The null hypothesis (H0) proposed by the friend is H0: p^ = 0.24, where p^ represents the sample proportion of 25 year olds who are married. The alternative hypothesis (Ha) is Hα: p^ ≠ 0.24.
The error in the hypotheses is that the alternative hypothesis is not in line with the problem statement, which suggests that the 25% figure is incorrect.
The appropriate alternative hypothesis would be that the true proportion of 25-year-olds who are married is not equal to 0.25. Therefore, the correct alternative hypothesis would be Ha: p^ ≠ 0.25.
In summary, the correct set of hypotheses for this problem would be:
H0: p^ = 0.25 (Null hypothesis)
Ha: p^ ≠ 0.25 (Alternative hypothesis)
We would use a significance level and statistical test to determine whether we have enough evidence to reject the null hypothesis in favor of the alternative hypothesis.
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A person hikes 4 miles in 2.5 hours. Then find the unit rate in hours per mile. 29 POINTS IF YOU ANSWER- PLEASE IM BEGGING
RESPOND IN FULL DETAIL ------- NOTE (IT IS NOT 1.6 MILES PER HOUR) THEIR ASKING FOR HOURS PER MILE!!!!!!!!
A survey was given to a random sample of 1050 voters in the United States to ask about their preference for a presidential candidate. Of those surveyed, 693 respondents said that they preferred Candidate A. Determine a 95% confidence interval for the proportion of people who prefer Candidate A, rounding values to the nearest thousandth
As per the confidence interval, the true proportion of people in the population who prefer Candidate A lies somewhere between 0.622 and 0.698, based on the results of the survey of 1050 voters.
To calculate the margin of error, we will use the following formula:
Margin of error = z x standard error
Where z is the critical value from the standard normal distribution corresponding to our desired confidence level (95%), and the standard error is given by:
Standard error = √[(sample proportion x (1 - sample proportion)) / sample size]
Using a z-table, we find that the critical value for a 95% confidence interval is 1.96
Plugging in the values we have, we get:
Standard error = √[(0.66 x 0.34) / 1050] = 0.0193
Margin of error = 1.96 x 0.0193 = 0.0378
Therefore, the 95% confidence interval for the proportion of people who prefer Candidate A is:
0.66 ± 0.0378
Rounded to the nearest thousandth, the lower bound of the confidence interval is 0.622 and the upper bound is 0.698.
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What is the Smallest Positive Integer with at least 8 odd Factors and at least 16 even Factors?
Therefore, the smallest positive integer with at least 8 odd factors and at least 16 even factors is N = 1800.
what is Combination?In mathematics, combination is a way to count the number of possible selections of k objects from a set of n distinct objects, without regard to the order in which they are selected.
The number of combinations of k objects from a set of n objects is denoted by [tex]nCk[/tex] or [tex]C(n,k),[/tex] and is given by the formula:
[tex]nCk = n! / (k! *(n-k)!)[/tex]
where n! denotes the factorial of n, i.e., the product of all positive integers up to n.
by the question.
Now, let's consider the parity (evenness or oddness) of the factors of N. A factor of N is odd if and only if it has an odd number of factors of each odd prime factor of N. Similarly, a factor of N is even if and only if it has an even number of factors of each odd prime factor of N. Therefore, the condition that N has at least 8 odd factors and at least 16 even factors can be expressed as:
[tex](a_{1} +1) * (a_{2} +1) * ... * (an+1) = 8 * 2^{16}[/tex]
Let's consider the factor 2 separately. Since N has at least 16 even factors, it must have at least 16 factors of 2. Therefore, we have a_i >= 4 for at least one prime factor p_i=2. Let's assume without loss of generality that p[tex]1=2[/tex] and [tex]a1 > =4.[/tex]
Now, let's consider the remaining prime factors of N. Since N has at least 8 odd factors, it must have at least 8 factors that are not divisible by 2. Therefore, the product (a2+1) * ... * (an+1) must be at least 8. Let's assume without loss of generality that n>=2 (i.e., N has at least three distinct prime factors).
Since a_i >= 4 for i=1, we have:
[tex]N > = 2^4 * p2 * p3 > = 2^4 * 3 * 5 = 240[/tex]
Let's now try to find the smallest such N. To minimize N, we want to make the product (a2+1) * ... * (an+1) as small as possible. Since 8 = 2 * 2 * 2, we can try to distribute the factors 2, 2, 2 among the factors (a2+1), (a3+1), (a4+1) in such a way that their product is minimized. The only possibility is:
[tex](a2+1) = 2^2, (a3+1) = 2^1, (a4+1) = 2^1[/tex]
This gives us:
[tex]N = 2^4 * 3^2 * 5^2 = 1800[/tex]
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A student has 30 minutes to complete an exam. There are 9 multiple choice questions worth 3 points each. There are also 3 short answer questions worth 5 points each. It takes about 2 minutes to answer a multiple choice question and about 6 minutes to complete a short answer question. How many multiple choice questions and short answer questions should the student answer to maximize his score in the time remaining (Use x = multiple choice; y = short answer.)State the Objective Function (S for score) in the linear programming problem givenA. S = 5x + 3y B. S = 3x + 5y C. S = 2x + 6y D. S = 6y + 2x
The Objective Function in the linear programming problem given in the above-stated scenario is:
B. S = 3x + 5y
Linear programming is a statistical technique used to find a maximum or minimum value of an equation in order to find a solution to a problem. It is used to calculate how much to produce to maximize profits, how to allocate resources, and determine which investments to make.
Linear programming problems include an objective function, which is the equation to be maximized or minimized, and constraints that must be followed. Linear programming problems can be solved graphically or algebraically. In order to solve a linear programming problem, we first need to identify the objective function and constraints.
Objective Function in the linear programming problem:
The score of the student is to be maximized in the given time frame by answering the maximum number of questions of both types.
Therefore, the objective function is: S = 3x + 5y
Answer: B. S = 3x + 5y
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Identify whether each of the following samples is a possible bootstrap sample from this original sample: 20, 24, 19, 23, 18 *Explain your answer (2pt cach) a. 20, 24, 21, 19, 18 b. 20, 20, 20, 20, 20
The two options are as follows: a. 20, 24, 21, 19, 18.
This sample is a possible bootstrap sample.
A bootstrap sample is created by taking samples from the original sample with replacement. Therefore, we can include the original data multiple times.
Here, 20, 24, 19, 23, 18 is the original sample.
Sample a includes 20, 24, 19, 23, and 18 which are part of the original data, and 21 which is not part of the original data. Since we can include the original data multiple times in bootstrap samples, a is possible.b. 20, 20, 20, 20, 20This sample is not a possible bootstrap sample.
In the original sample, we have five unique data points: 20, 24, 19, 23, and 18. Sample b includes only one unique data point, 20, repeated five times. We cannot create a bootstrap sample with the same data point repeated multiple times. Hence A is correct.
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Determine whether the following statement is true or false. If it is false, explain why. The probability that event A or event B will occur is P(A or B)= P(A) + P(B) - P(A or B). Choose the correct answer below. A. True B. False, the probability that A or B will occur is P(A or B)= P(A) middot P(B). C. False, the probability that A or B will occur is P(A or B)= P(A) + P(B). D. False, the probability that A or B will occur is P(A or B)= P(A) + P(B) - P(A and B).
False, the probability that A or B will occur is P(A or B) = P(A) + P(B) - P(A and B).
Define probabilityProbability refers to the measure of the likelihood or chance of a particular event occurring. It is expressed as a number between 0 and 1, where 0 indicates an impossible event, and 1 indicates a certain event.
This formula is known as the Addition Rule for Probability and states that to calculate the probability of either event A or event B occurring (or both), we add the probability of A happening to the probability of B happening, but then we need to subtract the probability of both A and B happening at the same time to avoid double counting.
Option A is not the correct answer because it is missing the subtraction of P(A and B), options B and C are incorrect because they omit the subtraction and only add the probabilities of the events. Option D is close, but it is missing the addition of the probabilities of A and B.To know more about event, visit:
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a card is drawn randomly from a standard 52-card deck. find the probability of the given event. write your answers as reduced fractions or whole numbers.
The probability of the given event is 11/13.
How to find the probabilityA card is drawn randomly from a deck of 52 cards.
The probability that the card drawn is neither an ace nor a king is a question that can be answered with probability.
We know that there are 4 aces and 4 kings in a deck of 52 cards.
The probability of drawing an ace or a king is P(Ace or King) = 4/52 + 4/52 = 8/52 = 2/13
There are four of each card rank in a deck of 52 cards, and there are a total of 52 cards. A player must choose one of the 52 cards at random.
The probability of drawing an ace or a king is P(Ace or King) = 4/52 + 4/52 = 8/52 = 2/13
The probability of at drawing neither an ace nor a king is:
P(Not Ace and Not King) = 1 - P(Ace or King) = 1 - 2/13 = 11/13
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Solve
2n > 20
Pls help really quick
Answer:
n > 10
Step-by-step explanation:
2n > 20
2n/2 > 20/2
n > 10
Kaitlin is on vacation at a tropical bay that has three islands. She rents a boat on Island A and plans to navigate to Island C, which is 13 miles away. Based on the figure below, at what angle should she navigate to go to Island C?
She should navigate at angle 76.86° to go to Island C.
How to find the angle for navigation?The cosine rule is for solving triangles which are not right-angled in which two sides and the included angle are given. The following are cosine rule formula:
cosA = (b² + c² - a²)/2bc
where a, b and c are the lengths and A, B and C are the angles for the islands respectively
Using the formula:
cosA = (b² + c² - a²)/2bc
cosA = (13² + 11² - 15²)/(2*13*11)
cosA = 5/22
A = arc cos(5/22)
A = 76.86°
Thus, θ = 76.86
Therefore, she should navigate at angle 76.86° to go to Island C.
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The perimeter of a rectangular map of the world is 270 cm. It is 90 cm in height. How wide is it?
Answer:
The perimeter of a rectangle is given by:
P = 2(L + W)
where P is the perimeter, L is the length, and W is the width.
In this case, we know that P = 270 cm and L = 90 cm, so we can solve for W as follows:
270 = 2(90 + w)
Divide both sides by 2:
135 = 90 + w
Subtract 90 from both sides:
w = 45
Therefore, the width of the map is 45 cm.
Step-by-step explanation:
At the grocery store, students measure the weight of three drinks in ounces. How many more ounces are the of water than orange juice?
Orange juice 7.9
Water 13.4
Milk 8.5
Answer:
2
Step-by-step explanation:
literally just 2 it's quite simple
Tamarisk company began operations on january 2, 2019. It employs 9 individuals who work 8-hour days and are paid hourly. Each employee earns 9 paid vacation days and 7 paid sick days annually. Vacation days may be taken after january 15 of the year following the year in which they are earned. Sick days may be taken as soon as they are earned; unused sick days accumulate. Additional information is as follows. Actual hourly wage rate vacation days used by each employee sick days used by each employee 2019 2020 2019 2020 2019 2020 $6 $7 0 8 5 6 tamarisk company has chosen to accrue the cost of compensated absences at rates of pay in effect during the period when earned and to accrue sick pay when earned
The total cost of compensated absences for Tamarisk Company for the years 2019 and 2020 was $348 + $1,399 = $1,747.
To calculate the cost of compensated absences for Tamarisk Company, we need to calculate the number of vacation days and sick days earned by the employees in 2019 and 2020, and then calculate the cost of the days earned but not taken.
Each employee earns 9 vacation days per year. As they can be taken after January 15th of the year following the year in which they are earned, the vacation days earned by the employees in 2019 can be taken in 2020. Therefore, in 2019, no vacation days were taken by any employee.
In 2020, the employees took a total of 8 vacation days. As there are 9 employees, the total vacation days taken in 2020 were 9 x 8 = 72.
Sick Days:
Each employee earns 7 sick days per year, and unused sick days accumulate. In 2019, the employees used a total of 5 sick days. Therefore, the unused sick days at the end of 2019 were 9 x 7 - 5 = 58.
In 2020, the employees used a total of 6 sick days, and the unused sick days at the end of 2020 were 58 + 9 x 7 - 6 = 109.
To find the cost of compensated absences. The unused sick days and vacation days must be multiplied to get the hourly wage rate in effect in a year.
In 2019, the cost of compensated absences was 58 x $6 = $348.
In 2020, the cost of compensated absences was (72 + 109) x $7 = $1,399.
Therefore, the total cost of compensated absences for Tamarisk Company for the years 2019 and 2020 was $348 + $1,399 = $1,747.
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Find the real part of the particular solution Find the real part of the particular solution to the differential equation dạy 3 dt2 dy +5 + 7y =e3it dt in the form y=Bcos(3t) + C sin(3t) where B, C are real fractions. = Re(y(t)) = = symbolic expression ?
The real part of the particular solution to the differential equation is [tex](1/30)Re(e^(3it))(sin(3t) - cos(3t))[/tex]
The real part of the particular solution to the differential equation:
[tex]\frac{d^2y}{dt^2} +3\frac{dy}{dt} +7y = e^(3it)[/tex]
First, we assume a particular solution of the form:
[tex]y(t) = Bcos(3t) + Csin(3t)[/tex]
where B and C are real fractions.
Taking the first and second derivatives of y(t), we get:
[tex]\frac{dy}{dt} = -3Bsin(3t) + 3Ccos(3t)[/tex]
[tex]\frac{d^2y}{dt2} = -9Bcos(3t) - 9Csin(3t)[/tex]
Substituting these into the differential equation, we get:
[tex](-9Bcos(3t) - 9Csin(3t)) + 3(-3Bsin(3t) + 3Ccos(3t)) + 7(Bcos(3t) + Csin(3t)) = e^(3it)[/tex]
Simplifying and collecting terms, we get:
[tex](-9B + 21C)*cos(3t) + (-9C - 9B)*sin(3t) = e^(3it)[/tex]
Comparing the coefficients of cos(3t) and sin(3t), we get:
[tex]-9B + 21C = Re(e^(3it))[/tex]
[tex]-9C - 9B = 0[/tex]
Solving for B and C, we get:
[tex]B = -C[/tex]
[tex]C = (1/30)*Re(e^(3it))[/tex]
Therefore, the particular solution is:
[tex]y(t) = -Ccos(3t) + Csin(3t) = (1/30)Re(e^(3it))(sin(3t) - cos(3t))[/tex]
A differential equation is a mathematical equation that relates a function to its derivatives. It is a powerful tool used in many fields of science and engineering to describe how physical systems change over time. The equation typically includes the independent variable (such as time) and one or more derivatives of the dependent variable (such as position, velocity, or temperature).
Differential equations can be classified based on their order, which refers to the highest derivative present in the equation, and their linearity, which determines whether the equation is a linear combination of the dependent variable and its derivatives. Solving a differential equation involves finding a function that satisfies the equation. This can be done analytically or numerically, depending on the complexity of the equation and the available tools.
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Pls solve 60 points if you do
Answer:
Cos2a - cot2a/sin2a - tan2a
Step-by-step explanation:
6. suppose that a brand of aa batteries reaches a significant milestone to their death on average after 7.36 hours, with standard deviation of 0.29 hours. assume that when this milestone occurs follows a normal distribution (a) calculate the probability that a battery does not reach this milestone in its first 8 hours of usage. (b) suppose that the company wants to sell a pack of n batteries of which (at least) 10 will last until after 7.5 hours of usage. if n12, what is the probability of this goal being met? (c) How many batteries n should be in the package in order for the probability to exceed 1%? Give the smallest number n which works.
The smallest number of batteries in the package for the probability to exceed 1% is a) 17. This can be calculated using the binomial distribution with parameters n=17 and b)p=0.2927 and number of batteries is c)4. (Where p is the probability from part a).
a) The probability that a battery does not reach the significant milestone after 8 hours of usage is 0.2927.
This can be calculated using the cumulative normal distribution function. The parameters are μ=7.36, σ=0.29, and x=8.
b) The probability that at least 10 batteries will last more than 7.5 hours is 0.7012.
This can be calculated using the binomial distribution with parameters n=12 and p=0.2927 (where p is the probability from part a).
c) The number of batteries should be in package is μ*4.2/7.5 = 4.
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An equation is given.
x² + 9 = 6x
What is one solution to the equation?
x=
Step-by-step explanation:
x²-6x+9=0
using the almighty formula where a=1 , b=-6 , c=9
(1 point) a bowl contains 6 red balls and 7 blue balls. a woman selects 4 balls at random from the bowl. how many different selections are possible if at least 3 balls must be blue?
The number of different selections possible if at least 3 balls must be blue is 35
1. Calculate the total number of possible selections (6 red + 7 blue = 13 total): 13C4 = 715
2. Calculate the number of possible selections that have fewer than 3 blue balls: 6C4 = 15
3. Subtract the number of possible selections with fewer than 3 blue balls from the total number of possible selections: 715 - 15 = 700
4. Divide the answer by the number of selections with exactly 3 blue balls: 700/7 = 100
5. Multiply the result by the number of selections with exactly 4 blue balls: 100 * 7 = 700
6. Finally, subtract the number of selections with 4 blue balls from the total number of possible selections: 715 - 700 = 35
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Give the coordinates for the translation of Rhombus ABCD with vertices A(-3,-2), B(0, 3),
C(5, 6), and D(2, 1).
Given the rule (x, y) = (x+2, y-6)
The new position of Rhombus ABCD after the translation can be described as follows: point A is now at (-1,-8), point B is at (2,-3), point C is at (7,0), and point D is at (4,-5).
To translate Rhombus ABCD using the rule (x, y) = (x+2, y-6), we add 2 to the x-coordinate and subtract 6 from the y-coordinate for each vertex.
Thus, the new vertices for the translated rhombus are:
A' = (-3+2, -2-6) = (-1, -8)
B' = (0+2, 3-6) = (2, -3)
C' = (5+2, 6-6) = (7, 0)
D' = (2+2, 1-6) = (4, -5)
Therefore, the coordinates for the translated Rhombus ABCD are A'(-1,-8), B'(2,-3), C'(7,0), and D'(4,-5).
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Solution of inequality 1/(x - 5) < 3 is
Step-by-step explanation:
1/(x - 5) < 3
1 < 3(x - 5)
1/3 < x - 5
1/3 + 5 < x
5 1/3 = 16/3 = 5.33333... < x
or
x > 5 1/3
The data in the table below shows the number of passengers and number of suitcases on various airplanes.
The line of best fit for the data is approximately y=-1.98x+7.97.
a) True
b) False
The line that best fits the data has a value of around y=-1.98x+7.97 is False.
Define straight lineA line is defined as a straight, one-dimensional figure that extends infinitely in both directions. It is a basic geometrical object that has no thickness or width, and can be defined as the set of all points that are equidistant from two fixed points, called the endpoints. It is represented by y=mx+c.
given y=-1.98x+7.97
Taking first caseNumber of passenger=75
Number of Suitcases=159
75=-1.98×159+7.97
75=-305.26
Hence, the line does not satisfy the data.
The line that best fits the data has a value of around y=-1.98x+7.97 is False.
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Please help me with my math!
Answer:
35 passengers produce the maximum revenue for the bus company.
Step-by-step explanation:
Define the variables:
Let "x" be the total number of passengers.Let "y" be the total revenue of the bus company (in dollars).If there are 30 or fewer passengers, each passenger will be charged $80. Therefore, the equation for the total revenue for 30 or fewer passengers is:
[tex]y = 80x,\quad x \leq 30[/tex]
If there are more than 30 passengers, each passenger will be charged $80 minus $2 for every passenger over 30. Therefore, the equation for the total revenue for more than 30 passengers is:
[tex]y = [80 - 2(x - 30)]x, \quad x > 30[/tex]
This simplifies to:
[tex]y = [80 - 2x + 60]x, \quad x > 30[/tex]
[tex]y = [140 - 2x]x, \quad x > 30[/tex]
[tex]y = 140x - 2x^2, \quad x > 30[/tex]
To maximize revenue, we need to find the value of x that maximizes the above equation.
Since this is a quadratic function, the maximum value occurs at the vertex of the parabola.
The x-value of the vertex of a parabola in the form y = ax² + bx + c is when x = - b/2a. Therefore, the x-value of the vertex is:
[tex]\implies x_{\sf vertex}=\dfrac{-140}{2(-2)}=\dfrac{-140}{-4}=35[/tex]
Therefore, the number of passengers that produce the maximum revenue for the bus company is 35.
show that a strictly diagonally dominant matrix is invertible. show furthermore that if all the diagonal entries are positive, then all the eigenvalues have positive real part.
Completing the proof that if A has strictly positive diagonal entries, then all of its eigenvalues have a positive real part.
For any square matrix A, a diagonal element is any element that is on the main diagonal, i.e., (i,i) for 1<=i<=n. If A has strictly diagonal dominance, then for each row i of A, the absolute value of the diagonal element of that row is more than the sum of the absolute values of the non-diagonal elements of that row.If A is invertible, then det(A) is nonzero. If all the diagonal entries are positive, then det(A) is also positive, which implies that all the eigenvalues of A have the same sign (either all are positive or all are negative).As a result, suppose all of A's diagonal entries are positive. Then all of A's eigenvalues are positive since they all have the same sign as det(A).
And if all of A's eigenvalues are positive, then all of their real parts are positive as well, hence completing the proof that if A has strictly positive diagonal entries, then all of its eigenvalues have a positive real part.
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NEED HELP DUE TODAY!!!!
2. How do the sizes of the circles compare?
3. Are triangles ABC and DEF similar? Explain your reasoning.
4. How can you use the coordinates of A to find the coordinates of D?
The triangle DEF is twice the size of the triangle ABC and the triangles are similar triangles
How do the sizes of the circles compare?Given the triangles ABC and DEF
From the figure, we have
AB = 1
DE = 2
This means that the triangle DEF is twice the size of the triangle ABC
Are triangles ABC and DEF similar?Yes, the triangles ABC and DEF are similar triangles
This is because the corresponding sides of DEF is twice the corresponding sides of triangle ABC
How can you use the coordinates of A to find the coordinates of D?Multipliying the coordinates of A by 2 gives coordinates of D
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simplify 2^(x+2) /2^(x-1)
Answer:
Step-by-step explanation:
When we divide two exponential expressions with the same base, we can subtract their exponents. Using this property, we can simplify the given expression as:
2^(x+2) /2^(x-1) = 2^x * 2^2 / 2^x * 2^(-1)
= 2^(x-1+2)
= 2^(x+1)
Therefore, the simplified expression is 2^(x+1).
What part of an hour passes from 1:35 P. M. To 2:15 P. M. ?
Answer:
2/3 rd or 0.667th part of an hour passes from 1:35 P.M to 2:15 P.M
To determine the fraction of an hour that passes from 1:35 P.M. to 2:15 P.M., we need to calculate the elapsed time between the two times and express it as a fraction of an hour.
The elapsed time between 1:35 P.M and 2:15 P.M is 40 minutes. To convert minutes to fractions of an hour, we can divide the number of minutes by 60.
40 minutes / 60 minutes per hour = 2/3 hours = 0.67 hours
Therefore, the fraction of an hour that passes from 1:35 P.M. to 2:15 P.M. is 0.67. This means that approximately two-thirds of an hour, or 67% of an hour, has passed between these two times.
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Katie's car averages 32 miles per hour. She has marked her distance traveled after 12, 13, and 14 hours. List the elements in the range of the function that would be used to determine how far Katie has traveled at each mark. Separate each elements from least to element with a comma and write the
greatest.
Pls hurry
To determine the distance traveled by Katie at different marks, we can use the formula Distance = Rate × Time.
Explanation:To determine how far Katie has traveled at each mark, we can use the formula:
Distance = Rate × Time
So, the elements in the range of the function that would be used to determine how far Katie has traveled at each mark are 384 miles, 416 miles, and 448 miles.
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The lengths of two sides of a triangle are 4 6. Which measurement cannot be the length of the third side?
Step-by-step explanation:
X = third side
due to the triangle side length rule: sum of any two sides must be greater than the third side
so
4 + 6 > x so x <10
x + 4 > 6 so x > 2
x + 6 > 4 and any value of x will work here
so
2 < x < 10 <=====use this to answer your question as you didn't list the choices in your post
At which values in the interval [0, 2π) will the functions f (x) = 2cos2θ and g(x) = −1 − 4cos θ − 2cos2θ intersect?
a: theta equals pi over 3 comma 4 times pi over 3
b: theta equals pi over 3 comma 5 times pi over 3
c: theta equals 2 times pi over 3 comma 4 times pi over 3
d: theta equals 2 times pi over 3 comma 5 times pi over 3
The values in the interval [0, 2π) for which the two points would intersect as required is; Choice C; theta equals 2 times pi over 3 comma 4 times pi over 3.
What values of θ make the two functions intersect?Recall from the task content; the given functions are;
f (x) = 2cos2θ and g(x) = −1 − 4cos θ − 2cos2θ
Therefore, for intersection; f (θ) and g(θ):
2 cos²θ = −1 − 4cos θ − 2cos²θ
4cos²θ + 4cosθ + 1 = 0
let cos θ = y;
4y² + 4y + 1 = 0
y = -1/2
Therefore; -1/2 = cos θ
θ = cos-¹ (-1/2)
θ = 2π/3, 4π/3.
Ultimately, the correct answer choice is; Choice C; theta equals 2 times pi over 3 comma 4 times pi over 3.
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How do you make 5. 4 in three different ways!!!?!?
Answer:
10.0 -4.610.8÷2√29.16Step-by-step explanation:
You want to make 5.4 in three different ways.
Arithmetic operationsYou can add, subtract, multiply, or divide numbers to obtain a result of 5.4:
2.9 +2.5 = 10.0 -4.6 = 2.0×2.7 = 10.8÷2 = 5.4
More complicated functions√29.16 = ∛157.464 = 5.4
[tex]\displaystyle\sum_{n=1}^\infty{10.8(3^{-n})}=5.4[/tex]
In Problems 1 through 6 you are given a homogeneous system of first- order linear differential equations and two vector-valued functions, X(1) and x(2) a. Show that the given functions are solutions of the given system of differential equations. b: Show that X = Cx(T) + C2x(2) is also a solution of the given system for any values of C1 and C2. C. Show that the given functions form a fundamental set of solutions of the given system.
The given functions form a fundamental set of solutions of the given system.
The solution of the given system of differential equations is shown below.a) To prove that the given functions X(1) and x(2) are the solutions of the given system of differential equations, we must substitute these functions into the given system to show that they satisfy the equations.In the given system, we have the following equations:
X_1' (t) = 2X_1 (t) - X_2 (t)
X_2' (t) = 4X_1 (t) - 2X_2 (t)
Now, let's substitute the given vector-valued functions X(1) and x(2) into the above equations and check if they satisfy these equations.
a. For X(1) = [1, 2]e^2t
Substituting X(1) into the given system, we get:
X_1' (t) = [1, 2] * 2e^2t = 2X_1 (t) - X_2 (t)
X_2' (t) = [1, 2] * 4e^2t = 4X_1 (t) - 2X_2 (t)
Therefore, the given function X(1) is a solution to the given system of differential equations.
b. To prove that X = C1x(1) + C2x(2) is also a solution of the given system for any values of C1 and C2, we need to X into the given system of equations and check if it satisfies the equations.
So, we have:
X = C_1[1, 2]e^2t + C_2[1, -1]e^-t
X_1 = C_1e^2t + C_2e^-t
X_2 = 2C_1e^2t - C_2e^-t
Differentiating X_1 and X_2 with respect to t, we get:
X_1' = 2C_1e^2t - C_2e^-t
X_2' = 4C_1e^2t + C_2e^-t
Substituting X_1 and X_2 into the given system, we get:
X_1' (t) = 2(C_1e^2t - C_2e^-t) = 2X_1 (t) - X_2 (t)
X_2' (t) = 4(C_1e^2t + C_2e^-t) = 4X_1 (t) - 2X_2 (t)
Therefore, the given function X = C1x(1) + C2x(2) is also a solution of the given system for any values of C1 and C2.
c. To show that the given functions form a fundamental set of solutions of the given system, we need to prove that they are linearly independent and that their Wronskian is non-zero.
We know that the vectors [1, 2] and [1, -1] are linearly independent, therefore the functions x(1) and x(2) are also linearly independent.
Also, the Wronskian of x(1) and x(2) is given by:
W(x1, x2) = | x1 x2 |
| x1' x2' |
Substituting x(1) and x(2) into the above equation, we get:
W(x1, x2) = | e^2t e^-t |
| 2e^2t -e^-t |
Simplifying the above equation, we get:
W(x1, x2) = 3e^(3t) ≠ 0
Therefore, the given functions form a fundamental set of solutions of the given system.
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