Work Shown:
21/35 = (7*3)/(7*5) = 3/5
Consider a medium with parameters € = 1.2 (10^-10 )F/m , n= 3(10^-3) H/m and sigma=0. Magnetic field intensity in the medium is given as R = 2cos (10^10t- 600x)äz Am.
Use Maxwell's equations to obtain the followings:
1) Magnetic flux density
These questions is circuit theory
Using Maxwell's equations, we can determine the magnetic flux density. One of the Maxwell's equations is:
[tex]\displaystyle \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}[/tex],
where [tex]\displaystyle \nabla \times \mathbf{H}[/tex] is the curl of the magnetic field intensity [tex]\displaystyle \mathbf{H}[/tex], [tex]\displaystyle \mathbf{J}[/tex] is the current density, and [tex]\displaystyle \frac{\partial \mathbf{D}}{\partial t}[/tex] is the time derivative of the electric displacement [tex]\displaystyle \mathbf{D}[/tex].
In this problem, there is no current density ([tex]\displaystyle \mathbf{J} =0[/tex]) and no time-varying electric displacement ([tex]\displaystyle \frac{\partial \mathbf{D}}{\partial t} =0[/tex]). Therefore, the equation simplifies to:
[tex]\displaystyle \nabla \times \mathbf{H} =0[/tex].
Taking the curl of the given magnetic field intensity [tex]\displaystyle \mathbf{R} =2\cos( 10^{10} t-600x)\hat{a}_{z}\, \text{Am}[/tex]:
[tex]\displaystyle \nabla \times \mathbf{R} =\nabla \times ( 2\cos( 10^{10} t-600x)\hat{a}_{z}) \, \text{Am}[/tex].
Using the curl identity and applying the chain rule, we can expand the expression:
[tex]\displaystyle \nabla \times \mathbf{R} =\left( \frac{\partial ( 2\cos( 10^{10} t-600x)) \hat{a}_{z}}{\partial y} -\frac{\partial ( 2\cos( 10^{10} t-600x)) \hat{a}_{z}}{\partial z}\right) \mathrm{d} x\mathrm{d} y\mathrm{d} z[/tex].
Since the magnetic field intensity [tex]\displaystyle \mathbf{R}[/tex] is not dependent on [tex]\displaystyle y[/tex] or [tex]\displaystyle z[/tex], the partial derivatives with respect to [tex]\displaystyle y[/tex] and [tex]\displaystyle z[/tex] are zero. Therefore, the expression further simplifies to:
[tex]\displaystyle \nabla \times \mathbf{R} =-\frac{\partial ( 2\cos( 10^{10} t-600x)) \hat{a}_{z}}{\partial x} \mathrm{d} x\mathrm{d} y\mathrm{d} z[/tex].
Differentiating the cosine function with respect to [tex]\displaystyle x[/tex]:
[tex]\displaystyle \nabla \times \mathbf{R} =-2( 10^{10}) \sin( 10^{10} t-600x)\hat{a}_{z} \mathrm{d} x\mathrm{d} y\mathrm{d} z[/tex].
Setting this expression equal to zero according to [tex]\displaystyle \nabla \times \mathbf{H} =0[/tex]:
[tex]\displaystyle -2( 10^{10}) \sin( 10^{10} t-600x)\hat{a}_{z} \mathrm{d} x\mathrm{d} y\mathrm{d} z =0[/tex].
Since the equation should hold for any arbitrary values of [tex]\displaystyle \mathrm{d} x[/tex], [tex]\displaystyle \mathrm{d} y[/tex], and [tex]\displaystyle \mathrm{d} z[/tex], we can equate the coefficient of each term to zero:
[tex]\displaystyle -2( 10^{10}) \sin( 10^{10} t-600x) =0[/tex].
Simplifying the equation:
[tex]\displaystyle \sin( 10^{10} t-600x) =0[/tex].
The sine function is equal to zero at certain values of [tex]\displaystyle ( 10^{10} t-600x) [/tex]:
[tex]\displaystyle 10^{10} t-600x =n\pi[/tex],
where [tex]\displaystyle n[/tex] is an integer. Rearranging the equation:
[tex]\displaystyle x =\frac{ 10^{10} t-n\pi }{600}[/tex].
The equation provides a relationship between [tex]\displaystyle x[/tex] and [tex]\displaystyle t[/tex], indicating that the magnetic field intensity is constant along lines of constant [tex]\displaystyle x[/tex] and [tex]\displaystyle t[/tex]. Therefore, the magnetic field intensity is uniform in the given medium.
Since the magnetic flux density [tex]\displaystyle B[/tex] is related to the magnetic field intensity [tex]\displaystyle H[/tex] through the equation [tex]\displaystyle B =\mu H[/tex], where [tex]\displaystyle \mu[/tex] is the permeability of the medium, we can conclude that the magnetic flux density is also uniform in the medium.
Thus, the correct expression for the magnetic flux density in the given medium is:
[tex]\displaystyle B =6\cos( 10^{10} t-600x)\hat{a}_{z}[/tex].
The darkness of the print is measured quantitatively using an index. If the index is greater than or
equal to 2.0 then the darkness is acceptable. Anything less than 2.0 means the print is too light and
not acceptable. Assume that the machines print at an average darkness of 2.2 with a standard
deviation of 0.20.
(a) What percentage of printing jobs will be acceptable? (4)
(b) If the mean cannot be adjusted, but the standard deviation can, what must be the new standard
deviation such that a minimum of 95% of jobs will be acceptable?
84.13% of the printing jobs will be acceptable.
The new standard deviation required to achieve a minimum of 95% of jobs acceptable is 0.121.
The darkness of the print is measured quantitatively using an index. If the index is greater than or equal to 2.0 then the darkness is acceptable. Anything less than 2.0 means the print is too light and not acceptable. The machines print at an average darkness of 2.2 with a standard deviation of 0.20.
The mean of the darkness of the print is µ = 2.2 and the standard deviation is σ = 0.20.Therefore, the z-score can be calculated as; `z = (x - µ) / σ`.The index required for acceptable prints is 2.0. Thus, the percentage of prints that are acceptable can be calculated as follows;P(X ≥ 2.0) = P((X - µ)/σ ≥ (2.0 - 2.2) / 0.20)P(Z ≥ -1) = 1 - P(Z < -1)Using the standard normal table, P(Z < -1) = 0.1587P(Z ≥ -1) = 1 - 0.1587= 0.8413.
To find the new standard deviation, we can use the z-score formula.z = (x - µ) / σz = (2.0 - 2.2) / σz = -1Therefore, P(X ≥ 2.0) = 0.95P(Z ≥ -1) = 0.95P(Z < -1) = 0.05Using the standard normal table, the z-score value of -1.645 corresponds to a cumulative probability of 0.05. Hence,z = (2.0 - 2.2) / σ = -1.645σ = (2.0 - 2.2) / -1.645= 0.121.
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Write the English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The product of 8 and a number, which is then subtracted from the product of 17 and the number.
The algebraic expression for the given phrase is: 17x - 8x. To simplify this expression, we can combine like terms by subtracting the coefficients of x. The simplified expression is: 9x.
In the given phrase, "The product of 8 and a number" can be represented as 8x, where x represents the number. Similarly, "The product of 17 and the number" can be represented as 17x. Since we are subtracting the product of 8x from the product of 17x, the algebraic expression becomes 17x - 8x.
To simplify the expression, we combine like terms. The coefficients of x are 17 and -8. Since we are subtracting 8x from 17x, we subtract the coefficient of 8x from the coefficient of 17x, resulting in 17x - 8x. Combining like terms gives us 9x.
In conclusion, the simplified expression for the phrase "The product of 8 and a number, which is then subtracted from the product of 17 and the number" is 9x.
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Bally Manufacturing sent Intel Corporation an invoice for machinery with a $13,100 list price. Bally dated the invoice August 01 with 3/10
EOM terms. Intel receives a 40% trade discount. Intel pays the invoice on August 14. On August 10, Intel Corporation returns $100 of the machinery due to defects. What does Intel pay Bally on August 14?
The Intel pays $7,760 to Bally Manufacturing on August 14.
The first step in calculating what Intel Corporation pays Bally on August 14 is to determine the net price of the machinery after the trade discount and the return of $100 due to defects.
The trade discount of 40% is calculated as follows:
Discount = List price × Discount rate
Discount = $13,100 × 0.40 = $5,240
So the net price of the machinery after the trade discount is:
Net price = List price - Discount
Net price = $13,100 - $5,240 = $7,860
After Intel returns $100 of machinery, the cost of the machinery is further reduced to:
Net price after return = Net price - Return
Net price after return = $7,860 - $100 = $7,760
Since the payment terms are 3/10 EOM (end of month), Intel receives a discount of 3% if payment is made within 10 days. The 10-day period begins on August 1 and ends on August 10 (the payment due date). Since Intel pays the bill on August 14, payment is late and the 3% discount does not apply.
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Waiting times (in minutes) of customers at a bank where all customers enter a single waiting line and a bank where customers wait in individual lines at three different teller windows are listed below. Find the coefficient of variation for each of the two sets of data, then compare the variation.
Bank A (single line) Bank B (individual lines)
6.5 4.0
6.6 5.4
6.7 5.9
6.7 6.2
7.1 6.8
7.4 7.7
7.5 7.7
7.7 8.5
7.7 9.4
7.7 9.8
A) The coefficient of variation for the waiting times at Bank A is
(Round to one decimal place as needed.)
B) The coefficient of variation for the waiting times at Bank B is
(Round to one decimal place as needed.)
C)Is there a difference in variation between the two data sets?
The coefficient of variation for Bank A is approximately 8.04%, while the coefficient of variation for Bank B is approximately 25.55%.
To find the coefficient of variation for each set of data, we need to calculate the mean and standard deviation for each set. The coefficient of variation is then calculated by dividing the standard deviation by the mean and multiplying by 100.
Let's calculate the coefficient of variation for each set of data:
Bank A (single line):
Mean: Calculate the mean of the data set.
Mean = (6.5 + 6.6 + 6.7 + 6.7 + 7.1 + 7.4 + 7.5 + 7.7 + 7.7 + 7.7) / 10 = 7.03 minutes
Standard deviation: Calculate the standard deviation of the data set.
Standard deviation = √[(6.5 - 7.03)² + (6.6 - 7.03)² + ... + (7.7 - 7.03)²] / 10 ≈ 0.565 minutes
Coefficient of variation:
Coefficient of variation = (0.565 / 7.03) * 100 ≈ 8.04%
Bank B (individual lines):
Mean: Calculate the mean of the data set.
Mean = (4.0 + 5.4 + 5.9 + 6.2 + 6.8 + 7.7 + 7.7 + 8.5 + 9.4 + 9.8) / 10 = 7.5 minutes
Standard deviation: Calculate the standard deviation of the data set.
Standard deviation = √[(4.0 - 7.5)² + (5.4 - 7.5)² + ... + (9.8 - 7.5)²] / 10 ≈ 1.916 minutes
Coefficient of variation:
Coefficient of variation = (1.916 / 7.5) * 100 ≈ 25.55%
Comparing the variation:
The coefficient of variation for Bank A is approximately 8.04%, while the coefficient of variation for Bank B is approximately 25.55%. Since the coefficient of variation measures the relative variability of the data, we can conclude that the waiting times at Bank B (individual lines) have a higher variation compared to Bank A (single line).
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The sum of five and twice a number 49. Find the number