean. Standard error = (Standard deviation) / sqrt(Number of items) Standard error = 1.6 / sqrt(25) = 1.6 / 5 = 0.32 years

Step 2: Find the z-score corresponding to the 8% probability. We look for the z-score in a standard normal distribution table, which tells us that 8% of the time (0.08 probability), the z-score is approximately -1.4.

Step 3: Use the z-score formula to find the mean life (x) that corresponds to this probability. Z = (x - Mean) / Standard error -1.4 = (x - 6.3) / 0.32

Step 4: Solve for x. x - 6.3 = -1.4 * 0.32 x = 6.3 - (1.4 * 0.32) x ≈ 5.852

The mean life of the 25 items will be less than 5.9 years (rounded to one decimal place) 8% of the time.

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The volume of the cylinder is 12π units³.

Option A is the correct answer.

We have,

The formula for the volume V of a cylinder with radius r and height h is:

V = πr²h

Now,

Radius = 2 units

Height = 3 units

Now,

Volume.

= πr²h

= π x 2 x 2 x 3

= 12π units³

Thus,

The volume of the cylinder is 12π units³.

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There are 59,049 possible 5-digit PINs where adjacent digits cannot be identical, and you are permitted to use digits 0-9.

To determine the number of possible 5-digit PINs where adjacent digits cannot be identical and using the digits 0-9, follow these steps:

Step 1: Consider the first digit. Since there are no restrictions, you have 10 choices (0-9).

Step 2: For the second digit, you can't have it identical to the first digit. Therefore, you have 9 choices left.

Step 3: For the third digit, it can't be identical to the second digit. So, you again have 9 choices.

Step 4: Similarly, for the fourth digit, you have 9 choices.

Step 5: Finally, for the fifth digit, you have 9 choices.

Now, multiply the choices for each digit together: 10 × 9 × 9 × 9 × 9 = 59,049.

So, there are 59,049 possible 5-digit PINs where adjacent digits cannot be identical, and you are permitted to use digits 0-9.

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The equivalent system of first-order differential equations for the given problem is: 1. dv1/dt = v2 2. dv2/dt = v3 3. dv3/dt = 2v2 - v1 + 1 + t*e ^t

Given differential equation: x''' - 2x'' + x' = 1 + t*e ^t

Step 1: Define new variables.
Let's introduce new variables:
v1 = x'
v2 = v1'
v3 = v2'

Now we have:
v1 = x'
v2 = v1'
v3 = v2'

Step 2: Rewrite the given equation using new variables.
Substitute the new variables into the given differential equation:
v3 - 2v2 + v1 = 1 + t*e ^t

Step 3: Write the equivalent system of first-order differential equations.
Now we have the following equivalent system of first-order differential equations:
dv1/dt = v2
dv2/dt = v3
dv3/dt = 2v2 - v1 + 1 + t*e ^t

So, the equivalent system of first-order differential equations for the given problem is:

1. dv1/dt = v2
2. dv2/dt = v3
3. dv3/dt = 2v2 - v1 + 1 + t*e ^t

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The likelihood of randomly selecting a terrier from the shelter would be unlikely. That is option B

The formula that can be used to determine the probability of a selected event is given as follows;

Probability = possible event/sample space.

The possible sample space for terriers = 15%

Therefore the remaining sample space goes for Labradors and Golden Retrievers which is = 75%

Therefore, the probability of selecting the terriers at random is unlikely when compared with other dogs.

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To solve an equation, you need to isolate the variable or get it alone on one side of the equation.

Finding the value of the variable is the fundamental goal when solving an equation. You can achieve this by placing the variable alone on one side of the equation or by isolating it. A number of mathematical procedures must be carried out in order to accomplish this while maintaining the equality of the equation and simplifying the expression containing the variable.

The secret is to alter the equation so that the variable term is on its own by adding, removing, multiplying, or dividing both sides by the same number. By using the necessary mathematical procedures, the solution can be derived after the variable has been isolated.

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The balloon that was farther from the town at the beginning, and which traveled more quickly is option D. Tasha's balloon was farther from the town at the beginning, but Henry's balloon traveled more quickly.  

In order for us to know or to figure out which balloon had a faster journey, we can employ the speed equation:

Speed: Distance divided by time

Note that from the question, Henry's balloon was one that covered a distance of  16 miles within a span of 2 hours resulting in its velocity being 8 miles per hour and  Tasha's balloon was situated y = 5x + 25 miles away from the town.

Theis mean that its distance from the town would be y = 5(2) + 25 = 35 miles, after a duration of 2 hours. So, Tasha's balloon covered a distance of 10 miles within a span of 2 hours, showing a speed of 5 mph.

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See text below

Two hot air balloons are traveling along the same path away from a town, beginning from different locations at the same time. Henry's balloon begins 15 miles from the town and is 31 miles from the town after 2 hours. The distance of Tasha's balloon from the town is represented by the function y = 5x+25.

Which balloon was farther from the town at the beginning, and which traveled more quickly?

A. Tasha's balloon was farther from the town at the beginning, and it traveled more quickly.

B. Henry's balloon was farther from the town at the beginning, and it traveled more quickly.

C. Henry's balloon was farther from the town at the beginning, but Tasha's balloon traveled more quickly.

D. Tasha's balloon was farther from the town at the beginning, but Henry's balloon traveled more quickly.

The value of the variable x for the arc angle m∠AB and the angle it subtends at the center of the circle C is equal to 73.

The angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference. Also the arc measure and the angle it subtends at the center of the circle are directly proportional.

Thus:

3x - 46 = x + 98

3x - x = 98 + 46 {collect like terms}

2x = 146

x = 146/2 {divide through by 2}

x = 73

Therefore, the value of the variable x for the arc angle m∠AB and the angle it subtends at the center of the circle C is equal to 73.

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Answer:

The slope is 5.

Step-by-step explanation:

We are given the points (-2, -5) and (0,5).

We want to find the slope between these 2 points.

The slope (m) is written with the formula [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], where [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] are points.

Let's label the values of the points to avoid any confusion and mistakes when calculating.

[tex]x_1=-2\\y_1=-5\\x_2=0\\y_2=5[/tex]

Now, substitute into the formula.

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m = \frac{5--5}{0--2}[/tex]

This can be simplified to:

[tex]m = \frac{5+5}{0+2}[/tex]

Add the values together.

[tex]m = \frac{10}{2}[/tex]

m = 5

The slope is 5.

The requested probabilities are: A. P(X > 1) ≈ 0.628;                                  B. P(X > 0.36) ≈ 0.844; C. P(X < 0.47) ≈ 0.226;                                              D. P(0.32 < X < 2.46) ≈ 0.524

The probability density function of an exponentially distributed random variable with parameter lambda is given by:

f(x) = lambda * e^(-lambda * x), for x >= 0

The cumulative distribution function (CDF) of X is given by:

F(x) = P(X <= x) = 1 - e^(-lambda * x), for x >= 0

Using the given value of lambda = 0.47, we can solve for each probability as follows:

A. P(X > 1) = 1 - P(X <= 1) = 1 - (1 - e^(-0.47 * 1)) = e^(-0.47) ≈ 0.628

B. P(X > 0.36) = 1 - P(X <= 0.36) = 1 - (1 - e^(-0.47 * 0.36)) = e^(-0.1692) ≈ 0.844

C. P(X < 0.47) = P(X <= 0.47) = 1 - e^(-0.47 * 0.47) ≈ 0.226

D. P(0.32 < X < 2.46) = P(X <= 2.46) - P(X <= 0.32) = (1 - e^(-0.47 * 2.46)) - (1 - e^(-0.47 * 0.32)) ≈ 0.524

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The proof that of the above expression on the condition of a/b = c/d = e/f is given below.

Given: a/b = c/d = e/f

Let e/b = e/c = k

Then, a/b = k and c/d = k, so a = kb and c = kd

Now we have:

√((a⁴ +  c⁴)/  (b⁴ + d⁴))   = √(((k b) ⁴ + ( kd )⁴ )/(b ⁴ + d ⁴)  )

= √ (k ⁴ *  (b⁴ + d⁴ ) / (b⁴ + d⁴))

= k²

Let p = 1 and q = k², then:

(p  a² + q * c²)/(p * b² + q * d²) = (a² + k² * c²)/(b² + k⁴ * d²)

= (k² *   b² + k² *    d ²)/(b ² +   k ⁴ * d ²)

= k ²

Therefore, we have shown that √ ((a ⁴ + c ⁴)/(b ⁴ + d ⁴))   = (p x a ² + q * c ²) / (p * b ² + q * d² )

if a/b =  c/ d =  e/f.

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If the first-year teacher wants to retire in 40 years and plans to invest in an account with a 5.67% annual interest rate compounded continuously, retiring with at least $100,000 in the account, they must initially invest A) $10,352 (present value).

The present value for continuous compounding is given by the formula: P = A / e^rt.

This present value that is required to earn a future value of $100,000 can be determined using an online finance calculator as follows:

Total P+I (A): $100,000.00

Annual Rate (R) = 5.67%

Time (t in years): 40 years

Result:

P = $10,351.9

= $10,352.

Thus, to have $100,000 in 40 years, the teacher should invest $10,352 now.

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The probability of the compound event of spinning a 5 and flipping heads is 1/12.

Assuming that the spinner is fair and each outcome is equally likely, the probability of spinning a 5 is:

P(spinning 5) = number of ways to get 5 / total number of outcomes

P(spinning 5) = 1 / 6

Now, assuming that the coin is fair and has an equal probability of landing on heads or tails, the probability of flipping heads is:

P(flipping heads) = number of ways to get heads / total number of outcomes

P(flipping heads) = 1 / 2

To find the probability of the compound event of spinning 5 and flipping heads, we multiply the probability of spinning 5 by the probability of flipping heads:

P(spinning 5 and flipping heads) = P(spinning 5) x P(flipping heads)

P(spinning 5 and flipping heads) = (1/6) x (1/2)

P(spinning 5 and flipping heads) = 1/12.

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The complete question:

You spin the spinner and flip a coin. Find the probability of the compound event. The probability of spinning number 5 and flipping heads is__.

And spinner sample space is {1, 2, 3, 4, 5, 6}

Answer:

I believe (3, 5) and (10, 1) is the answer

Step-by-step explanation:

West Virginia is needed to estimate the true proportion of children living in poverty within 3% with 95% confidence

To estimate the required sample size, we can use the formula:

n = (Z^2 * p * (1-p)) / E^2

where:

Z = the Z-score associated with the desired level of confidence (95% confidence corresponds to a Z-score of 1.96)

p = the estimated proportion of the population with the characteristic of interest (in this case, the estimated proportion of children under age 6 living in poverty in West Virginia, which is 0.3)

E = the desired margin of error (in this case, 0.03)

Substituting the given values, we get:

n = (1.96^2 * 0.3 * (1-0.3)) / 0.03^2

Simplifying:

n = 601.78

Rounding up to the nearest whole number, we get:

n = 602

Therefore, a sample of at least 602 children under age 6 from West Virginia is needed to estimate the true proportion of children living in poverty within 3% with 95% confidence.

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Answer:

option B.

Step-by-step explanation:

The original price of Casey's favorite pizza was $5.50, but now it costs $6. To find the percentage markup, we can use the formula:

(markup / original price) * 100%

The markup is the difference between the new price and the original price:

$6.00 - $5.50 = $0.50

So the markup is $0.50.

Using the formula above:

(markup / original price) * 100% = ($0.50 / $5.50) * 100% = 9.09%

Therefore, the pizza was marked up by about 9%, which is option B.

The general solution of the given differential equation is:
y = (r^(1/2)) * (1 + Ce^(2r^(1/2))) or y = (r^(1/2)) * (-1 + Ce^(2r^(1/2)))
where C is the constant of integration.

To find the general solution of r = y^2 - 1 dr, we need to separate the variables and integrate both sides. We can start by rearranging the equation as:
dr/(y^2 - 1) = dy/r

Now, we can integrate both sides. On the left side, we can use partial fractions to make the integration easier. We can write:
dr/(y^2 - 1) = [1/(2*(y-1))] - [1/(2*(y+1))] dy

Integrating both sides, we get:
1/2 * ln|y-1| - 1/2 * ln|y+1| = ln|r| + C
where C is the constant of integration.

We can simplify this as:
ln|(y-1)/sqrt(r)| - ln|(y+1)/sqrt(r)| = 2C

Using logarithmic properties, we can simplify further as:
ln|[(y-1)/sqrt(r)] / [(y+1)/sqrt(r)]| = 2C
ln|[(y-1)/(y+1)]| = 2C

Exponentiating both sides, we get:
|[(y-1)/(y+1)]| = e^(2C)

Taking the positive and negative cases separately, we get:
(y-1)/(y+1) = e^(2C)

or
(y-1)/(y+1) = -e^(2C)

Solving for y in each case, we get the general solution as:

y = (r^(1/2)) * (1 + Ce^(2r^(1/2))) or y = (r^(1/2)) * (-1 + Ce^(2r^(1/2)))

where C is the constant of integration. This is the general solution of the given differential equation.

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The equation of the plane passing through point P with normal vector 2i + 6j + 7k is x + 3y + 4z = 48.

A: True.

The equation of a plane in 3D space is given by Ax + By + Cz = D, where A, B, C are the components of the normal vector and D is the distance from the origin to the plane along the direction of the normal vector.

In this case, the normal vector is 2i + 6j + 7k, so A = 2, B = 6, and C = 7. To find D, we can substitute the coordinates of the given point P into the equation of the plane:

2(1) + 6(3) + 7(4) = D

2 + 18 + 28 = D

D = 48

Therefore, the equation of the plane passing through point P with normal vector 2i + 6j + 7k is x + 3y + 4z = 48.

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Clark would spend about $420 on these items in a year.

Given that Clark finds that in an average month, he spends $35 on things he really doesn't need and can't afford.

If Clark spends $35 on things he doesn't need or can't afford in an average month, then he would spend:

$35/month x 12 months/year = $420/year

Therefore, Clark would spend about $420 on these items in a year.

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