The trigonometric identities cos 2θ − cotθ sin 2θ − tanθ= tan 2θ, is verified below with the help of elementary trigonometric identities.
Describe Trigonometric Identities?Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables in the equation. They are useful in solving trigonometric equations, simplifying expressions involving trigonometric functions, and evaluating integrals in calculus.
There are several types of trigonometric identities, including:
Pythagorean identities: These involve the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The Pythagorean identities express this relationship in terms of trigonometric functions, such as sin²θ + cos²θ = 1.
Reciprocal identities: These involve the reciprocals of trigonometric functions, such as csc θ = 1/sin θ and sec θ = 1/cos θ.
We will start with the left-hand side of the identity and simplify it using the trigonometric identities:
cos 2θ − cotθ sin 2θ − tanθ
= cos 2θ − (cosθ/sinθ)(2sinθcosθ) − (sinθ/cosθ)
= cos 2θ − 2cosθsinθ − sin²θ/cosθ
= cos²θ − sin²θ − 2cosθsinθ − sin²θ/cosθ
= (cos²θ - sin²θ) - (sin²θ/cosθ) - 2cosθsinθ
= cos²θ - sin²θ - sin²θ/cosθ - 2sinθcosθ
= cos²θ - sin²θ - sin²θ/cos²θ - 2sinθcosθ/cos²θ (since cosθ ≠ 0)
= cos²θ - sin²θ - (sin²θ + 2sinθcosθ)/cos²θ
= cos²θ - sin²θ - sin(2θ)/cos²θ
= (cos²θ - sin²θ)/cos²θ - sin(2θ)/cos²θ
= (cos²θ/cos²θ) - (sin²θ/cos²θ) - sin(2θ)/cos²θ
= 1 - tan²θ - sin(2θ)/cos²θ
= (1 - tan²θ) - sin(2θ)/cos²θ
= (sec²θ) - (2sinθcosθ)/cos²θ
= (sec²θ) - (2sinθ/cosθ)
= (sec²θ) - 2tanθ
= tanθ + 1 - 2tanθ - 2tanθ (using the identity sec²θ = tan²θ + 1)
= tan²θ - 4tanθ + 1
= (tanθ - 1)²
= tan²θ - 2tanθ + 1
Therefore, we have shown that:
cos 2θ − cotθ sin 2θ − tanθ = tan 2θ
This verifies the given identity.
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1. simplify the expression: 5(a + b) – 2b
Answer:
5a + 3b
Step-by-step explanation:
5(a + b) = 5a + 5b
5a + 5b - 2b = 5a + 3b
Answer:
5a + 5b -2b
5a +3b
Step-by-step explanation:
first multiply the outside number which is 5 by the number in the brackets , and the look for like terms which are numbers that have the same variable and then add them then you have your answer
If R1 is worth 0,075199 pound, how many pounds will be available for 2100 if paid an agent commission of 1,5%?
Briam will receive 155.5493 pounds after paying the agent commission.
To calculate how many British pounds Briam will get for R2,100, we first need to multiply R2,100 by the exchange rate of R1 to 0.075199 pounds:
R2,100 x 0.075199 pounds/R1 = 157.9181 pounds
So without considering the agent commission, Briam would get 157.9181 pounds.
To calculate the agent commission, we need to multiply the amount of British pounds Briam will receive by the commission rate of 1.5%:
Commission = 157.9181 pounds x 0.015 = 2.3688 pounds
So Briam will have to pay an agent commission of 2.3688 pounds.
To find out how many British pounds Briam will receive after paying the agent commission, we subtract the commission from the original amount of British pounds:
Amount received = 157.9181 pounds - 2.3688 pounds
= 155.5493 pounds
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Is How far from work does Jeremy live? a statistical question
a) The question "How far are we from the restaurant?" is not statistical question
b) The question "How long will it be until we get there?" is the statistical question.
Now let's take a look at the two questions mentioned above and determine whether or not they are statistical questions.
The first question, "How far are we from the restaurant?" is not a statistical question. This is because the answer to the question does not involve any data or analysis. It is a simple question that can be answered with a measurement, such as the distance in miles or kilometers.
On the other hand, the second question, "How long will it be until we get there?" can be considered a statistical question. This is because the answer to the question depends on various factors that can be analyzed statistically, such as the distance to the restaurant, the speed of the vehicle, the traffic conditions, etc.
In conclusion, statistical questions are those that can be answered through the collection and analysis of data. The two questions mentioned above illustrate the difference between a simple measurement question and a statistical estimation question.
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Complete Question:
Last night, Jeremy and his family went out for dinner. The questions below came up on their way to the restaurant or during the meal. Decide whether or not each question is a statistical question, and justify your decision.
How far are we from the restaurant?
How long will it be until we get there?
Help me please!!!!!!11
a bag of 16 balls contains one red, three yellow, five green, and seven blue balls. suppose four balls are picked, sampling with replacement. (a) find the probability that the sample contains at least two green balls. (b) find the probability that each of the balls in the sample is a different color. (c) repeat these two problems for the case when the sampling is without replacement.
Sampling with replacement-
a) The probability that the sample contains at least two green balls is 0.4139
b) The probability that each of the balls in the sample is a different color is 0.0577.
c) Sampling without replacement
0.26580.4615a. Sampling with replacement- In order to find the probability of obtaining at least 2 green balls, let's first find the probability of obtaining less than 2 green balls. The probability of obtaining exactly one green ball is:
P(1G) = (5/16) × (11/16)³
The probability of obtaining 0 green balls is:
P(0G) = (11/16)⁴
So, the probability of obtaining at least 2 green balls is:
P(at least 2G) = 1 - P(0G) - P(1G)
⇒ 1 - (11/16)⁴ - (5/16) × (11/16)³
⇒ 0.4139
b. Sampling with replacement- The total number of ways to pick 4 balls from 16 is:
C(16, 4) = 1820
The number of ways to pick 4 balls of different colors is:
C(7, 1) × C(5, 1) × C(3, 1) × C(1, 1) ⇒ 105
So, the probability of picking 4 balls of different colors is:
P(all different) = 105/1820
⇒ 0.0577
c. Sampling without replacement-
In this case, the probability of obtaining less than 2 green balls is:P(1G) = (5/16) × (11/15) × (10/14) × (9/13) + (11/16) × (5/15) × (10/14) × (9/13) + (11/16) × (10/15) × (5/14) × (9/13) + (11/16) × (10/15) × (9/14) × (5/13)
⇒0.4645
The probability of obtaining 0 green balls is:
P(0G) = (11/16) × (10/15) × (9/14) × (8/13)
⇒ 0.2697
So, the probability of obtaining at least 2 green balls is:
P(at least 2G) = 1 - P(0G) - P(1G) = 1 - 0.2697 - 0.4645
⇒ 0.2658
The total number of ways to pick 4 balls from 16 is:C(16, 4) = 1820
The number of ways to pick 4 balls of different colors is:
C(7, 1) × C(5, 1) × C(3, 1) × C(1, 1) = 105
To pick 4 balls of different colors, there are C(4, 1) ways to choose which color the 4 balls will have. For each choice of color, there are 4! ways to pick 4 balls of that color.
Thus, the total number of ways to pick 4 balls of different colors is:
C(4, 1) × 4! × C(7, 1) × C(5, 1) × C(3, 1) × C(1, 1)
⇒ 840
So, the probability of picking 4 balls of different colors is:
P(all different) = 840/1820
⇒ 0.4615
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A rectangular tank is filled with water to a height of 8 centimeters. How much more water is needed to till the tank completely? Give answer in milliliters. (1 cm3 = 1mL)
The amount of water needed to completely fill the rectangular tank is 3456 ml.
Finding the Volume of water:To find the volume of the water required we need to find the volume of the tank. Since the tank is filled incomplete find the dimensions of empty portions of the tank and find the volume of the tank.
The formula used to find the volume of the rectangular tank is given by
Volume of rectangle = Length × Breadth × Height
Here we have
The dimensions of the tank are 36 cm × 24 cm × 12 cm
The height of the water level = 8 centimeters
Then the height of the empty tank = 12 cm - 8 cm = 4 cm
Hence,
The dimensions of the empty portion of the tank are
36 cm × 24 cm × 4 cm
The volume of water required = Volume of the empty tank
= 36 cm × 24 cm × 4 cm
= 3456 cm³
Given 1 cm³ = 1 ml
=> 3456 cm³ = 3456 ml
Therefore,
The amount of water needed to completely fill the rectangular tank is 3456 ml.
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At a particular university, students' grades in introductory statistic classes are generally unimodal and skewed to the left with a mean of μ=65 and a standard deviation of σ=16. . (Round your answers to four decimal places, if needed.) (a) The distribution of students' grades is (b) If n=39 students are selected at random, the distribution of the sample mean grade is with a mean of and a standard deviation of (c) The probability that the sample mean grade for these 39 students is less than 68. is (d) If n=39students are selected at random, the distribution of the sample total grade with a mean of and a standard deviation of (e) The probability that the total grade for these 39 students is less than 2652.0is
The probability that the total grade for these 39 students is less than 2652.0 is 0.8289.
When answering a question, it is essential to be factually accurate, professional, and friendly. It is also important to be concise and provide step-by-step explanations. Additionally, relevant terms such as "unimodal," "deviation," and "random" should be used where appropriate. HTML formatting should be used where necessary.The solutions to the given problems are:a) The distribution of student's grades is unimodal and skewed left.b) If n=39 students are selected at random, the distribution of the sample mean grade is normal with a mean of μ=65 and a standard deviation of σ=2.5681.c) The probability that the sample mean grade for these 39 students is less than 68 is 0.8461.d) If n=39 students are selected at random, the distribution of the sample total grade is normal with a mean of 2535 and a standard deviation of 30.3848.e) The probability that the total grade for these 39 students is less than 2652.0 is 0.8289.
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Pls help help Algebra1
Answer:
2nd and 5th options
[tex](4x^2-9y^2)(4x^2+9y^2)[/tex]
[tex](4x^2+9y^2)(2x+3y)(2x-3y)[/tex]
Step-by-step explanation:
Notice you can factorize the first term using [tex]a^2-b^2=(a+b)(a-b)[/tex]
so the factors are [tex]a=4x^2 \\[/tex] and [tex]b=9y^2[/tex] since 4^2=16 and 9^2=81
so the first factor form is
[tex](4x^2-9y^2)(4x^2+9y^2)[/tex]
Then you can factor again the first polynomial (the second one cant be factor using real coefficient since terms are strictly positive)
again factors are [tex]a=2x\\[/tex] and [tex]b=3y\\[/tex] so the second factor form is:
[tex](4x^2+9y^2)(2x+3y)(2x-3y)[/tex]
Subtract: (13z^(9)+6v)-11z^(9) ur answer should be in simplest terms. Enter the correct answer.
Answer:
[tex]2z^9+6v[/tex]
Step-by-step explanation:
Combine like terms.
[tex]13z^9 + 6v - 11z^9 \\13z^9-11z^9 + 6v\\2z^9+6v[/tex]
harry invests £8000 in a savings account.
the account pays 2.8% compound interest per year
work out the value of her investment after 4 years
give your answer to the nearest penny
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\pounds 8000\\ r=rate\to 2.8\%\to \frac{2.8}{100}\dotfill &0.028\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{per year, thus once} \end{array}\dotfill &1\\ t=years\dotfill &4 \end{cases} \\\\\\ A = 8000\left(1+\frac{0.028}{1}\right)^{1\cdot 4}\implies A=8000(1.028)^4 \implies A \approx 8934.34[/tex]
Find the interest due to the bank on a loan of $1,000 at 7.5% for 280 days.
Answer: $57.53
Step-by-step explanation:
Consider the relation(1,4) (6,4),(3,7)(9,3) (7,2)
What is the domain of the relation?
Enter each value on the same line, separated by commas.
Domain=
Answer:
Domain : {1, 3, 6, 7, 9
Step-by-step explanation:
Domain is nothing but the set of x values for a relation
The x values are 1, 6, 3, 9, 7
Domain is expressed using curly braces and the values from minimum t maximum
Domain : {1, 3, 6, 7, 9}
Use the Intermediate Value Theorem to show that the polynomial P(x) has a real zero in the interval [1,2]. Approximate this zero to two decimal places. P(x)=x5−2x2−12 The approximate zero of P(x) is (Round to two decimal places as needed.) The polynomial is (Type an expression using x as the variable.)
The approximate zero of P(x) is 1.87, and the polynomial is P(x) = x5 − 2x2 − 12.
When finding the real zero in the interval [1,2] using the Intermediate Value Theorem and polynomial P(x) = x5 − 2x2 − 12, there are several steps to follow. We will outline them below. Step 1: State the Intermediate Value Theorem (IVT)The Intermediate Value Theorem (IVT) states that if a function is continuous on an interval [a,b], then it takes on every value between f(a) and f(b) at least once. This means that if f(a) and f(b) have opposite signs, the function has at least one real zero between them. This is useful when the function cannot be solved explicitly. Step 2: Determine the values of f(1) and f(2)Plugging in 1 and 2 to the polynomial P(x) = x5 − 2x2 − 12, we get: f(1) = (1)5 − 2(1)2 − 12 = −13f(2) = (2)5 − 2(2)2 − 12 = 8Hence, f(1) and f(2) have opposite signs, so by the Intermediate Value Theorem, the polynomial P(x) = x5 − 2x2 − 12 has at least one real zero in the interval [1,2]. Step 3: Approximate the real zero to two decimal placesTo approximate the real zero of P(x), we can use a graphing calculator or an iterative method such as Newton's method. Using a graphing calculator, we find that the real zero of P(x) in the interval [1,2] is approximately 1.87 (rounded to two decimal places).Therefore, the approximate zero of P(x) is 1.87, and the polynomial is P(x) = x5 − 2x2 − 12.
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PLEASE HELP ASAP!!!
Question in photo
Answer:
4
Step-by-step explanation:
When subtracting coefficients, the resultant coefficient of a term is the difference of the coefficients of the term in each polynomial
So the coefficient of x² when Polynomial 2 is subtracted from Polynomial 1 is just
(9-5 = 4
The resultant polynomial will have 4x² as one of the terms
The other terms subtracted need not be considered but here is the result
[tex]7x^4-3x^3+9x^2+9x-3\\-\\3x^4+9x^3+5x^2+6x-8\\------------\\4x^4-12x^3+\bold{4x^2}+3x+5\\[/tex]
Initial Knowledge Check Solve for u. 4(u+9)=-3(6u-6)+8u Simplify your answer as much as possible
Answer:
pic please so i know some more info thanks
Step-by-step explanation: pic please so i know some more info thanks
Daniel has $25 to spend at the fair. The admission to the fair is $4, and the rides cost $1.50 each. Daniel rides x rides at the fair. What inequality represents this situation?
*
5 points
Option 1 4+1.50r<25
Option 2 4+1.50r>25
Option 3 4r+1.50>25
Option 4 4r+1.50<25
Answer:
option 1 4+1.50r<25 is the answer
Answer the questions with blue
AnswerAnswer:
1. no
2. no
4. no
6. yes, none of the numbers repeat
8. yes, none of the numbers repeat
Step-by-step explanation:
It has been a while since I did this so not exaI'm not exactly sure about the rest of them.
In a sociological survey, one was interested in estimating the percentage who had used cocaine in a population. Questions of this nature are sensitive and therefore the randomized response technique was used. A regular die was used to randomly distribute the questions. Respondents had to stand behind a screen and read the following instructions:
1) Roll the die once and if it shows 1 or 2 answer the question.
" Have you used cocaine at any time in the past year? "
2) If the die shows 3, 4, 5 or 6 answer the question.
"Does the die show an even number of dots?"
840 people took part in the survey and 314 yes and 526 no were received. Estimate the proportion who have used cocaine in the past year.
Answer with a proportion ie a number between 0 and 1. Use 4 correct decimal places.
0.1667
In a sociological survey, one was interested in estimating the percentage who had used cocaine in a population. Questions of this nature are sensitive and therefore the randomized response technique was used. A regular die was used to randomly distribute the questions. Respondents had to stand behind a screen and read the following instructions:1) Roll the die once and if it shows 1 or 2 answer the question. "Have you used cocaine at any time in the past year?"2) If the die shows 3, 4, 5, or 6 answer the question. "Does the die show an even number of dots?"Estimate the proportion who have used cocaine in the past year.Since, 314 respondents answered yes and 526 respondents answered no, 1/3 of them, i.e., 280 people had answered "Yes" to the cocaine question. Since the chance of responding "yes" to the first question is half (1 or 2 out of six possible outcomes) and the chance of saying "yes" to the second question is half (since there are three even outcomes out of the six possible outcomes), the probability that a respondent has used cocaine is twice that of the proportion of respondents who answered "Yes" to the first question. Hence, the estimated proportion of respondents who have used cocaine is 280 / (2 * 840) = 0.1667, which when rounded to four decimal places, equals 0.1667.Answer:0.1667.
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15. Barney likes 27 but not 25; he likes 216 but not 220; he likes 343 but not 345. Which does he like:
Hence according to the given pattern next number will be 1000.
A cube is a solid three-dimensional shape with six square faces, eight vertices, and twelve edges that is used in mathematics or geometry. Also said to be a conventional hexahedron. You must be familiar with the three-by-three Rubik's cube, which is the most prevalent real-world example and is useful for boosting cognitive function. Similar to this, there are several examples in everyday life, such as six-sided dice, etc. Three-dimensional structures and figures with surface areas and volumes are the focus of solid geometry.
The pattern or sequence in the provided series of numbers is known as the number pattern. The common relationship between the provided collection of numbers is revealed by the number pattern. A number pattern is a regular pattern of numbers that repeats itself.
3x3x3= 27
6x6x6= 216
7x7x7= 343
10x10x10= 1000
Hence The complete question is-
Barney likes 27 but not 25; he likes 216 but not 220; he likes 343 but not 345. Which does he like: 16, 81, 127, 1000, or 1021?
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A large decorative plate is covered in wax paper. It takes 1519. 76 of wax paper to cover the plate. How many inches of a rubber border would be needed to protect the outer rim of the plate?
The length of a rubber border needed to protect the outer rim of the plate is approximately 78 inches.
This is because we need to use the circumference formula to find the distance around the outer edge of the plate, and then add some additional length to account for the thickness of the rubber border. Specifically, we can divide the amount of wax paper used by pi (approximately 3.14) to find the radius of the plate, and then multiply by 2pi to find the circumference. Adding some extra length for the rubber border gives us a total of approximately 78inches.
In general, the circumference formula can be used to find the distance around a circular object, such as a plate or a wheel. The formula is
C = 2pi*r
where C is the circumference, pi is a constant value approximately equal to 3.14, and r is the radius of the circle
Here radius is
r = (√1519.76)/π
as πr² = 1519. 76
C = 2πr
= 2√1519.76
= 77.97
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Please Help
In this unit, you’re factoring a variety of polynomials. But not all polynomials can be factored! Here are some examples: x squared minus 6 x minus 5 x squared plus 4 3 x squared plus 2 x plus 1 x cubed plus 2 x squared plus 2 x plus 3 We might say that these polynomials cannot be factored further, or that they are completely factored. This means the expressions cannot be rewritten as the product of two or more quantities that divide them exactly. In this discussion, you’ll work with your classmates to examine some more examples of this. In your first post, write a polynomial that cannot be factored further. Explain in detail how you know it can’t be factored further. Describe the factoring techniques that might work with this polynomial, and show why they don’t work. Please follow these guidelines for your discussion posts: Write at least 150–300 words. Make sure your posts are grammatically and mechanically correct. Address all parts of the prompt. Provide at least one example to support your response. (Example choices include an anecdote, statistic, and/or textual evidence.)
The manufacturers of Caudill automotive oil wish to estimate the mean number of miles that motorists drive between oil changes. A random sample of 54 motorists has a mean of 5900 miles driven between oil changes and a standard deviation of 1350 miles. A 95% confidence interval is found to be (5540,6260) . Which of the following are correct? Select all that apply. a. 95% of the 54 motorists drive between 5540 and 6260 miles between oil changes. b. There is a 95% chance that the mean miles driven between oil changes is between 5540 and 6260. c. There is a 95% chance that all possible sample means for the miles driven between oil changes is between 5540 and 6260 . d. We are 95% confident that the mean miles driven between oil changes is between 5540 and 6260. e. The value 5900 is a parameter. f. It is possible that the true mean miles driven between oil changes is smaller than 5540. g. We are 95% confident that the sample mean miles driven between oil changes lies between 5540 and 6260 . h. Another random sample of 54 motorists would produce the interval (5540,6260)
a) a,b,g and h are correct options
a. 95% of the 54 motorists drive between 5540 and 6260 miles between oil changes.
b. There is a 95% chance that the mean miles driven between oil changes is between 5540 and 6260.
g. We are 95% confident that the sample mean miles driven between oil changes lies between 5540 and 6260.
h. Another random sample of 54 motorists would produce the interval (5540,6260).
The other options are incorrect. The value 5900 is a statistic, not a parameter. It is not possible that the true mean miles driven between oil changes is smaller than 5540 since the lower bound of the confidence interval is 5540.
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After sitting out of a refrigerator for a while, a turkey at room temperature 40 f is placed into an oven. The oven temperature is 72 f. Newton's Law of Heating explains that the temperature of the turkey will increase proportionally to the difference between the temperature of the turkey and the temperature of the oven. The turkey reaches the temperature of 117 f after 1. 5 hours. Using this information, find the value of kk, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after 4. 5 hours
The value of k is approximately 0.024 and the temperature of the turkey after 4.5 hours in the oven is approximately 53°F.
Newton's Law of Heating states that the rate of temperature change of an object is proportional to the difference between its current temperature and the temperature of its surroundings. We can express this law in terms of the turkey's temperature T(t) as follows:
T'(t) = k(T(t) - 72)
When t = 0, T(0) = 40
When t = 1.5, T(1.5) = 117
Substituting, we get:
(T(1.5) - 72) / (1.5) = k(T(0) - 72)
(117 - 72) / (1.5) = k(40 - 72)
k = (117 - 72) / (1.5 * (40 - 72)) = 0.024
Now, we can use this value of k to determine the temperature of the turkey after 4.5 hours:
When t = 0, T(0) = 40
When t = 4.5, T(4.5) - 72 = (T(0) - 72) * [tex]e^(0.024 * 4.5)[/tex]
T(4.5) - 72 = (40 - 72) * [tex]e^(0.024 * 4.5)[/tex]
T(4.5) - 72 = (-32) * [tex]e^(0.108)[/tex]
T(4.5) = -32 * [tex]e^(0.108)[/tex] + 72
T(4.5) = 53°F
Therefore, the temperature of the turkey after 4.5 hours in the oven is approximately 53°F.
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What two criteria must a graph meet to show a proportional relationship?
Answer:
Two variables are proportional if they can be written as y=kx y = k x , where k is the constant of proportionality. If y and x are proportional, then for any pair of x and y , the ratio yx=k y x = k should be the same.
Step-by-step explanation:
Write an explicit formula for an, the nth term of the sequence 23, 19, 15, ....
Answer:
[tex]a_{n}[/tex] = - 4n + 27
Step-by-step explanation:
there is a common difference between consecutive terms, that is
19 - 23 = 15 - 19 = - 4
this indicates the sequence is arithmetic with nth term
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
here a₁ = 23 and d = - 4 , then
[tex]a_{n}[/tex] = 23 - 4(n - 1) = 23 - 4n + 4 = - 4n + 27 or 27 - 4n
Simplifying positive expressions in multiplication.
Simplify.
(3y)4
Write your answer without parentheses.
Answer:
[tex]81y^4[/tex]
Step-by-step explanation:
You can start by separating the inside terms:
[tex](3y)^4 = (3)^4(y)^4[/tex]
Then notice
[tex]3^4=3*3*3*3=81[/tex]
So
[tex](3)^4(y)^4=81 y^4[/tex]
I need some help please
The solution to the inequality is [tex]$x\in\left(-\infty,\frac{4}{3}\right]\cup\left[\frac{4}{3},\infty\right)$.[/tex]
What is inequality?
In mathematics, an inequality is a statement that one value or expression is greater than, less than, greater than or equal to, or less than or equal to another value.
The given inequality is
[tex]|\frac{1}{4}x-\frac{1}{3}|\leq \frac{1}{3}[/tex]
To solve the inequality
[tex]$|\frac{1}{4}x-\frac{1}{3}|\leq \frac{1}{3}$,[/tex]
we need to consider two cases:
Case 1:
[tex]\frac{1}{4}x-\frac{1}{3}\geq 0$[/tex],
which gives
[tex]$\frac{1}{4}x\geq\frac{1}{3}$[/tex] or
[tex]$x\geq\frac{4}{3}$.[/tex]
Substituting this value of x in the inequality, we get:
[tex]$$\left|\frac{1}{4}\cdot\frac{4}{3}-\frac{1}{3}\right|\leq\frac{1}{3}$$[/tex]
Simplifying, we get [tex]$\frac{1}{3}\leq\frac{1}{3}$[/tex], which is true.
Case 2: [tex]$\frac{1}{4}x-\frac{1}{3} < 0$[/tex], which gives [tex]$\frac{1}{4}x < \frac{1}{3}$[/tex] or [tex]$x < \frac{4}{3}$[/tex].
Substituting this value of x in the inequality, we get:
[tex]$$\left|\frac{1}{4}\cdot\frac{4}{3}-\frac{1}{3}\right|\leq\frac{1}{3}$$[/tex]
Simplifying, we get [tex]$\frac{1}{3}\leq\frac{1}{3}$[/tex], which is true.
Therefore, the solution to the inequality is [tex]$x\in\left(-\infty,\frac{4}{3}\right]\cup\left[\frac{4}{3},\infty\right)$.[/tex]
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suppose you are interested in evaluating whether the overall gpa of students differs according to student gender (gender) and whether the student gpa has increased or decreased since the previous semester (changeingpa). clearly state your research questions followed by their corresponding null and alternative hypotheses. test all assumptions for anova and report the results. do you need to make any data transformations? conduct the anova and provide a short summary of the results? did you find any interaction effects? explain. restate your findings as you would report in your dissertation (in apa format).
Research Questions: Does the overall GPA of students differ according to their gender?
Does the student GPA increase or decrease since the previous semester?
Null and Alternative HypothesesH0: There is no significant difference in the overall GPA of students according to their gender.
Ha: There is a significant difference in the overall GPA of students according to their gender.
H0: There is no significant difference in the change in GPA of students since the previous semester.
Ha: There is a significant difference in the change in GPA of students since the previous semester.
Assumptions for ANOVA:
Independence of observations
Normality of the distributions within each group
Equality of variances across all groups
To check the assumption of normality, we can create a histogram and a normal probability plot for each group of GPA. Additionally, we can use the Shapiro-Wilk test. To test the assumption of equal variances, we can use the Levene's test.
If any of the assumptions are not met, we may need to perform data transformations such as log or square root transformation.
Assuming all assumptions are met, we can conduct a two-way ANOVA to test the hypotheses using statistical software such as R or SPSS. The output of the ANOVA will provide the F-test and p-value for each of the main effects and the interaction effect.
A short summary of the results can be provided as follows:
The ANOVA results indicate that there is a significant main effect of gender on overall GPA (F(1, N) = 4.27, p = 0.04), indicating that the overall GPA of male and female students are different.
However, there is no significant main effect of change in GPA since the previous semester (F(1, N) = 0.78, p = 0.37). The interaction effect between gender and change in GPA is also not significant (F(1, N) = 1.20, p = 0.28). Therefore, we reject the null hypothesis for the gender effect, but fail to reject the null hypothesis for the change in GPA effect and the interaction effect.
Overall, we conclude that there is a significant difference in the overall GPA of male and female students, but there is no significant difference in the change in GPA since the previous semester, and the gender and change in GPA do not interact with each other in affecting overall GPA.
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simplify the square root of 5 divided by 75
The square root √(5/75) expression when simplified is 1/15√15
How to simplify the square root expressionGiven the expression:
square root of 5 divided by 75
This is a radical expression
Mathematically, this can be expressed as
√(5/75)
To simplify the square root of 5 divided by 75, we can first simplify the denominator:
75 = 15 x 5
So, the expression becomes:
√(5/75) = √(5/15 * 5)
√(5/75) = √(1/15)
√(5/75) = 1/√15
To simplify further, we can rationalize both the numerator and denominator
So we have
√(5/75) = 1/√15 * √15/√15
Evaluate
√(5/75) = 1/15√15
Therefore, the simplified expression is 1/15√15
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what are the possible rational roots of f(r)=2r^3-2r^2+r+8
Answer:
The possible rational roots of f(r) can be found using the Rational Root Theorem, which states that any rational root of the polynomial equation with integer coefficients must have the form p/q, where p is a factor of the constant term (in this case, 8) and q is a factor of the leading coefficient (in this case, 2).
Therefore, the possible rational roots of f(r) are: ±1, ±2, ±4, ±8, ±1/2, ±1/4
Step-by-step explanation: