Answer:
10 times
Step-by-step explanation:
1km=1000m
3*1000=3000m
3000/300=10
you will need to run 10 times around the field
Evaluate for f(8), show all your work: f(x)=x2+10
Answer:
[tex]f(8)=74[/tex]
Step-by-step explanation:
[tex]f(8)=8^2+10=64+10=74[/tex]
Multiply the additive inverse of -1/14 with the multiplicative inverse of 21/49.
Chapter : Rational Numbers
Answer:
1/6
Step-by-step explanation:
When we talk of additive inverse, we mean the value we will add to a certain number to give 0
for example, the additive inverse of 4 is -4
Hence, the additive inverse of -1/14 is 1/14
The
multiplicative inverse is the number multiplied by another that gives 1
For example, the multiplicative inverse of 4 is 1/4
Thus, we have it that;
multiplicative inverse of 21/49 is 49/21
multiplying;
1/14 * 49/21 = 1/6
Answer:
Answer:
1/6
Step-by-step explanation:
When we talk of additive inverse, we mean the value we will add to a certain number to give 0
for example, the additive inverse of 4 is -4
Hence, the additive inverse of -1/14 is 1/14
The
multiplicative inverse is the number multiplied by another that gives 1
For example, the multiplicative inverse of 4 is 1/4
Step-by-step explanation:
6р + 22 – 10
Help me please
Answer:
P = -2
Step-by-step explanation:
6p +22 =10
-22 -22
6p= -12
6 6
p= -2
Im pretty sure thats it.
“Calculate the lengths of the 2 unlabeled sides”
Answer:
NL = 4.33
NM = 5
Step-by-step explanation:
tan 60° = NL/2.5
tan 60° = 1.7321
so:
1.7321 = NL/2.5
NL = (2.5)(1.7321)
NL = 4.33
cos 60° = 2.5/NM
cos 60° = 0.5
so:
0.5 = 2.5.NM
NM = 2.5/0.5
NM = 5
It is commonly recognized that a poor environment (for example, poverty, low educational level of parents) and the presence of stressors (for example divorce of parents, abuse) are associated with lower IQs in children. Recent research indicates that a child's IQ is related to the number of such risk factors in the children's background, regardless of which specific factors are present. Below are the numbers of risk factors and the IQs of 12 children.
Child
Risk factors (x)
IQ (y)
A
3
112
B
6
82
C
0
105
D
5
102
E
1
115
F
1
101
G
5
94
H
4
89
I
3
98
J
3
109
K
4
91
L
2
107
A. Draw a scatterplot of the data. What does the plot tell you? Determine the regression equation. Be sure to include an interpretation of each parameter in the regression equation.
B. What IQ do you predict for James, who has 5 risk factors, versus Elizabeth, who has only 2?
A) The scatterplot of the data is as follows:What can we interpret from the scatterplot of the given data?We can observe that the higher the number of risk factors, the lower the IQ of the children. This fact is supported by the trend line that shows a negative correlation. This is what the plot is telling us.A regression equation is given by the formula: $y=\beta_0+\beta_1x$Where, $y$ is the IQ of the child, $x$ is the number of risk factors, $\beta_0$ is the y-intercept and $\beta_1$ is the slope of the line.Both $\beta_0$ and $\beta_1$ can be calculated using the following formulae: $$\beta_1=r_{xy}\frac{s_y}{s_x}$$$$\beta_0=\bar{y}-\beta_1\bar{x}$$Where, $r_{xy}$ is the correlation coefficient between $x$ and $y$, $s_x$ and $s_y$ are the standard deviations of $x$ and $y$ respectively, and $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$ respectively.From the scatterplot, we can observe that $r_{xy}$ is negative. Therefore, there exists a negative correlation between $x$ and $y$.Let us calculate the values of $\beta_1$, $\beta_0$ using the above formulae.$$r_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})}{\sqrt{\sum(x-\bar{x})^2\sum(y-\bar{y})^2}}$$$$r_{xy}=\frac{(3)(112)+(6)(82)+(0)(105)+(5)(102)+(1)(115)+(1)(101)+(5)(94)+(4)(89)+(3)(98)+(3)(109)+(4)(91)+(2)(107)}{\sqrt{(3^2+6^2+0^2+5^2+1^2+1^2+5^2+4^2+3^2+3^2+4^2+2^2)(112^2+82^2+105^2+102^2+115^2+101^2+94^2+89^2+98^2+109^2+91^2+107^2)}}$$$$r_{xy}=-0.835$$Now, let's calculate the standard deviations of $x$ and $y$:$$s_x=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}$$$$s_y=\sqrt{\frac{\sum(y-\bar{y})^2}{n-1}}$$$$s_x=\sqrt{\frac{(3-2)^2+(6-2)^2+(0-2)^2+(5-2)^2+(1-2)^2+(1-2)^2+(5-2)^2+(4-2)^2+(3-2)^2+(3-2)^2+(4-2)^2+(2-2)^2}{12-1}}$$$$s_x=\sqrt{\frac{44}{11}}=2$$$$s_y=\sqrt{\frac{\sum(y-\bar{y})^2}{n-1}}$$$$s_y=\sqrt{\frac{(112-99.17)^2+(82-99.17)^2+(105-99.17)^2+(102-99.17)^2+(115-99.17)^2+(101-99.17)^2+(94-99.17)^2+(89-99.17)^2+(98-99.17)^2+(109-99.17)^2+(91-99.17)^2+(107-99.17)^2}{12-1}}$$$$s_y=\sqrt{\frac{5626.1}{11}}=9.025$$Finally, let's calculate $\beta_1$ and $\beta_0$:$$\beta_1=r_{xy}\frac{s_y}{s_x}$$$$\beta_1=-0.835\frac{9.025}{2}=-3.762$$$$\beta_0=\bar{y}-\beta_1\bar{x}$$$$\beta_0=99.17-(-3.762)(3.08)=112.91$$Therefore, the regression equation is $y=112.91-3.762x$.Here, $\beta_0=112.91$ represents the expected IQ of a child with zero risk factors, while $\beta_1=-3.762$ represents the decrease in IQ per risk factor.B) James has 5 risk factors. Substituting $x=5$ in the regression equation, we get:$$y=112.91-3.762x$$$$y=112.91-3.762(5)$$$$y=93.13$$Therefore, the IQ predicted for James is $93.13$.Similarly, Elizabeth has 2 risk factors. Substituting $x=2$ in the regression equation, we get:$$y=112.91-3.762x$$$$y=112.91-3.762(2)$$$$y=105.39$$Therefore, the IQ predicted for Elizabeth is $105.39$.
At what number of days will the cost of attendance be the same for both preschools?
Answer:
Forgot the pic ?
Step-by-step explanation:
volume of a cylinder that has a diameter of 16 and a height of 7
Answer:
volume of a cylinder = 1408
hope this answer helps you...
Answer:
1407.616
Step-by-step explanation:
Volume of cylinder=πr²h
V=3.142×8×8×7
V=1407.616
NOTE: radius =diameter/2=16/2
giving 8.
I WILL MARK BRAINLIEST
Answer:
A.
Step-by-step explanation:
I need help quick please
i’ll tell you on discord, add me there my user is ri#0511
Answer: 0.1562 I believe
Find the Volume. 5 1/2 x. 3 x 6 1/3
Answer:6
1/3
Step-by-step explanation:
HELP PLEASE!!! WILL CROWN BRAINLIEST....
The temperature on Sunday was 7 C. The temperature dropped 8 C on Monday
and dropped twice as much on Tuesday. What was the temperature on Tuesday?
Answer: - 17 C
Step-by-step explanation: 7 - 8 is negative 1 and twice as much of 8 is 16. -1 minus 16 is negative 17 (-17).
shoes
11 cm
8.8 cm
18.4 cm
WILL GIVE BRAINLIEST FOR BEST ANSWER NO LINKS!!!!*Use the data from the dot plot below to answer the question. The data shows games of a soccer season and the number of goals scored in each game.
Which goals were scored the fewest amount of times during the soccer season?
Question 5 options:
3
0, 7
1, 4, 5, 6
2
0, 7
0 and 7 have the least amount of dots.
Given 2 non-empty Languages A,B⊆ {a,b}∗, give an example of A* = B* and A != B
A is not equal to B because they have different initial strings. A contains strings composed of 'a', while B contains strings composed of 'b'.
Can you provide an example where two non-empty languages A and B, both subsets of {a, b}∗, satisfy the condition A* = B* but A is not equal to B?Let's consider the following example:
A = {a, aa}
B = {b, bb}
In this case, A* represents the Kleene closure (or Kleene star) of language A, which includes all possible concatenations and repetitions of strings in A, including the empty string ε. So A* would be {ε, a, aa, aaa, ...}.
Similarly, B* would be {ε, b, bb, bbb, ...}.
In this example, we can see that A* is equal to B* because both languages contain strings of varying lengths formed by repeating their respective symbols (a and b).
To summarize:
A* = {ε, a, aa, aaa, ...}
B* = {ε, b, bb, bbb, ...}
A != B
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Someone pls help me solve this problem. No guessing
Use the photo below to solve for x
Answer:
x = 6
Step-by-step explanation:
9/x = 12/8
12x = 72
/12 /12
x = 6
hope this helps ^^
if you want to be 99onfident of estimating the population mean to within a sampling error of ±3 and the standard deviation is assumed to be 14, what sample size is required?
The sample size required is_________
The sample size required is 357.
Here's how to solve the problem:
Given that we need to be 99% confident of estimating the population mean to within a sampling error of ±3.
So, the margin of error (E) = 3 z-score for 99% confidence level = 2.58 (from standard normal distribution table)
The formula for sample size is:n = [z² * σ²] / E²
Where, n = sample size, σ = standard deviation of the population,
E = margin of error, and z = z-score
For a 99% confidence level, the z-score = 2.58
Substitute the given values in the formula:n = [2.58² * 14²] / 3²= 357.15≈ 357
Therefore, the sample size required is 357.
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Let A be a connected and compact Jordan region with |A| > 0 and let ƒ: A → R be a function continuous on A. Prove that there exits xo E A such that 1 f(x₁) = = // f(x)dx. |A| A
The presence of xo in A with the end goal that 1/|A| ∫ f(x)dx = f(xo). This finishes the confirmation.
To demonstrate the presence of a point xo in A to such an extent that 1/|A| ∫ f(x)dx = f(xo), where A will be an associated and minimized Jordan district with |A| > 0 and ƒ: A → R is a nonstop capability, we can involve the Mean Worth Hypothesis for Integrals.
In the first place, we should characterize a capability F: A → R as F(t) = 1/|A| ∫ f(x)dx - f(t), where t is a point in A. We need to show that there exists xo in A to such an extent that F(xo) = 0.
Since A will be an associated and minimal Jordan locale, it is likewise a shut and limited subset of R^n. Subsequently, A will be a smaller set. We realize that consistent capabilities on minimized sets accomplish their greatest and least qualities.
Since F is a consistent capability on the minimized set A, it accomplishes its most extreme and least qualities. Let M = max{F(t) : t in A} and m = min{F(t) : t in A}.
We have two cases to consider:
Case 1: In the event that M ≤ 0 and m ≥ 0, F(t) = 0 for all t in A, including xo. For this situation, we have demonstrated the presence of xo to such an extent that 1/|A| ∫ f(x)dx = f(xo).
Case 2: If either M > 0 or m < 0, we accept without loss of over-simplification that M > 0. Since M is the greatest worth of F on A, there exists a point t1 in A with the end goal that F(t1) = M. Essentially, we expect to be that m < 0, and there exists a point t2 in A with the end goal that F(t2) = m.
Consider the consistent way γ(t) from t1 to t2 in A. Since An is associated, such a way exists. Presently, characterize another capability G: [0, 1] → R as G(s) = F(γ(s)).
We have G(0) = F(γ(0)) = F(t1) = M > 0, and G(1) = F(γ(1)) = F(t2) = m < 0. In this way, by the Halfway Worth Hypothesis, there exists a point s0 in [0, 1] with the end goal that G(s0) = 0.
Let xo = γ(s0). Since G(s0) = F(γ(s0)) = 0, we have F(xo) = 0. Subsequently, we have demonstrated the presence of xo in A to such an extent that 1/|A| ∫ f(x)dx = f(xo).
In the two cases, we have shown the presence of xo in A with the end goal that 1/|A| ∫ f(x)dx = f(xo). This finishes the confirmation.
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The lumen output was determined for each of k = 3 different brands of lightbulbs having the same wattage, with n_j = 8 bulbs of each brand tested (this is the number of observations in each treatment group). The sums of squares were computed as MSTr = 297.850 and MSE = 227.619. State the hypotheses of interest (including word definitions of parameters).
µ_j = sample average lumen output for brand j bulbs
µ_o : µ_1≠µ_2≠µ_3
H_a: all three µ_j's are equal
µ_j = sample average lumen output for brand i bulbs
µ_o : µ_1=µ_2=µ_3
H_a: all three µ_j's are unequal
µ_j = true average lumen output for brand i bulbs
µ_o : µ_1≠µ_2≠µ_3
H_a: at least two µ_j's are equal
µ_j= true average lumen output for brand i bulbs
µ_o : µ_1=µ_2=µ_3
H_a: at least two µ_j's are unequal
Use the Single Factor ANOVA F test with (α = 0.05) to decide whether there are any differences in true average lumen outputs among the three brands for this type of bulb. Calculate the F test statistic then use software to find your p-value, Recall the p-value from an F test is always the area to the right of the F test statistic.
f statistic = _______ (Round your answer to two decimal places.)
p-value = ________(Round your answer to four places.)
State the conclusion in the problem context.
Fail to reject H_o. There are statistically significant differences in the lumen output.
Fail to reject H_o. There are no statistically significant differences in the lumen output.
Reject H_o. There are statistically significant differences in the lumen output. Reject H_o. There are no statistically significant differences in the lumen output.
Using the Single Factor ANOVA F test with a significance level of α = 0.05, the F test statistic can be calculated to determine if there are any differences in the true average lumen outputs among the three brands of lightbulbs.
The p-value is then obtained from the software. Based on the conclusion derived from the p-value, either the null hypothesis (H0) is rejected, indicating statistically significant differences in the lumen output, or it is failed to be rejected, suggesting no statistically significant differences.
To determine if there are any differences in the true average lumen outputs among the three brands of lightbulbs, a Single Factor ANOVA F test is conducted. The null hypothesis (H0) assumes that there are no differences, while the alternative hypothesis (Ha) suggests that there are differences among the means.
The F-test statistic is calculated by dividing the mean square between treatments (MSTr) by the mean square error (MSE). The F-test statistic is not provided in the question, so it needs to be calculated using the given information.
The p-value, which represents the probability of obtaining test results as extreme as observed or more extreme, is obtained using software. The p-value is the area to the right of the F-test statistic in the F-distribution.
Based on the obtained p-value and a significance level of α = 0.05, the conclusion is made. If the p-value is less than 0.05, the null hypothesis (H0) is rejected, indicating statistically significant differences in the lumen output among the three brands. If the p-value is greater than or equal to 0.05, the null hypothesis (H0) is failed to be rejected, suggesting no statistically significant differences.
The conclusion should be stated based on the calculated p-value and the significance level. It could either be "Reject H0. There are statistically significant differences in the lumen output" or "Fail to reject H0. There are no statistically significant differences in the lumen output."
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Which of the following is typical of the starting work geometry in sheet metal processes: (a) high volume to area ratio or (b) low volume to area ratio?
The correct answer is (b) low volume to area ratio is typical of the starting work geometry in sheet metal processes.
Explanation: Sheet metal processing is the method of shaping metal objects to create complex shapes and designs.
Cutting, punching, stamping, and forming are the most common sheet metal processes.
The workpiece's original shape and size are critical considerations in sheet metal processing.
The workpiece's geometry plays an important role in sheet metal processes.
The term volume-to-area ratio refers to the quantity of metal in a part divided by the surface area of that part.
The ratio is low in the case of sheet metal processing. Sheet metal workpieces have a high surface area-to-volume ratio.
This is due to the fact that sheet metal workpieces are often quite thin.
Because of this, sheet metal workpieces are lightweight.
Furthermore, thin sheets are more pliable and can be easily formed into various forms.
This geometry is ideal for sheet metal work since it allows for greater flexibility during processing.
In sheet metal processing, low volume-to-area ratios are typical of the starting work geometry.
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π (pi) is an unending decimal. Find the Circumference of the circle below using exact π (pi).
Step-by-step explanation:
Given radius, r = 5,
Circumference of Circle, C
[tex] = 2\pi \: r \\ = 2\pi(5) \\ = 10\pi \: units[/tex]
Answer:
[tex]10\pi[/tex]
Step-by-step explanation:
As the hint below says:
[tex]C = 2\pi r[/tex]
We also know that
r = 5
So thus:
[tex]C = 2 \pi \cdot 5 = 10\pi[/tex]
Which answer choice below lists only personal financial assets?
А
car, savings, mortgage loan
B
personal savings, college fund, 401K retirement fund
C
credit cards, home equity loan, retirement fund
D
boat, personal loans, savings
Answer:
c
Step-by-step explanation:
Answer:
c
Step-by-step explanation:
hi hottie, pls help me !
Mrs. St. Louis purchased an iPhone 12 Pro. The decreasing value of her phone, after t years, is modeled by the function V(t)=1,200(0.65)^t
Fill in the numbers from the options to correctly complete the statements interpreting this function.
The phone initially cost $______
The value of the phone is depreciating at a rate of _____ % each year.
Answer:
1200
0.65
The answers are in the function
tyler wants to buy a new television that costs $312.
Answer:
ummm is there more to the problem?!
Cool.... the question???
Conservative change function is given in terms of dimensionless variables as:
-dg/dt + (1-g^2)dg/dx =0, - [infinity]0
a) Which condition must g provide for function to oppose the traffic rules.
b) What is vehicles’ maximum velocity (Umax)=?
c) inital condition is given as: g(x,0)= 1-x. Find the value of g(1,1)=?
To oppose the traffic rules, the function g must satisfy |g(x)| > 1. The maximum velocity of the vehicles is Umax = 0. Finally, the value of g(1,1) is 0 based on the given initial condition.
a) For the function g to oppose the traffic rules, it must satisfy the condition |g(x)| > 1. In other words, the absolute value of g must be greater than 1. This condition indicates that the function represents a vehicle moving in the opposite direction of traffic flow.
b) To determine the maximum velocity of the vehicles (Umax), we can analyze the equation -dg/dt + (1-g^2)dg/dx = 0. By setting dg/dx = 0, we can find the critical points where the velocity is maximum. In this case, when g = ±1, the term (1-g^2) reaches its maximum value of 0. Therefore, the maximum velocity of the vehicles is Umax = 0.
c) Given the initial condition g(x,0) = 1 - x, we can find the value of g(1,1) by substituting x = 1 into the function. Thus, g(1,1) = 1 - 1 = 0. Therefore, the value of g(1,1) is 0.
In summary, to oppose the traffic rules, the function g must satisfy |g(x)| > 1. The maximum velocity of the vehicles is Umax = 0. Finally, the value of g(1,1) is 0 based on the given initial condition.
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Find the volume of the cylinder whose base area is (4x)2 and height is (5x)2.
Answer:
Step-by-step explanation:
The computation of the volume of the cylinder is shown below:
As we know that
The Volume of the cylinder is
= base × height
= (4x)^2 + (5x)^2
=16x^2 + 25x^2
= 41x^3
Hence, the volume of the cylinder is 41x^3
The volume of the cylinder will be [tex]20x^2[/tex] .
What is volume ?Volume is the amount of space that something contains or fills.
Volume [tex]=Length*Breadth*Height[/tex]
We have,
Base area [tex]=(4x)^2[/tex]
Height [tex]= (5x)^2[/tex]
So,
Volume of the cylinder [tex]=Base\ Area*Height[/tex]
[tex]= (4x)^2* (5x)^2[/tex]
Volume of the cylinder [tex]=20x^2[/tex]
Hence, we can say that the volume of the cylinder will be [tex]20x^2[/tex] .
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Using the Excel file Weddings, apply the Excel Regression tool using the wedding cost as the dependent variable and attendance as the independent variable. Interpret all key regression results, hypothesis tests, and confidence intervals in the output. Analyze the residuals to determine if the assumptions underlying the regression analysis are valid. Use the standard residuals to determine if any possible outliers exist. If a couple is planning a wedding for 175 guests, how much should they budget?
Using the Excel Regression tool on the Weddings dataset with wedding cost as the dependent variable and attendance as the independent variable, the key regression results, hypothesis tests, and confidence intervals should be interpreted.
After applying the Excel Regression tool with wedding cost as the dependent variable and attendance as the independent variable, the key regression results should be examined. These results typically include coefficients, standard errors, t-values, p-values, and confidence intervals. The coefficient for the independent variable (attendance) represents the estimated change in the wedding cost for each unit increase in attendance. Hypothesis tests and confidence intervals help assess the statistical significance and precision of the estimated coefficients. The t-value indicates the strength of evidence against the null hypothesis of no relationship between the variables, and the p-value indicates the probability of observing the estimated coefficient under the null hypothesis. Lower p-values suggest stronger evidence against the null hypothesis. To validate the assumptions of the regression analysis, the residuals (the differences between the observed and predicted values) should be analyzed. Residual plots can be examined to check for patterns or deviations from assumptions, such as linearity, independence, and constant variance. Outliers can be identified by examining the standardized residuals. Values with high absolute values indicate potential outliers. To estimate the budget for a wedding with 175 guests, the regression model can be used to predict the corresponding cost. The attendance value of 175 can be plugged into the regression equation, and the predicted wedding cost can be obtained.
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Which of the following statements about group decision making is true? If enough time is available, groups usually make higher-quality decisions than most individuals. If enough time is available, most individuals usually make higher-quality decisions than a group. There are far more disadvantages than advantages to group decision making. Individual decisions are generally more difficult to reach than group decisions. Group decisions should rarely be used to address significant business problems.
The statement "If enough time is available, groups usually make higher-quality decisions than most individuals" is true.
Group decision-making has both advantages and disadvantages, but when enough time is available, groups tend to make higher-quality decisions compared to most individuals. This is due to several reasons. First, groups offer diverse perspectives and expertise, allowing for a broader range of ideas and insights.
Different individuals bring unique knowledge and experiences to the table, leading to a more comprehensive examination of the problem. Second, group decision-making involves collective scrutiny and evaluation of options, which helps in identifying potential flaws or biases in individual opinions.
Group discussions allow for critical analysis, debate, and challenging of assumptions, leading to a more thorough decision-making process. However, it is important to note that time constraints can impact the effectiveness of group decision-making. When time is limited, individual decision-making may be more efficient.
Additionally, the success of group decision-making also depends on factors such as group dynamics, effective communication, and skilled facilitation. Therefore, while groups have the potential for making higher-quality decisions, it is essential to consider the specific context and constraints when determining the most appropriate approach to decision making.
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ASAP HELP MATH!!!!!!!! 1st PERSON GETS BRAINLIEST
Given the following sets, find the set (A U B) O (AUC). U = {1, 2, 3, ..., 10) 3 , A = {1, 2, 3, 7} B = {1, 3, 10} C = {1, 2, 3, 6, 8}
The set (A U B) O (AUC) is {1, 2, 3, 6, 7, 8, 10}.
The following steps can be used to find the set (A U B) O (AUC):
Step 1: Find A U B {1, 2, 3, 7} U {1, 3, 10} = {1, 2, 3, 7, 10}
Step 2: Find (A U B) U C({1, 2, 3, 7, 10} U {1, 2, 3, 6, 8}) = {1, 2, 3, 6, 7, 8, 10}
A set is a group of things. The objects (or elements) that make up a set are often listed within curly brackets, and sets are typically identified by a letter. It's important to keep in mind that a set is an unordered collection of objects, meaning that it doesn't matter what order the elements are listed in. are viewed as being equivalent.
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Solve the following: 6 sin² x Your answer [APPL- 6 marks] 5 cos x 20 for 0 ≤ x ≤ 2π
The solutions are x = 0, π, and 2π for the trigonometric equation 6 sin² x = 5 cos x + 20.
To solve the given equation:
6 sin² x = 5 cos x + 20
We can use the trigonometric identity:
sin² x + cos² x = 1
Multiplying both sides by 6, we get:
6 sin² x + 6 cos² x = 6
Substituting 1 - sin² x for cos² x, we get:
6 sin² x + 6 (1 - sin² x) = 6
Simplifying the equation, we get:
6 - sin² x = 6
sin² x = 0
Taking the square root of both sides, we get:
sin x = 0
x = nπ, where n is an integer.
Substituting this value in the original equation, we get:
6(0)² = 5(cos(nπ)) + 20
0 = (-1)n + 4
n must be even for the equation to hold true. Therefore, the solutions are:
x = 0, π, and 2π.
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