Please help me on a math problem it’s for a grade
Thank You
Answer:
Angle 4 and angle 8 are corresponding angles because they are the same angle on the same side.
John is making apple pies and apple cobblers to sell at his stand at the Farmer's Market.
A pie uses 4 cups of apples and 3 cups of flour.
A cobbler uses 2 cups of apples and 3 cups of flour.
John has 16 cups of apples and 15 cups of flour.
When John sells the pies and cobblers at the Farmer's Market, he will make $3.00 profit per pie and $2.00 profit per cobbler.
Let x = the number of pies John makes.
Let y = the number of cobblers John makes.
Enter the four constraints into the graphing calculator.
What are the vertices of the feasible region?
Hint: input your answers from questions 3, 4, and 5 into Desmos to find the vertices.
Answers from questions 3, 4, and 5
[tex]4x+2y≤16\\3x+3y≤15\\x≥0\\y≤0[/tex]
Answer:huh,
Step-by-step explanation:I don’t understand you’re saying
shaina makes a container of lemonades. her brother drinks 1/4 of it. her father then drinks 2/3 of the remaining 3/4. how much of the container did her father drink
Shaina made a container of lemonades and her brother drank 1/4 of it. Her father then drank 2/3 of the remaining 3/4, which is equivalent to drinking 1/2 of the container.
Shaina made a container of lemonades and her brother drank 1/4 of it, leaving 3/4 in the container. Her father then drank 2/3 of the remaining 3/4, which is equivalent to drinking 2/3 x 3/4 = 6/12 = 1/2 of the container. In other words, her father drank 1/2 of the container.
We can also solve this problem using algebra. Let 'x' represent the total amount of lemonade in the container. Then, her brother drank 1/4x, leaving 3/4x in the container. Her father then drank 2/3 of the remaining lemonade, which is equal to 2/3 x (3/4 x) = 6/12 x = 1/2x. Since her father drank 1/2x of the container, we can conclude that x = 1/2x, which means that x = 2/2x = 1. Therefore, the total amount of lemonade in the container was 1 and her father drank 1/2 of it.
To summarize, Shaina made a container of lemonades and her brother drank 1/4 of it. Her father then drank 2/3 of the remaining 3/4, which is equivalent to drinking 1/2 of the container. This can be expressed algebraically as 2/3 x (3/4 x) = 1/2x, where x = 1, the total amount of lemonade in the container.
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what are the appropriate measures of relative standing for ordinal data? percentiles and quartiles quartiles none
The appropriate measures of relative standing for ordinal data are percentiles and quartiles.
What is ordinal data?Ordinal data is a type of categorical data in which the categories are ordered, as in the case of survey data in which respondents are given the option of selecting "poor," "fair," "good," or "excellent" to describe a product, service, or other object. The ordering of the data allows for comparisons between the data sets to be made with greater accuracy than with nominal data, which simply labels data points without any order or ranking.
What are the appropriate measures of relative standing for ordinal data?Percentiles and quartiles are appropriate measures of relative standing for ordinal data. The percentile is a measure of location that specifies the percentage of observations in the distribution that fall below the value. Quartiles are measures of location that divide a dataset into four equal parts. The first quartile (Q1) represents the 25th percentile of the distribution, the second quartile (Q2) represents the 50th percentile, and the third quartile (Q3) represents the 75th percentile.
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A rope of length L is clamped at both ends. Which one of thefollowing is not a possible wavelength for standing waves on thisrope?
a. L/2
a. 2L/3
c. L
d. 2L
e. 4L
Please show all work or explain so that I can learn fromyou.
Thanks...
The not a possible wavelength for standing waves on this rope is 4L
Hence the answer is 'e'
Explanation:
A rope of length L is clamped at both ends. The wavelength of a wave on a string depends on the frequency of the wave and the speed of the wave on the string.
What is the wavelength of the wave?The wavelength of a wave on a string depends on the frequency of the wave and the speed of the wave on the string. The speed of the wave is given by
:v = √(F/u
)where v is the speed of the wave, F is the tension on the string, and u is the linear mass density of the string (mass per unit length).The frequency of the wave is given by:
f = n*v/2L
where f is the frequency of the wave, n is the number of antinodes in the standing wave pattern, and L is the length of the string.The wavelength of the wave is given by:λ = 2L/n
What is the possible wavelength for standing waves on this rope?The possible wavelengths for standing waves on this rope are given by the formula:
λ = 2L/n
where L is the length of the rope and n is a positive integer.
Therefore, the possible wavelengths for standing waves on this rope are:L/22L/32LL2L4L
The wavelength that is not possible for standing waves on this rope is option e) 4L, because it is not an integer multiple of L/2.
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State whether you agree with the following statement. Explain your reasoning.
A temperature of -11 degrees Fahrenheit is warmer than a temperature of -15 degrees Fahrenheit.
Answer:
I disagree with the statement that a temperature of -11 degrees Fahrenheit is warmer than a temperature of -15 degrees Fahrenheit.
Temperatures are usually compared in terms of their absolute values. In the case of Fahrenheit, the absolute value of a temperature can be obtained by adding 459.67 to the Fahrenheit temperature. Using this conversion, we can see that:
•A temperature of -11 degrees Fahrenheit is equal to 471.47 degrees absolute.
•A temperature of -15 degrees Fahrenheit is equal to 456.67 degrees absolute.
Since 471.47 is greater than 456.67, we can conclude that a temperature of -11 degrees Fahrenheit is colder than a temperature of -15 degrees Fahrenheit in terms of their absolute values. Therefore, a temperature of -15 degrees Fahrenheit is actually warmer than a temperature of -11 degrees Fahrenheit.
why do we use the paired t test for these data? in other words, why these two samples are matched pairs? select the most suitable answer.
Two samples are matched pairs because there are two samples and each pair in the two samples have about the same GPA indicating they have similar characteristics. Correct option is A.
This is because the paired t-test is used when we have two related samples or when the same sample is measured twice, such as before and after a treatment.
In this case, the two samples being compared are students' GPAs before and after a particular intervention or treatment. Since the same group of students are being measured twice, they are matched pairs and the paired t-test is appropriate to use.
Option B is not a suitable answer because the fact that the samples are both MBA majors does not necessarily make them matched pairs. Option C is also not a suitable answer because having equal numbers of data does not necessarily make the samples matched pairs.
Option D is not a suitable answer because it does not provide any explanation as to why the paired t-test is being used.
Therefore, correct option is A.
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Complete question is:
Why do we use the paired t test for these data? In other words, why these two samples are matched pairs? Select the most suitable answer. A. There are two samples and each pair in the two samples have about the same GPA indicating they have similar characteristics.
B. There two samples of MBA majors.
C. There are two samples with equal number of data.
D. None of these.
HELP The expression 1 × e(0.06t) models the balance, in thousands of dollars, of an account t years after the account was opened. 1. What is the account balance: a. when the account is opened? b. after 1 year? c. after 2 years? 2. Diego says that the expression In 5 represents the time, in years, when the account will have 5 thousand dollars. Do you agree? Explain your reasoning. 3. Suppose you opened this account at the beginning of this year. Assume that you deposit no additional money and withdraw nothing from the account. Will the account balance reach $1,000,000 and make you a millionaire by the time you reach retirement? Show your reasoning.
a. The account balance is $1000 when opened.
b. $1,060 is the balance after 1 year.
c. 42.98 years as per compound interest.
What is compound interest?Compound interest is a type of interest that is calculated not only on the principal amount of a loan or investment but also on the accumulated interest from previous periods. This means that interest is added to the principal amount, and the new total becomes the basis for calculating interest in the next period.
In the given question,
a. When the account is opened, t = 0. Therefore, the balance is:
1 × e(0.06 × 0) = 1 × e⁰ = 1
The account balance is $1,000 when it is opened.
b. After 1 year, t = 1. Therefore, the balance is:
1 × e(0.06 × 1) ≈ 1.06
The account balance is approximately $1,060 after 1 year.
c. After 2 years, t = 2. Therefore, the balance is:
1 × e(0.06 × 2) ≈ 1.123
The account balance is approximately $1,123 after 2 years.
Diego's statement is not accurate. The expression In 5 represents the natural logarithm of 5, which is approximately 1.609. It does not represent the time, in years, when the account will have 5 thousand dollars.
To find the time when the account will have a balance of $5,000, we need to solve the equation:
1 × e(0.06t) = 5
Dividing both sides by 1:
e(0.06t) = 5
Taking the natural logarithm of both sides:
ln(e(0.06t)) = ln(5)
0.06t = ln(5)
t = ln(5) / 0.06 ≈ 11.55
Therefore, the account will have a balance of $5,000 after approximately 11.55 years.
We can use the same equation as in part 2 to find out if the account balance will reach $1,000,000. We need to solve the equation:
1 × e(0.06t) = 1000
Dividing both sides by 1:
e(0.06t) = 1000
Taking the natural logarithm of both sides:
ln(e(0.06t)) = ln(1000)
0.06t = ln(1000)
t = ln(1000) / 0.06 ≈ 42.98
Therefore, the account balance will reach $1,000,000 after approximately 42.98 years.
Whether or not this will make you a millionaire by the time you reach retirement depends on when you plan to retire and how much money you need for retirement. If you plan to retire in less than 43 years or you need more than $1,000,000 for retirement, then this account alone will not make you a millionaire.
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Please help work this out with workings out
give the number of total electron groups, the number of bonding groups, and the number of lone pairs for geometry (a). express your answer as integers separated by commas.
Hence, the answer is (4, 3, 1).In conclusion The answer provided above is concise and ng factually correct, and it addresses the question directly.
In order to determine the number of total electron groups, bondiroups, and lone pairs for geometry (a), we need to use the VSEPR theory. According to this theory, the electron groups around a central atom in a molecule will arrange themselves in a way that minimizes their repulsion. The total number of electron groups includes both the bonding and lone pairs of electrons.To determine the number of electron groups for geometry (a), we first need to determine the molecular geometry of the molecule.
From the given name, we can assume that geometry (a) is tetrahedral. In a tetrahedral molecule, there are four electron groups: three bonding groups and one lone pair. Therefore, the number of total electron groups for geometry (a) is four, the number of bonding groups is three, and the number of lone pairs is one.
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Solve for Z =
Please answer
Answer:
z = 32°
vertical angles are equal to each other. 32° ≅ z°
Find the average rate of change for the function
Answer:
average rate of change on [-1,3] = 11
Step-by-step explanation:
avg rate of change = [tex]\frac{f(b)-f(a)}{b-a}[/tex]
[tex]\frac{f(3)-f(-1)}{3-(-1)}[/tex]
1. find your y-values.
we are given the x values from the interval [-1,3]. Plug each into the equation to get the y-value of the coordinates.
[tex]f(-1)=4(-1)^2+3(-1)-4\\f(-1)=4(1)-3-4\\f(-1)=-3\\[/tex]
coordinate: (–1,3)
[tex]f(3)=4(3)^2+3(3)-4\\f(3)=4(9)+9-4\\f(3)=36+9-4\\f(3)=41[/tex]
coordinate: (3, 41)
2. plug into the slope formula
[tex]m=\frac{y_{2} -y_{1} }{x_{2} -x_{1} } \\m=\frac{41-(-3)}{3-(-1)} \\m=\frac{41+3}{4} \\m=\frac{44}{4} \\m=11[/tex]
A nationwide coffee chain is downsizing its branches in six states It has already told the store man
the nationwide coffee chain is downsizing its branches in six states, and has already informed the store managers. The process should involve analyzing the reasons for downsizing, identifying the branches to be downsized, developing a transition plan, implementing the plan, and monitoring the progress and results.
The question asks about a nationwide coffee chain that is downsizing its branches in six states and has informed the store managers.
To answer this question, we can discuss the process of downsizing branches and the possible reasons behind it.
Analyze the reasons for downsizing
The coffee chain may be downsizing due to financial challenges, declining sales, or a shift in strategic focus. Understanding the reasons for downsizing will help in determining the best approach to manage the process.
Identify the branches to be downsized
The coffee chain must have a clear plan for which branches in the six states will be downsized. This can be based on factors such as sales performance, location, and overall profitability of each branch.
Inform the store managers
The coffee chain has already taken this step by informing the store managers about the downsizing. This is important to ensure that they are aware of the changes and can communicate this information to their employees.
Develop a transition plan
The coffee chain should develop a comprehensive plan to manage the downsizing process. This could involve offering support to affected employees, such as providing severance packages, job placement assistance, or opportunities to transfer to other branches.
Implement the downsizing plan
Once the plan has been developed, the coffee chain can begin the process of downsizing the identified branches. This may involve closing or merging some locations, reducing staff, and reallocating resources.
Monitor the progress and results
After the downsizing process is complete, the coffee chain should closely monitor the outcomes and make any necessary adjustments to ensure the desired results are achieved.
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The initial number of bacteria in a culture is 12,000 the culture doubles each day write an exponential function to model the population y of bacteria after X days which we used to determine how many bacteria present after 15 days 
The population of bacteria after 15 days is 384,000.
Describe Equation?An equation is a mathematical statement that expresses the equality of two expressions, typically separated by an equal sign (=). It consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, exponents, and roots. Equations are used to solve problems in various fields of study, such as physics, engineering, economics, and mathematics. They are also used in everyday life, such as calculating the cost of groceries or determining the time it takes to complete a task. Solving an equation involves finding the values of the variables that make the equation true.
The exponential growth model for this scenario can be written as:
y = a * (2ˣ)
Where:
y = population of bacteria after x days
a = initial population of bacteria = 12,000
x = number of days
Substituting the given values into the equation, we get:
y = 12,000 * (2ˣ)
To determine the population of bacteria after 15 days, we simply substitute x = 15 into the equation:
y = 12,000 * (2¹⁵)
y = 12,000 * 32
y = 384,000
Therefore, the population of bacteria after 15 days is 384,000.
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James deposited 10000 into an account that earns 5.5% compound interest, compounded semiannually. How much interest will James earn after 10 years?
James will earn approximately $6,639.12 in interest after 10 years.
What is simple interest?
Simple Interest (S.I.) is the method of calculating the interest amount for a particular principal amount of money at some rate of interest.
To solve this problem, we can use the formula for compound interest:
[tex]A = P(1 + r/n)^{(nt)}[/tex]
where:
A is the final amount (including interest)
P is the principal amount (initial deposit)
r is the annual interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the time (in years)
Plugging in the given values, we get:
[tex]A = 10000(1 + 0.055/2)^{(2*10)}[/tex]
[tex]A = 10000(1.0275)^{20}[/tex]
A = 10000(1.664)
A ≈ 16639.12
To find the amount of interest earned, we subtract the principal amount from the final amount:
Interest = A - P
Interest = 16639.12 - 10000
Interest ≈ 6639.12
Therefore, James will earn approximately $6,639.12 in interest after 10 years.
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here is a cylinder with hight 4 units and diameter 10 units
what is the volume of the cylinder's base?
what is the volume of this cylinder's?
Step-by-step explanation:
Diameter = 10 units then radius, r = 5 inches
Cylinder's base AREA = pi r^2 = pi (5)^2 = 25 pi = 78.54 units^2
Base area * height = volume = 25 pi * 4 = 100 pi =314.2 units^3
A walkway forms one diagonal of a square playground. The walkway is 22m long. How long is a side of the playground?
Answer:
15.556
Step-by-step explanation:
The length of the diagonal of a square is equal to the length of one side of the square multiplied by the square root of 2
22=x√2
a pizza that is 12 in diameter is cut into six slices. find the length of each arc intercepted by each slice (length of the crust). find the area of each slice.
The length of each arc intercepted by each slice is 2.09 inches.
The area of each slice 18.85 square inches .
To find the length of each arc intercepted by each slice and the area of each slice of a pizza that is 12 inches in diameter, follow the steps below. Length of each arc intercepted by each slice
The formula to find the length of each arc is given by:
L = θ/360 * 2πr
WhereL = length of arc
θ = angle of arc in degrees
r = radius of circle (diameter/2)
Given, diameter of pizza = 12 inches
So, radius of pizza = 12/2 = 6 inches
Since the pizza is cut into 6 equal slices, each slice will have an angle of 360/6 = 60 degrees.
Now, substituting the given values in the formula:
L = 60/360 * 2π* 6L = 1/6 * 12πL = 2π/3 ≈ 2.09 inches
The formula to find the area of each slice is given by:
A = θ/360 * πr²
Given, diameter of pizza = 12 inches
So, radius of pizza = 12/2 = 6 inches
Since the pizza is cut into 6 equal slices, each slice will have an angle of 360/6 = 60 degrees.
Now, substituting the given values in the formula:
A = 60/360 * π* 6²A = 1/6 * 36π
A = 6π ≈ 18.85 square inches
Thus, the length of each arc intercepted by each slice is approximately 2.09 inches and the area of each slice is approximately 18.85 square inches.
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PLEASEEE HELP ASAP MIGHT BE EASYY
Answer:
3rd choice down (the one already selected)
Step-by-step explanation:
to prove it, plug each value of x in the table into f(x) = x - 6 and check the answers with what is in the y column.
f(-5) = -5 - 6 = -11
f(-8) = -8 - 6 = -14
f(-7) = -7 - 6 = -13
all the numbers in the 3rd choice check out
The form of federalism favored by Chief Justice Roger Taney in which national and state governments are seen as distinct entities providing separate services. This model limits the power of the national government.
This model of federalism ensures that both the national and state governments are equal and that neither has the power to override the other.
The form of federalism favored by Chief Justice Roger Taney was a dual-sovereignty system, in which the national and state governments are seen as distinct entities, providing separate services and limiting the power of the national government.
Taney argued that each government should remain supreme within its own sphere, and that there should be a strict division of authority between the two levels of government.
He argued that the Constitution was a compact between states, each of which had the right to govern itself without interference from the other, and that the Constitution created the federal government only to manage matters that could not be handled by the states.
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in how many positive four-digit integers that are not multiples of $1111$ do the digits form an arithmetic sequence? (the digits must form an arithmetic sequence, in order. for example, the number $5137$ does not count.)
$8\times9\times8\times9 = 5184$
There are a total of $8\times9\times8\times9 = 5184$ positive four-digit integers that are not multiples of 1111 and form an arithmetic sequence. To explain further, the first digit can be any integer from 1 to 8 (not 0 as the number must be four-digits long). Once the first digit is chosen, the second digit can be any integer from 1 to 9. Once both the first and second digits are chosen, the third digit can be any integer from 1 to 8, and finally the fourth digit can be any integer from 1 to 9. Therefore, the total number of four-digit integers that are not multiples of 1111 and form an arithmetic sequence is $8\times9\times8\times9 = 5184$.
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shen wants to earn more than $33 trimming trees. he charges $8 per hour and pays $7 in equipment fees. what are the possible numbers of hours shen could trim trees?
The answer is 4.375 hours.
Since this is not a whole number, Shen will need to trim trees for at least 5 hours in order to earn more than 33.
In order to answer this question, we need to know how much Shen is wanting to earn and what his rate of pay and equipment fees are. You've stated that Shen wants to earn more than 33 trimming trees and that he charges 8 per hour and pays 7 in equipment fees.
The formula to solve this problem is:
Earnings = Rate of Pay x Number of Hours – Equipment Fees
Therefore, we can calculate the number of hours Shen needs to trim trees in order to earn more than 33.
Number of Hours = (Earnings + Equipment Fees) / Rate of Pay
Substituting in the values provided in the question:
Number of Hours = (33 + 7) / 8
+ 4.375 hours.
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20 out of 21 students want pancakes and the rest want waffles. What is the ratio of the number of students who want waffles to the total number of students?
Answer: 20:21 or 20 to 21
Step-by-step explanation:
Answer:
If 20 out of 21 students want pancakes, then only 1 student wants waffles.
The ratio of the number of students who want waffles to the total number of students is therefore:
1 waffle student : 21 total students
or simplified as:
1 : 21
So the ratio is 1 to 21
which of the following is a discrete data random variable? cars finished in a factory each day a person's height each birthday a person's weight on each year the volume of water in a swimming pool each day
A discrete data random variable is a random variable that can only take on a finite or countable number of values.
Out of the options given, the number of cars finished in a factory each day is a discrete data random variable. This is because the number of cars finished can only take on a finite number of values, such as 0, 1, 2, 3, and so on. It cannot take on any value between these integers, and there are only a finite number of possible values.
In contrast, a person's height and weight are continuous data random variables, as they can take on any value within a certain range (for example, a person's height can take on any value between 4 feet and 7 feet).
Similarly, the volume of water in a swimming pool can take on any value within a certain range, and is also a continuous data random variable.
Therefore, the only discrete data random variable out of the options given is the number of cars finished in a factory each day.
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Proportions
Two plus x divided by twelve equals one dived by three. Solve for x.
Two plus x divided by twelve equals one divided by three
Case 1 :
Rewrite into numbers : 2 + x /12 = 1/3
-> x/12 = 1/3 - 2 = -5/3
-> x = -5/3 x 12 = -20
Case 2 :
Rewrite into numbers : (2 + x)/12 = 1/3
-> 2 + x = 1/3 x 12 = 4
-> x = 4 - 2 = 2
i dont know if you meant it the right way or the wrong way but ill just put them both
x=2
Step-by-step explanation:
(2+x)/12=1/3
3(2+x)=12
2+x=4
x=4-2
x=2
What is the MAD of 12, 10, 12, 6, 8, 4, 2, 12,
The mean absolute deviation (MAD) of the given set of data is 3.875.
To calculate the mean absolute deviation, we first need to find the mean of the data set by adding up all the numbers and dividing by the total number of values:
Mean = (12 + 10 + 12 + 6 + 8 + 4 + 2 + 12) / 8
Mean = 8
Next, we find the absolute deviation of each number from the mean. To do this, we subtract the mean from each number and take the absolute value:
|12 - 8| = 4
|10 - 8| = 2
|12 - 8| = 4
|6 - 8| = 2
|8 - 8| = 0
|4 - 8| = 4
|2 - 8| = 6
|12 - 8| = 4
Then, we find the mean of the absolute deviations:
Mean absolute deviation = (4 + 2 + 4 + 2 + 0 + 4 + 6 + 4) / 8
Mean absolute deviation = 26 / 8
Mean absolute deviation = 3.875
Therefore, the mean absolute deviation of the given set of data is 3.875.
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Complete Question:
what is the mean absolute deviation of the following set of data 12 10 12 6 8 4 2 12.
you are skiing down a mountain with a vertical height of 1200 feet. the distance from the top of the mountain to the base is 2400 feet. what is the angle of elevation from the base to the top of the mountain?
The angle of elevation from the base to the top of the mountain is 26.56°.
Given, The vertical height of a mountain = 1200 feet
The distance from the top of the mountain to the base = 2400 feet
Now, we need to find the angle of elevation from the base to the top of the mountain.
Hence, we can find the angle of elevation by using the formula
tanθ = Perpendicular/Base
Now, let's apply the formula,
tanθ = Perpendicular/Base 1200/2400= 0.5 tanθ = 0.5
Now, we need to find the value of θθ = tan⁻¹ 0.5θ = 26.56°
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find the Lcm of the two numbers 294 and 714 using their Hcf
Answer:
Step-by-step explanation:
i need help pleaseee
1/10 will be the result of the function j(7).
The input-output pair (-10,7) is on function h
The input-output pair 7,-10) is on function j
An inverse function is what?An inverse function is described as a function where two variables affect one another and the value of an independent variable is used to solve the function.
h and j are inverses
x=-10 so h(x)=7
h(-10)=7
H is inverse of j
h(x)=j-1(x)
7=j-1(x)
j(7)=x
Hence j(7)=-10
F(x) = 2x
y = 2x
The inverse function is calculated as:-
x = 2y
y = x / 2
F'(x) = x / 2
Given that Functions hand j are inverses. For x = -10, the value of h(x), or h (-10) = 7, is 7.
As, h(x) is 7
The input-output pair (-10,7) is on function h
The input-output pair (7,-10) is on function j
Using the supplied data, the value of the function j(7) will be 1/10.
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a(n) ? is a device that indicates whether two ac sources to be connected in parallel are in the correct phase relationship.
The device that indicates whether two AC sources to be connected in parallel are in the correct phase relationship is called a synchronizing device.
A synchronizing device is a mechanism that ensures that two AC sources are in sync when they are connected in parallel. It's used to match the voltage, frequency, and phase angle of two alternating current (AC) sources.
It guarantees that the power supplied by both generators is synchronized, allowing them to be combined into a single electrical system without disrupting the balance of the current or causing a short circuit.
As a result, it is critical to the safe and efficient operation of power systems. A phase sequence indicator (PSI) or a synchroscope is often used as a synchronizing device. It works by providing an indication of the voltage difference, the phase angle difference, and the frequency difference between two AC sources that are to be synchronized.
Therefore, a synchronizing device is an instrument that determines whether two alternating current (AC) sources to be connected in parallel are in the appropriate phase relationship.
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