what advances in mathematics have helped form a better approximation of pi

a. The magnitude of the magnetic field at a distance x from the middle of the dipole along the axis connecting the north and south poles is given by \(\[ B = \frac{{\mu_0}}{{4\pi}} \frac{{2p}}{{(x^2 + d^2)^{3/2}}} \]\).

b. The magnetic flux Φ through the area of the coil in the one-dimensional solenoid is \(\[ \Phi = \frac{{\mu_0 N^2 A}}{{2a}} \]\)), where N is the number of loops, A is the cross-sectional area, and a is the half-length of the solenoid.

a. To determine the magnitude of the magnetic field at a distance x from the middle of the dipole along the axis connecting the north and south poles, we can use the formula for the magnetic field due to a magnetic dipole.

The magnitude of the magnetic field B at that point is given by:

\(\[ B = \frac{{\mu_0}}{{4\pi}} \frac{{2p}}{{(x^2 + d^2)^{3/2}}} \]\)

where p is the magnetic strength of each pole, d is the distance between the poles, and μ0 is the permeability of free space.

b. For the solenoid, to calculate the magnetic flux Φ through the area of the coil, we can use the formula for the magnetic flux through a solenoid.

Assuming one-dimensional geometry along the x-axis, the magnetic flux Φ is given by:

\(\[ \Phi = \frac{{\mu_0 N^2 A}}{{2a}} \]\)

where N is the number of loops, A is the cross-sectional area of the solenoid, a is the half-length of the solenoid, and μ0 is the permeability of free space.

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Step-by-step explanation:

If t and w are inversely proportional, it means that as one variable increases, the other variable will decrease by the same factor. In this case, we have the equation t = 4/w, which means that if we multiply w by a certain factor, t will be divided by the same factor.

If we multiply w by 6, the new value of w will be 6 times the original value:

w' = 6w

To find the new value of t, we can substitute w' into the equation:

t' = 4/w'

t' = 4/(6w)

t' = 2/3w

So we see that t has been divided by 6/1 = 6. Therefore, the value of t will decrease by a factor of 6 if the value of w is multiplied by 6.

Answer:

40meters per minutes

Step-by-step explanation:

Answer:

40m/min

Step-by-step explanation:

200m/5min

40m/min

Hope this helps!

• Domain: All real numbers

• Range: (-∞, ∞)

• To find the local minimum/maximum and intervals of increase/decrease, we need to take the first and second derivatives of the function:

First derivative: -4x^3 + 6x + 2

Setting this equal to zero and solving for x, we get critical points at x ≈ -1.23, x ≈ 0.65, and x ≈ 1.23.

Second derivative: -12x^2 + 6

• At x ≈ -1.23, the second derivative is negative, so we have a local maximum.

• At x ≈ 0.65, the second derivative is positive, so we have a local minimum.

• At x ≈ 1.23, the second derivative is negative, so we have a local maximum.

• As x —> -∞, the function approaches ∞.

• As x —> ∞, the function approaches ∞.

• To find the x-intercepts, we set the function equal to zero and solve for x:

-x^4+3x^2+2x+2 = 0

This is a quartic equation that can be solved using various methods, such as factoring or using the quadratic formula. One solution is x ≈ -1.38. The other three solutions are complex numbers.

• To find the y-intercept, we set x = 0:

-y^4 + 2 = 0

Solving for y, we get y ≈ ±1.19. So the y-intercepts are (0, 1.19) and (0, -1.19).

Answer:

Domain- (−∞,∞),{x|x∈R}

Range-  (−∞,17+6√3(over)4], {y∣y≤17+6√3(over)4}

Local Minimum/Maximum- (−1,2) is a local maxima

(1+√3(over)2 ,17+6√3(over)4) is a local maxima

(1−√3(over)2 ,17−6√3(over)4)is a local minima

Intervals of Increase/Decrease- Increasing on: (−∞,−1)(1−√3(over)2,1+√3(over)2)

Decreasing on: (−1,1−√3(over)2),(1+√3(over)2,∞)

Step-by-step explanation:

i hope this helps !

The Length of the conjugate axis is equal to 2b. The transverse axis is an essential feature of a hyperbola, as it determines the overall shape of the hyperbola.

In a hyperbola, the name of the green segment is called the transverse axis. The transverse axis is the longest distance between any two points on the hyperbola, and it passes through the center of the hyperbola. It divides the hyperbola into two separate parts called branches.

The transverse axis of a hyperbola lies along the major axis, which is perpendicular to the minor axis. Therefore, it is also sometimes called the major axis.

The other axis of a hyperbola is called the conjugate axis or minor axis. It is perpendicular to the transverse axis and passes through the center of the hyperbola. The length of the conjugate axis is usually shorter than the transverse axis.In the hyperbola above, the green segment is the transverse axis, and it is represented by the letters "2a". Therefore, the length of the transverse axis is equal to 2a.

The blue segment is the conjugate axis, and it is represented by the letters "2b".

Therefore, the length of the conjugate axis is equal to 2b.The transverse axis is an essential feature of a hyperbola, as it determines the overall shape of the hyperbola. In particular, the distance between the two branches of the hyperbola is determined by the length of the transverse axis.

If the transverse axis is longer, then the branches of the hyperbola will be further apart, and the hyperbola will look more stretched out. Conversely, if the transverse axis is shorter, then the branches of the hyperbola will be closer together, and the hyperbola will look more compressed.

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Answer: Maggie went to the store and bought 3 cartons of organic eggs for $23, and a bag of flour for $7. She had $15 dollars in her bank account. But the checkout total was $30. So the bank charged her credit card of her $15 dollars. Now she has a -$15 dollar fine that needs to be paid to her card.

Step-by-step explanation:

Maggie went to the store and bought 3 cartons of organic eggs for $23, and a bag of flour for $7. She had $15 dollars in her bank account. But the checkout total was $30. So the bank charged her credit card of her $15 dollars. Now she has a -$15 dollar fine that needs to be paid to her card.

To order the values ​​from least to greatest, we must take into account some important rules, for example.

1.a whole number is greater than a decimal number

2. The greater the decimal number that has more in its integer part.

3.The number placed furthest to the right of the number line is greater. For example +2 is greater than -1. -2 is greater than -3

Now taking into account these rules, we are going to order the numbers given in the exercise from lowest to highest;

-35

-4.3

1.21

2.7

III ;which is equal to 3 integers

4.6

Answer: The difference between 125 million dollars and 312.5 million dollars is, 187.5 Million Dollars.

Answer:

-26= 15 x -15

Step-by-step explanation:

Sorry the question wasn't really clear??

Answer:

(-13)^2= (15* -11 + -4) = -169

Step-by-step explanation:

Replace all occurrences of + − with a single − . A plus sign followed by a minus sign has the same mathematical meaning as a single minus sign

because1 x −1 = −1

(−13)^2=15 x −11 −4

Raise − 13 to the power of 2 .

169=15 x −11 −4

Multiply 15 by −11.

169 = −165 − 4

and finally Subtract 4 from−165.

169 = −169

if its (-13)^2 / (15*-11+-4) as the orginal post did not say divided, the answer would be -1 hope his helps.

Answer:

The finance charge is rounded to the nearest cent.

Answer:

Step-by-step explanation: The length of q to the nearest 10th of a centimeter is 0.2 cm.

Describe Law of Sines?

The Law of Sines is a trigonometric formula used to find the missing sides or angles of a non-right triangle (a triangle that does not have a 90-degree angle). It states that the ratio of the length of a side of a triangle to the sine of the opposite angle is the same for all three sides and angles:

a/sin(A) = b/sin(B) = c/sin(C)

where:

a, b, c are the lengths of the sides of the triangle, and

A, B, C are the measures of the angles opposite to those sides.

This formula is useful when we have some information about the sides or angles of a triangle, but not enough to solve it completely. By using the Law of Sines, we can set up and solve equations to find the missing values.

Using the Law of Sines, we have:

q / sin(2°) = p / sin(149°)

Solving for q:

q = p * sin(2°) / sin(149°)

= 6.3 * sin(2°) / sin(149°)

≈ 0.23 cm

Therefore, the length of q to the nearest 10th of a centimeter is 0.2 cm.

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To model 9/6 visually, you can draw a rectangle or a circle and divide it into six equal parts. Then, shade in 9 of those parts to represent 9/6. Alternatively, you could draw three halves (1/2 + 1/2 + 1/2) and group them together to represent 3/2. Then, multiply 3/2 by 6 to get 9/6.

what is rectangle?

A rectangle is a four-sided flat shape with opposite sides that are parallel and equal in length. It is a type of quadrilateral, which means it has four sides. The opposite sides of a rectangle are parallel and have the same length, while the other two sides are also equal in length but not parallel. The angles between the sides are all right angles (90 degrees). A rectangle is often represented as ABCD, where AB and CD are parallel and equal in length, and BC and AD are also parallel and equal in length. The area of a rectangle is calculated by multiplying the length and the width of the rectangle.

what is circle?

A circle is a simple closed shape in Euclidean geometry, consisting of all points in a plane that are a fixed distance, called the radius, from a given point, called the center. The distance around a circle is known as its circumference, and the distance across a circle through its center is known as its diameter. The ratio of the circumference of a circle to its diameter is a constant value known as pi (π), which is approximately 3.14159. Circles are a fundamental geometric shape and are used in many areas of mathematics, science, and everyday life.

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Answer:

The answer is "0.35"

Step-by-step explanation:

Please find the complete question in the attached file.

In the given table the distributions of probabilities provided is a continuous spread for both the random variable c.

They need to have a cumulative chance of receiving claims of 4000 or higher to determine the probability of an insurance company ending let's give claims of at least 4000.

So, The right equation for the answer will then be \(0.13 + 0.22=0.35\)

Answer: Sixteen is a rational number

If 66 sixth graders are randomly selected, the probability that the sample mean would be greater than 92.25 wpm is approximately 0.2142 or 21.42%.

If the sample size is sufficient, we can apply the theorem of central limitation to determine the pattern of distribution of sample means. Since n = 66 is a big enough number in this situation, we may use a normal distribution to roughly comparable the distribution of the sample means.

The general population's mean, which is 89 wpm, is the same as the mean value of the sample means. The following formula can be used to identify the average deviation of the sample means, commonly referred to as the standard deviation or error of the mean:

Standard error is equal to standard deviation squared.(sample size) Standard deviation: 16 squared(66) standard deviation: 1.969 Using the following formula, we can now standardise the sample mean: The formula for z is (sample mean - population mean) / standard error. z = (92.25 - 89) / 1.969 z = 1.732

We may determine the probability when a standard normal random variable is greater than 1.732 using the standard normal distribution table or calculator. This likelihood is roughly 0.0429, or 4.29%. we must remember that rather than just looking for any random variable, we are seeking for the probability that a sample mean is higher than 92.25 wpm. As a result, we must use the sample mean distribution, which we roughly categorized as a normal distribution.

The z-score must then be converted using the following formula to its original units of measurement: Sample mean equals population mean plus z times the standard error. 89 + 1.732 * 1.969 is the sample mean.

92.25 is the sample mean.

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Answer:

They used 6 vans and 3 buses

Step-by-step explanation:

x + y= 9

13x + 40y= 198

x= 9 - y

13(9 - y) + 40y= 198

117 - 13y + 40y= 198

27y= 81

y= 3

9-3= 6

9514 1404 393

Answer:

  x² -11x +18 = (x -9)(x -2)

Step-by-step explanation:

The x-coefficient is the sum of the constants in the binomial factors; the constant term is their product. It usually works well to look for factors of the constant that have the right sum. Here, both must be negative.

  18 = (-1)(-18) = (-2)(-9) = (-3)(-6)

The factors -2 and -9 have a sum of -11, so those are the binomial constants of interest. The factorization is ...

  x² -11x +18 = (x -9)(x -2)

Answer: 27 students

Step-by-step explanation:

First take 45 + 90 = 135 students total

20% = 0.2

Then take 135 times 0.2 = 27 students

The solid figure that has 8 edges and four triangular faces is a pyramid.

Answer:

fx) = \(3x+4\)

Step-by-step explanation:

1. step:

solve the bracket

fx) = 3(\(x\) - 1) 2 + 5

\(fx) = 3x-1+2+5\)

2. step:

use the BEDMAS form.

fx) = \(3x-3+7\)

fx) = \(3x+4\)

Answer:

2y

Step-by-step explanation:

2/3 y  +  4/3 y

= (2/3 + 4/3) y

= y (2 + 4) / 3

= y(6)/3

= 2y

The area to the right of z = 2.01 is 0.0222, which can be found using a standard normal distribution table.

Explanation:

To find the area to the right of z = 2.01, we need to use a standard normal distribution table. A standard normal distribution has a mean of 0 and a standard deviation of 1. The table gives the area under the curve to the left of a given z-score.

Since we want the area to the right of z = 2.01, we need to subtract the area to the left of z = 2.01 from 1. Using the table, we can find that the area to the left of z = 2.01 is 0.9778. Therefore, the area to the right of z = 2.01 is 1 - 0.9778 = 0.0222.

The area to the right of z = 2.01 represents the probability that a standard normal random variable takes on a value greater than 2.01 standard deviations above the mean. This can be used in various applications, such as hypothesis testing or confidence interval estimation, where we want to find the probability of observing a sample mean that is a certain number of standard deviations away from the population mean.

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In a coordinate system, a vector is oriented at an angle this statement is false.

In a coordinate system, vectors are typically represented by their components along the coordinate axis . These components determine the direction and magnitude of the vector. The orientation of a vector is determined by the angles it makes with the coordinate axis or by the direction cosines or direction angles that specify its direction in space.

The direction of a vector can be represented using angles, such as the azimuthal angle (horizontal angle) and inclination angle (vertical angle) in spherical coordinates or the angles of inclination from the positive x-axis and positive z-axis in cylindrical coordinates.

Therefore, it is incorrect to say that a vector is oriented at an angle in a coordinate system. The orientation of a vector is defined by its components or direction angles relative to the coordinate axis.

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True. In a coordinate system, a vector can be oriented at any angle with respect to the reference axis.

In a coordinate system, a vector can be oriented at any angle with respect to the reference axis. The angle at which a vector is oriented is measured with respect to a reference axis, usually the positive x-axis. This angle is typically measured in degrees or radians.

The orientation of a vector can be described using trigonometric functions such as sine, cosine, and tangent. These functions relate the angle of orientation to the coordinates of the vector in the coordinate system. For example, the x-coordinate of a vector can be determined using the cosine function, while the y-coordinate can be determined using the sine function.

The orientation of a vector can also be represented using direction angles. In two-dimensional space, the orientation of a vector can be described using a single angle. In three-dimensional space, the orientation of a vector can be described using direction angles, which are the angles between the vector and each of the coordinate axes.

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Step-by-step explanation:

\( {x}^{2} - 3x + 4 + 4 {x}^{2} - (x + 12)\)

\(5 {x}^{2} - 4x - 8\)

Answer:

$30

Step-by-step explanation:

12.50/5=2.50

2.50*12=30

The area of the segment of the circle is 13.9cm²

To calculate the area of segment of the circle, we can use the formula which is given as;

A = (1/2) * r² * [(π/180)θ - sinθ]

In the problem, our data given are;

Substituting the values into the formula;

A = (1/2) * 6² * [(π/180)45 - sin45]

A = 0.5 * 36 * [(3.14/180)45 - sin45]

A = 0.5 * 36 * 0.773

A = 13.9cm²

The segment area is 13.9 squared meters

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The partial derivative diffusivity equation for spherical flow is ∂²p/∂r² + (1/r) ∂p/∂r = ϕμC/k ∂p/∂t, and for hemispherical flow, it is the same equation.

1. The partial derivative diffusivity equation for spherical flow is derived from the spherical coordinate system and applies to radial flow in a spherical geometry. It can be expressed as ∂²p/∂r² + (1/r) ∂p/∂r = ϕμC/k ∂p/∂t.

2. The partial derivative diffusivity equation for hemispherical flow is derived from the hemispherical coordinate system and applies to radial flow in a hemispherical geometry. It can be expressed as ∂²p/∂r² + (1/r) ∂p/∂r = ϕμC/k ∂p/∂t.

1. For the derivation of the partial derivative diffusivity equation for spherical flow, we consider a spherical coordinate system with the radial direction (r), the azimuthal angle (θ), and the polar angle (φ). By assuming steady-state flow and neglecting the other coordinate directions, we focus on radial flow. Applying the Laplace operator (∇²) in spherical coordinates, we obtain ∇²p = (1/r²) (∂/∂r) (r² ∂p/∂r). Simplifying this expression, we arrive at ∂²p/∂r² + (1/r) ∂p/∂r.

2. Similarly, for the derivation of the partial derivative diffusivity equation for hemispherical flow, we consider a hemispherical coordinate system with the radial direction (r), the azimuthal angle (θ), and the elevation angle (ε). Again, assuming steady-state flow and neglecting the other coordinate directions, we focus on radial flow. Applying the Laplace operator (∇²) in hemispherical coordinates, we obtain ∇²p = (1/r²) (∂/∂r) (r² ∂p/∂r). Simplifying this expression, we arrive at ∂²p/∂r² + (1/r) ∂p/∂r.

In both cases, the term ϕμC/k ∂p/∂t represents the source or sink term, where ϕ is the porosity, μ is the fluid viscosity, C is the compressibility, k is the permeability, and ∂p/∂t is the change in pressure over time.

These equations are commonly used in fluid mechanics and petroleum engineering to describe radial flow behavior in spherical and hemispherical geometries, respectively.

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Answer:

x = -4 1/5

Step-by-step explanation:

5x + 14 = -7

5x = -21

-21/5 = -4 1/5

Hope this is correct. Have a great day.

Answer:

\( \frac{ - 21}{5} \: or \: - 4 \frac{1}{5} \: or \: - 4.2\)

Step-by-step explanation:

\(5x+14=−7 \\ 5x=−7−14 \\ 5x=−21 \\ x = \frac{ - 21}{5} \\ x = - 4.2\)

In Mixed fraction.

\(x = - 4 \frac{1}{5} \)