The width of the counter is 3 meters or 9/3 meters as a fraction considering the condition that its value in meters is a number such that its power squared added to its power to the fourth will give 90.
Let's represent the width of the counter in meters as "x". As per the given condition, we can form the equation:
x² + [tex]x^4[/tex] = 90
Simplifying the equation, we get:
[tex]x^4[/tex] + x² - 90 = 0
Now, we can factor the equation as:
(x² - 9)(x² + 10) = 0
So, the possible values of "x" are:
x² = 9 or x² = -10
Since the width of the counter cannot be negative, we can ignore the second solution. Therefore, we have:
x² = 9
x = ±√9
x = ±3
Since the width of the counter cannot be negative, the only possible solution is:
x = 3
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The question is -
In the entrance hall, there is a counter for the sale of tickets, the chief architect will ask one of your assistants to determine how wide it is with the following conditions: its value in meters is a number such that its power squared added to its power to the fourth will give 90. Does the counter have to measure the width?
find the point on the curve r(t)=(2cost 2sint e^t)
To find the point on the curve r(t)=(2cost, 2sint, e^t) we need to evaluate the x, y, and z coordinates at a specific value of t. Let's choose t=0 for simplicity.
So, plugging in t=0 we have:
r(0) = (2cos(0), 2sin(0), e^0)
r(0) = (2, 0, 1)
Therefore, the point on the curve at t=0 is (2, 0, 1).
To find the point on the curve r(t) = (2cos(t), 2sin(t), e^t) for a specific value of t, you can plug in the value of t into the parameterized equation:
1. Replace t with the specific value you're interested in (e.g., t = a).
2. Compute 2cos(a) for the x-coordinate.
3. Compute 2sin(a) for the y-coordinate.
4. Compute e^a for the z-coordinate.
The point on the curve will be (2cos(a), 2sin(a), e^a).
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In each of the following find the pdf of Y.
(a) Y = X2 and fX(x) = 1, 0 < x < 1
(b) Y = −log X and X has pdf
The PDF of Y is1/(2*sqrt(y)), for 0 < y < 1 and fX(e^(-Y)) * |-e^(-Y)|.
(a) To find the pdf of Y when Y = X² and fX(x) = 1, 0 < x < 1, we first find the inverse function of Y = X², which is X = sqrt(Y). Next, we compute the derivative of X with respect to Y: dX/dY = 1/(2*sqrt(Y)). Now, we apply the change of variables formula to find the pdf of Y:
fY(y) = fX(x) * |dX/dY| = 1 * |1/(2*sqrt(y))| = 1/(2*sqrt(y)), for 0 < y < 1.
(b) To find the pdf of Y when Y = -log(X) and X has pdf, we first find the inverse function of Y = -log(X), which is X = e^(-Y). Next, we compute the derivative of X with respect to Y: dX/dY = -e^(-Y). We need the pdf of X to proceed, which is not provided in the question. Assuming we have the pdf of X as fX(x), we apply the change of variables formula to find the pdf of Y:
fY(y) = fX(x) * |dX/dY| = fX(e^(-Y)) * |-e^(-Y)|.
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you want to determine the minimum total sample size needed to detect the desired effect of 50 mg/l lower than the true mean water hardness of the stock ponds using a significance level of 0.05 and a power of 0.8.
To determine the minimum total sample size needed to detect a desired effect of 50 mg/l lower than the true mean water hardness of the stock ponds, you would use a significance level of 0.05 and a power of 0.8. This ensures the statistical test has adequate sensitivity to identify the effect while maintaining a low risk of false-positive findings.
The minimum total sample size needed to detect the desired effect of 50 mg/l is lower than the true mean water hardness of the stock ponds, a significance level of 0.05 and a power of 0.8 are required. The sample size calculation requires knowledge of the expected effect size, variability of the data, and significance level
. Using statistical software, the calculation can be done to obtain the minimum sample size required to achieve the desired level of significance and power. It is important to ensure that the sample size is sufficient to detect the desired effect size and to minimize the risk of a type II error.
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1. for a data set f x( ) to be modeled by an exponential function, which relationship is necessary between successive values of f x( ) for equally spaced values of x? a the first differences are approximately equal. b the second differences are approximately equal. c the ratios are approximately equal.
To model a data set f(x) by an exponential function for equally spaced values of x, the necessary relationship between successive values of f(x) is:
c) The ratios are approximately equal.
For a data set f x( ) to be modeled by an exponential function, the necessary relationship between successive values of f x( ) for equally spaced values of x is that the ratios are approximately equal. This means that the ratio of any two consecutive values of f x( ) is approximately the same, regardless of the specific values of x chosen. This property is known as exponential growth or decay, and it is a fundamental characteristic of many natural and social phenomena. Therefore, if we observe that the ratios of successive values of f x( ) are approximately equal, we can conclude that the data set is likely to be modeled by an exponential function.
To model a data set f(x) by an exponential function for equally spaced values of x, the necessary relationship between successive values of f(x) is:
c) The ratios are approximately equal.
In other words, when the values of x are equally spaced, the ratio of f(x+1)/f(x) should be approximately the same for each successive pair of points in the data set. This constant ratio is a characteristic of exponential functions.
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if $f(x)$ is a polynomial of degree $7$, and $g(x)$ is a polynomial of degree $7$, then what is the product of the minimum and the maximum possible degrees of $f(x) g(x)$? (assume that $f(x) g(x)$ is nonzero.)
the product of the minimum and maximum possible degrees of $f(x)g(x)$ is $2 \times 14 = \boxed{28}$.
The product of the minimum and maximum possible degrees of $f(x)g(x)$ is equal to the degree of the product polynomial.
If $f(x)$ is a polynomial of degree 7, then it can be written as:
$f(x) = a_7x^7 + a_6x^6 + \cdots + a_1x + a_0$
Similarly, if $g(x)$ is a polynomial of degree 7, it can be written as:
$g(x) = b_7x^7 + b_6x^6 + \cdots + b_1x + b_0$
The product of $f(x)$ and $g(x)$ is given by:
$f(x)g(x) = (a_7x^7 + a_6x^6 + \cdots + a_1x + a_0)(b_7x^7 + b_6x^6 + \cdots + b_1x + b_0)$
When we expand this product using the distributive property, we get a polynomial of degree 14, which is the sum of all possible products of the terms of $f(x)$ and $g(x)$.
The degree of a term in the product polynomial is the sum of the degrees of the terms being multiplied. Therefore, the degree of the product polynomial will be at most $7+7=14$.
In order to obtain the minimum possible degree of the product polynomial, we need to construct a scenario where the highest degree terms in $f(x)$ and $g(x)$ multiply to give the highest degree term in the product polynomial, and similarly for the lowest degree terms.
Thus, we want to choose $f(x)$ and $g(x)$ such that $a_7b_7 \neq 0$ and $a_0b_0 \neq 0$. In this case, the highest degree term in the product polynomial will be $a_7b_7x^{14}$, and the lowest degree term will be $a_0b_0$.
Therefore, the minimum possible degree of the product polynomial is 2.
On the other hand, the degree of the product polynomial will be exactly 14 if $a_7b_7 \neq 0$ and $a_0b_0 \neq 0$. This can be seen from the fact that the sum of the degrees of all terms in the product polynomial is 14.
Therefore, the maximum possible degree of the product polynomial is 14.
Hence, the product of the minimum and maximum possible degrees of $f(x)g(x)$ is $2 \times 14 = \boxed{28}$.
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Jace wrote a sentence as an equation.
56 is 14 more than a number.
14 + p = 56
Which statement best describes Jace’s work?
Jace is not correct. The phrase more than suggests using the symbol > and Jace did not use that symbol.
Jace is not correct. He was correct to use addition, but the equation should be 56 + p = 14.
Jace is not correct. The first number in the sentence is 56, so the equation should start with 56.
Jace is correct. The phrase more than suggests addition, so Jace showed that 14 plus a variable equals 56.
The required, Option D "Jace is correct. The phrase more than suggests addition, so Jace showed that 14 plus a variable equals 56." is correct.
Jace is correct. The sentence "56 is 14 more than a number" implies that you can start with a certain number (which is unknown) and add 14 to it to get 56. Jace correctly used addition to represent this relationship in equation 14 + p = 56, where p represents the unknown number. The phrase "more than" does not necessarily suggest the use of the symbol >, as it can also be interpreted as an additional relationship.
Therefore, Jace's work is accurate and correctly represents the relationship between 56 and a certain number that is 14 less than 56.
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lu vaccines are not thought to make people more susceptible to other respiratory infections.
A 2012 study external icon suggested that flu vaccination might make people more susceptible to other respiratory infections. After that study was published, many experts looked into this issue further and conducted additional studies to see if the findings could be replicated. No other studies have found this effect. It’s not clear why this finding was detected in the one study, but the majority of evidence suggests that this is not a common or regular occurrence and that flu vaccination does not, in fact, make people more susceptible to other respiratory infections.
Prompt: Which hypothesis testing-related concept may explain why only one research study found that getting the influenza vaccine might make people more susceptible to other respiratory infections, but none of the studies that followed found that connection?
The concept that may explain this is the possibility of a Type I error in the initial study. A Type I error occurs when a hypothesis is rejected when it is actually true.
The concept that may explain why only one research study found that getting the influenza vaccine might make people more susceptible to other respiratory infections, while none of the studies that followed found that connection, is called "Type I error" in hypothesis testing.
Type I error occurs when a study incorrectly rejects a null hypothesis that is actually true, leading to a false positive result. In this case, the initial study may have found a significant effect due to chance, while subsequent studies did not find the same effect, suggesting that the initial finding was likely a Type I error. In this case, it is possible that the initial study had a false positive result, and subsequent studies that failed to find the same effect were more accurate.
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Use long division to determine the decimal equivalent of fraction with numerator 7 and denominator 9
The decimal equivalent of a fraction with 7 as the numerator and 9 as the denominator is 0.77
To convert a fraction into decimals we have to divide the numerator and the denominator and the quotient we get is the answer.
First, we have to write the numerator as the Dividend and the denominator as the divisor. Thus, we get 7 ÷ 9.
Since the divisor is greater than the dividend, we add a zero after it and in the quotient, we add a decimal. Thus we get 70 ÷ 9
We write the largest multiple of 9 which is smaller than 70 which is 63
and add 7 in the quotient as 7 * 9 is 63 and we get the quotient as 0.7 and the remainder of 70 - 63 = 7.
We continue the above steps until two places of decimals or as much as required. And we get 0.77 as the answer.
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Marissa is selecting a sports ball she selects one at random replaces it and then selects another ball what is the probability that Marissa selects a football both times
The probability that Marissa selects a football both times is 4/81.
We have,
Number of American Football = 4
Number of Football = 2
Number of Basketball = 1
Number of Baseball = 2
So, the probability that Marissa selects a football both times
= 2/ 9 x 2/9
= 4/ 81
Thus, the required probability is 4/81.
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A triangle has an angle that measures 110°. The other two angles are in a ratio of 3:11. What are the measures of those two angles?
The measure of the other two angles of the triangle in the ratio 3:11 are 15° and 55° respectively.
What is ratioA ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. It can be used to express one quantity as a fraction of the other ones.
We recall that the sum of interior angles of a triangle is 180° and given one angle 110° and the ratio of the other two as 3: 11, then by comparison using the variable x;
110° + 3x + 11x = 180°
110° + 14x = 180°
14x = 180° - 110° {collect like terms}
14x = 70°
x = 70/14 {divide through by 14}
x = 5
so the two angles will be:
3(5) = 15°
11(5) = 55°
Therefore, the measure of the other two angles of the triangle in the ratio 3:11 are 15° and 55° respectively.
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For each of the following statements, identify the number that appears in boldface type as the value of either a population characteristic or a statistic.
(a) A department store reports that 83% of all customers who use the store's credit plan pay their bills on time.
Population characteristic
Statistic
(b) A sample of 100 students at a large university had a mean age of 24.3 years.
Population characteristic
Statistic
(a) A department store reports that 83% of all customers who use the store's credit plan pay their bills on time.
Your answer: Population characteristic (b) A sample of 100 students at a large university had a mean age of 24.3 years.
Your answer: Statistic
(a) Population characteristic: There is no number in boldface type in statement (a) that represents a population characteristic.
Statistic: The number in boldface type is 83%, which represents the percentage of customers who use the store's credit plan and pay their bills on time. This is a statistic because it is calculated from a sample of customers who use the credit plan and does not represent the entire population of customers who shop at the department store.
(b) Population characteristic: The number in boldface type in statement (b) does not represent a population characteristic because the statement only refers to a sample of students, not the entire population of students at the university.
Statistic: The number in boldface type is 24.3 years, which represents the mean age of the sample of 100 students. This is a statistic because it is calculated from a sample of students and does not represent the entire population of students at the university.
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(30 points)
A research company is performing an observational study of a certain endangered species of desert salamander to determine whether the species can survive in a new habitat.
Which reason provides a good rationale for avoiding randomization in this observational study?
(a). Placing a number of the endangered species in a new habitat where they may or may not survive is unethical.
(b). The observational study is too expensive to run.
(c). Food sources in the new habitat would not be the same as those in the species' present habitat.
(d). The species population may exceed expectations in the new habitat.
Placing a number of the endangered species in a new habitat where they may or may not survive is unethical. Option A
What does it mean to randomize a study?The random technique of assigning participants to treatment and control groups makes the assumption that each participant has an equal chance of being assigned to any group.
An observational study would involve the researchers observing the species in both its old and new habitats without altering any variables or introducing any novel situations. The researchers could gather data on the behavior and survival rates of the species in both habitats using this method without putting them in risk.
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00.110.220.0530.65find the mean of this probability distribution. round your answer to one decimal place.
The mean of this probability distribution is 2.3 when rounded to one decimal place.
finding mean:To find the mean of this probability distribution, you'll need to multiply each value by its corresponding probability and then sum the products.
1. Multiply each value by its probability:
- 0 * 0.1 = 0
- 1 * 0.2 = 0.2
- 2 * 0.05 = 0.1
- 3 * 0.65 = 1.95
2. Sum the products:
- 0 + 0.2 + 0.1 + 1.95 = 2.25
So, the mean of this probability distribution is 2.3 when rounded to one decimal place.
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Simplify completely 3(2x+y+10)+2(x-y)
Answer: Simplified answer is 8x+y+30
I'm lazy no judging
Find the Volume of this solid
Answer:
Step-by-step explanation:
21x9
181
Which expression is equivalent to (65. 85)3?
Answer: 197.55
Step-by-step explanation:
multiply the two numbers 65.85 times 3
Based on the simulations from parts (A) thru (E), what is the shape of the distribution of "heads"? (This is a reading assessment question. Be certain of your answer because you only get one attempt on this question.) Choose the correct shape of the distribution below. A. Skewed left B. Bell-shaped C. Skewed right
Based on the simulations from parts (A) thru (E), the shape of the distribution of "heads" is bell-shaped. Therefore, the correct answer is B. Bell-shaped.
What is the most probable distribution's shape?
Bell-shaped distributions, sometimes referred to as normal or Gaussian distributions in math and science, are the most significant probability distribution shapes since they are typically the result of sufficiently big data sets from naturally occurring random variables.
The mathematical idea known as the normal distribution, and also referred to as the Gaussian distribution, is commonly defined by the mound form.
When data points are utilized to plot a line for a particular item that satisfies the requirements of the normal distribution, a form called a mound is produced.
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A random sample of 100 automobile owners shows that in the state ofVirginia, an automobile is driven on the average 23,500 kilometersper year with a standard deviation of 3900 kilometers.
b) What can we assert with 99% confidence about the possible sizeof error if we estimate the average number of kilometers driven bycar owners in Virginia is to be 23500 per year?
Based on the information provided, we can assert with 99% confidence that the possible size of error in estimating the average number of kilometers driven by car owners in Virginia to be 23,500 kilometers per year is within a range of +/- 774.14 kilometers. This is calculated using the formula:
Margin of error = z-score x (standard deviation / square root of sample size)
Where the z-score for a 99% confidence level is 2.576. Plugging in the values, we get:
Margin of error = 2.576 x (3900 / square root of 100)
Margin of error = 2.576 x 390
Margin of error = 1004.64 / 2
Margin of error = +/- 774.14
Therefore, we can assert that the estimated average number of kilometers driven by car owners in Virginia is within a range of 22,725.86 kilometers to 23,274.14 kilometers with 99% confidence.
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2) Write MN as the sum of unit vectors for M(- 3/4, 5, 2/3) and N(6, - 9, ⅗)
The sum of unit vectors for M(- 3/4, 5, 2/3) and N(6, - 9, ⅗) is (27/4√(8329/900))i - (14/√(8329/900))j - (1/15√(8329/900))k.
To write MN as the sum of unit vectors, we first need to find the vector MN, which is the difference between the position vectors of M and N. Then, we divide this vector by its magnitude to obtain a unit vector in the same direction. Finally, we express MN as the sum of these unit vectors.
MN = N - M = (6 - (-3/4), -9 - 5, 3/5 - 2/3) = (27/4, -14, -1/15)
|MN| = √((27/4)² + (-14)² + (-1/15)²) = √(8329/900)
Unit vector in the direction of MN = MN/|MN| = (27/4√(8329/900), -14/√(8329/900), -1/15√(8329/900))
Therefore, MN can be written as the sum of unit vectors:
MN = (27/4√(8329/900))i - (14/√(8329/900))j - (1/15√(8329/900))k
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a white number cube, a red number cube, and a green number cube, each with faces numbered 1 to 6, are tossed at the same time. what is the probability of all three cubes landing on odd numbers given that the white cube landed on three?
the probability of all three cubes landing on odd numbers given that the white cube landed on three is 1/24
Since the white cube landed on 3, we only need to consider the red and green cubes.
The probability of each cube landing on an odd number is 1/2, since there are 3 odd numbers out of 6 total on each cube.
The probability of both the red and green cubes landing on odd numbers is the product of their individual probabilities:
P(red and green both odd) = P(red odd) x P(green odd) = 1/2 x 1/2 = 1/4
Therefore, the probability of all three cubes landing on odd numbers given that the white cube landed on three is:
P(white=3 and red and green both odd) = P(red and green both odd | white=3) x P(white=3)
= (1/4) x (1/6) = 1/24
So the probability of all three cubes landing on odd numbers given that the white cube landed on three is 1/24
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When there is a treatment or behavior for which researchers want to study risk, they often compare it to the _______ risk, which is the risk without the treatment or behavior.
When there is a treatment or behavior for which researchers want to study risk, they often compare it to the baseline risk, which is the risk without the treatment or behavior.
The baseline risk provides a reference point for evaluating the effect of the treatment or behavior on the outcome of interest.
For example, if researchers want to study the risk of a certain disease among smokers compared to non-smokers, the baseline risk would be the risk of the disease among non-smokers. By comparing the risk of the disease among smokers to the baseline risk among non-smokers, researchers can determine the extent to which smoking increases the risk of the disease.
Similarly, if researchers want to study the risk of a certain side effect associated with a medication, the baseline risk would be the risk of the side effect in the absence of the medication. By comparing the risk of the side effect among people taking the medication to the baseline risk among people not taking the medication, researchers can determine the extent to which the medication increases the risk of the side effect.
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Find the absolute maximum and absolute minimum values of the function
f(x)= x3 + 6x2 −63x +8
over each of the indicated intervals.
(a) Interval = [−8,0].
The absolute minimum value of the function is 120 which occurs at x = -8. To find the absolute maximum and minimum values of the function f(x) = x^3 + 6x^2 - 63x + 8 over the interval [-8, 0], you need to first find the critical points by taking the first derivative and setting it to zero, and then evaluate the function at the critical points and the endpoints of the interval.
1. Take the derivative of f(x):
f'(x) = 3x^2 + 12x - 63
2. Set f'(x) to zero and solve for x:
3x^2 + 12x - 63 = 0
Divide by 3:
x^2 + 4x - 21 = 0
Factor:
(x+7)(x-3) = 0
So, the critical points are x = -7 and x = 3.
However, only x = -7 is within the interval [-8, 0].
3. Evaluate f(x) at the critical point x = -7 and at the endpoints of the interval, x = -8 and x = 0:
f(-7) = (-7)^3 + 6(-7)^2 - 63(-7) + 8 = 120
f(-8) = (-8)^3 + 6(-8)^2 - 63(-8) + 8 = 64
f(0) = 0^3 + 6(0)^2 - 63(0) + 8 = 8
Comparing the values of f(x) at these points, we find:
Absolute maximum: f(-7) = 120
Absolute minimum: f(0) = 8
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How many miles can i walk in 110 minutes if i walk 3 miles in 70 minutes?
The number of miles that one can walk in 110 minutes with such a pace that they cover 3 miles in 70 minutes is 4.72.
The unitary method refers to the process in which by dividing we find the value of one by dividing. And then finding the value of a specific quantity by multiplying the one by the given number.
Therefore, we are given the number of miles traveled in 70 minutes and we can find the number of miles covered in 1 minute by division.
Number of miles in 70 minutes = 3 miles
Number of miles in 1 minute = 3 ÷ 70
= 0.042 miles
According to the question, we have to find miles covered in 110 minutes. To find it we multiply the miles in one minute by 110.
Number of miles in 110 minutes = 0.042 * 110
= 4.72 miles
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The team won 3/8 of its games and lost the rest. What was the team’s win-loss ratio?
Based on the fractional winning of ³/₈, the team's win-loss ratio is 3:5.
What is the ratio?The ratio shows the relative size that one quantity or value has when compared to another quantity or value.
We depict ratios in decimals, fractions, or percentages. We can also use the standard ratio form (:) to show ratios.
The fraction of games won by the team = ³/₈
The fraction of games lost by the ream = ⁵/₈ (1 - ³/₈)
The implication is that for every 8 games, the team won 3 but lost 5 games.
The ratio of win-loss = 3:5
The sum of ratios = 8 (3 + 5)
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Identify why this assignment of probabilities cannot be legitimate: P(A) = 0.4, P(B) = 0.3, and P( AB=0.5 (A) A and B are not given as disjoint events (B) A and B are given as independent events (GP(A and B) cannot be greater than either P(A) or P(B) (D) The assignment is legitimate
The assignment of probabilities cannot be legitimate because of option (C): P(A and B) cannot be greater than either P(A) or P(B). In this case, P(AB) = 0.5, which is greater than both P(A) = 0.4 and P(B) = 0.3. For probabilities to be valid, the intersection of two events (A and B) must not exceed the individual probabilities of each event.
The reason why this assignment of probabilities cannot be legitimate is because of option B - A and B are given as independent events, but option A - A and B are not given as disjoint events. If A and B are independent events, then the probability of their intersection (AB) should be equal to the product of their individual probabilities, which is not the case here (0.5 ≠ 0.4 x 0.3). Therefore, option D - The assignment is legitimate is incorrect. Additionally, option C - GP(A and B) cannot be greater than either P(A) or P(B) is also violated, but it is not the main reason why the assignment is illegitimate.
The assignment of probabilities cannot be legitimate because of option (C): P(A and B) cannot be greater than either P(A) or P(B). In this case, P(AB) = 0.5, which is greater than both P(A) = 0.4 and P(B) = 0.3. For probabilities to be valid, the intersection of two events (A and B) must not exceed the individual probabilities of each event.
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Ayana measured a house and its lot and made a scale drawing. She used the scale 14 millimeters : 5 meters. In the drawing, the back patio is 84 millimeters long. What is the length of the actual patio?
The actual length of the back patio is 30 meters.
Given information:
Ayana measured a house and its lot and made a scale drawing. She used the scale 14 millimeters: 5 meters.
In the drawing, the back patio is 84 millimeters long.
If 14 millimeters in the drawing represent 5 meters in reality, then we can set up a proportion to find the length of the actual patio:
14 mm : 5 m = 84 mm : x
Cross-multiplying, we get:
14 mm (x) = 5 m (84 mm)
To simplify, we have:
x = (5 m 84 mm) / 14 mm
x = 30 m
Therefore, x = 30 m.
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what is the distance between -27 and 30 on a number line
Answer:
57
Step-by-step explanation:
Get the absolute value of subtracting -27 from 30 (absolute value essentially meaning you make the answer positive no matter what). -27-30 = -57, and its absolute value is 57, which is the distance.
The absolute value is there because the distance can't be negative, as in you can't travel negative 2 miles to the grocery store. Hope this helps.
for each of the figures write an absolute value equation that has the following solution set
ASAP PLS
The absolute value equation that has a solution set of -1/2 and 3/2 is [tex]|x - \frac{1}{2} | = -1[/tex].
We are given the solution sets of the absolute value equation and we have to find the absolute value equation. The solution sets given are
x = { -1/2, 3/2}. The absolute value of the equation can be represented as [tex]| x -x_{1} | - x_{2} = 0[/tex] in which [tex]x_{1}[/tex] is the mean of the given two numbers in the set and [tex]x_{2}[/tex] is the difference between the two numbers divided by 2.
Firstly, we will find the mean of the two numbers
[tex]x_{1}[/tex] = (-1/2 + 3/2)/ 2
[tex]x_{1}[/tex] = 1/2
Now, we will find the value of [tex]x_{2}[/tex]
[tex]x_{2}[/tex] = (-1/2 -3/2)/2
[tex]x_{2}[/tex] = -2/2
[tex]x_{2}[/tex] = -1
We will substitute the values of [tex]x_{1}[/tex] and [tex]x_{2}[/tex] in the absolute value equation.
[tex]| x -x_{1} | - x_{2} = 0[/tex]
[tex]| x - 1/2| - (-1) = 0[/tex]
[tex]| x - \frac{1}{2}| + 1 = 0[/tex]
[tex]| x - \frac{1}{2}| = -1[/tex]
Therefore, the absolute value equation that has a solution set of -1/2 and 3/2 is [tex]|x - \frac{1}{2} | = -1[/tex].
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For the following exercises, set up and evaluate each optimization problem. Find two positive integers such that their sum is 10, and minimize and maximize the sum of their squares.
The two positive integers that maximize the sum of their squares while satisfying the constraint x + y = 10 are x = y = 5.
To set up the optimization problem, let x and y be the two positive integers we are trying to find. Then, we want to minimize and maximize the sum of their squares, which can be expressed as:
Minimize: [tex]f(x, y) = x^2 + y^2[/tex]
Subject to: x + y = 10 and x, y > 0
Maximize: [tex]g(x, y) = x^2 + y^2[/tex]
Subject to: x + y = 10 and x, y > 0
Note that the constraint x, y > 0 means that we are looking for positive integers, and the constraint x + y = 10 means that their sum must be 10.
To solve the minimization problem, we can use the method of Lagrange multipliers. The Lagrangian function is:
L(x, y, λ) = [tex]x^2 + y^2 + λ(10 - x - y)[/tex]
Taking the partial derivatives with respect to x, y, and λ, we get:
∂L/∂x = 2x - λ = 0
∂L/∂y = 2y - λ = 0
∂L/∂λ = 10 - x - y = 0
Solving these equations simultaneously, we get:
x = y = 5
λ = 10
Therefore, the two positive integers that minimize the sum of their squares while satisfying the constraint x + y = 10 are x = y = 5. The minimum value of the sum of their squares is:
[tex]f(5, 5) = 5^2 + 5^2 = 50[/tex]
To solve the maximization problem, note that the function x^2 + y^2 is a continuous and increasing function of x and y. Since x and y must sum to 10 and be positive integers, the maximum value of their sum of squares occurs when one of them is as large as possible, which is 5. Thus, the maximum value of the sum of their squares is:
[tex]g(5, 5) = 5^2 + 5^2 = 50[/tex]
Therefore, the two positive integers that maximize the sum of their squares while satisfying the constraint x + y = 10 are x = y = 5.
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pls help asap will give points
The area of the given composite figure is: 156 sq. units
How to find the area of the composite figure?The formula for the area of a rectangle is expressed as:
Area = L * W
Where:
W is width
L is Length
The formula for the area of a circle is:
Area = πr²
Thus:
Area of composite figure = 13 * 12 = 156 sq. units
This is because the semi circle added is exactly the same area with the one removed.
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