Thus, the two positive whole numbers that are greater than one multiply to 576 and share only the number one in common is 5.
Explain about the positive whole numbers?One of the categories that mathematicians have created for numbers is titled whole numbers, and it contains all of the numbers from 0 to infinity.
The specific class or collection of numbers known as whole numbers includes:
Are all positive numbers with no fractional or decimal parts, including zero.Positive numbers with no decimal or fractional portions are referred to as whole numbers, including zero. They are numerical representations of whole, unbroken things. The set of whole numbers is mathematically represented by the numbers 0 through 9.Two positive whole numbers greater than 1 are 2 and 3.
2 and 3 does not have any common factor except 1.
sum of 2 and 3 = 5
Thus, the two positive whole integers that are greater than one multiply to 576 and share only the number one in common is 5.
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What are domain restrictions of rational functions? What do domain restrictions
mean for the graph of a rational function?
an certain brand of upright freezer is available in three different rated capacities: 16 ft^3 , 18ft^3 what is the variance of the price paid by the next customer
As per the given probability, the values of
(a) E (X) is 17.4 ft³, E (X²) is 323.2 ft⁶, and V (X) is 2.16 ft⁶.
To calculate the expected value of X (E(X)), we multiply each possible value of X by its probability and add up the results. Using the probability mass function given, we have:
E(X) = (16)(0.3) + (18)(0.5) + (20)(0.2) = 17.4
This means that on average, the rated capacity of a freezer sold at this store is 17.4 ft³.
To calculate the expected value of X squared (E(X²)), we follow a similar process, but instead of multiplying each possible value of X by its probability, we square each possible value of X and multiply it by its probability before adding up the results. Using the probability mass function given, we have:
E(X²) = (16²)(0.3) + (18²)(0.5) + (20²)(0.2) = 323.2
This means that on average, the square of the rated capacity of a freezer sold at this store is 323.2 ft⁶.
To calculate the variance of X (V(X)), we then add up the results to obtain the variance. Using the probability mass function given, we have:
V(X) = (16 - 17.4)²(0.3) + (18 - 17.4)²(0.5) + (20 - 17.4)²(0.2) = 2.16
This means that the variance of the rated capacity of a freezer sold at this store is 2.16 ft⁶. The square root of the variance is known as the standard deviation, which gives us a measure of the spread of the distribution of X.
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Complete Question:
A certain brand of upright freezer is available in three different rated capacities: 16 ft³, 18 ft³ , and 20 ft³ . Let X = the rated capacity of a freezer of this brand sold at a certain store. Suppose that X has the following probability mass function:
x 16 18 20
p(x) 0.3 0.5 0.2
(a) Compute E (X) , E (X²) , and V (X) .
write a polynomial function in standard form with real coefficients whose zeros include 2,8i,-8i
If the zeros are 2, 8i, and -8i, the the polynomial has factors of (x-2), (x-8i), and (x+8i).
If we multiply those factors, we'll get our polynomial:
(x-2)·(x-8i)·(x+8i) = (x-2)·(x^2+64)
= x^3 - 2x^2 + 64x - 128
surface-finish defects in a small electric appliance occur at random with a mean rate of 0.1 defects per unit. find the probability that a randomly selected unit will contain at least one surface-finish defect.
The probability that a randomly selected unit will contain at least one surface-finish defect is 0.0952.
The number of surface finish flaws in this issue will follow a Poisson distribution, and since the mean is given, it is necessary to obtain the probability. More solutions will be found by using the Poisson distribution's normal approximation.
The following formula can be used to determine the likelihood that a randomly chosen unit will have at least one surface finish flaw:
The distribution's average and standard deviation are λ = 0.1
P(X ≥ 1) = 1 − P(X < 1)
P(X ≥ 1) = 1 − P(X = 0)
P(X ≥ 1) = 1 − [tex]\frac{e^{-0.1}(0.1)^0}{0!}[/tex]
After simplification
P(X ≥ 1) = 0.0952
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A 6 cm by 10 cm rectangle is dilated by a factor of 3.
What is the area of the dilated rectangle?
*Show your work on the sketch pad
area =
Answer: 540 cm²
Step-by-step explanation:To find the area of the dilated rectangle, we need to first find the dimensions of the new rectangle after it has been dilated by a factor of 3.
The length of the new rectangle will be 3 times the original length, which is:
10 cm x 3 = 30 cm
The width of the new rectangle will also be 3 times the original width, which is:
6 cm x 3 = 18 cm
Therefore, the area of the dilated rectangle will be:
30 cm x 18 cm = 540 cm²
The number of shoes sold varies inversely with the price two thousand shoes can be sold at the price of $250
The phrase "the number of shoes sold changes inversely with price" suggests that as the price of the shoes rises, so does the number of shoes sold, and vice versa.
We know that 2000 shoes can be sold for $250 in this circumstance. Using the inverse variation formula, we can write: k/Price = number of shoes sold where k is a proportionality constant. We may use the following information to calculate the value of k: 2000 = k/250 When we multiply both sides by 250, we get: k = 500000 In this case, the equation for the inverse variation is: Price/number of shoes sold = 500000 We may use this equation to compute the number of shoes sold at various prices. For instance, if the price is If the price is $300, the number of shoes sold is: The number of shoes sold is 500000/300, which is 1666.67. As a result, at a price of $300, about 1666.67 pairs of shoes would be sold.
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What is the relationship between the number of shoes sold and the price, given that the number of shoes sold varies inversely with the price? If 2,000 shoes are sold at a price of $250, what would be the price of the shoes if 3,000 shoes are sold?
. determine the value of x (namely, the smallest natural number that makes the statement true) and prove the claim using strong induction. claim: every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps
Answer:
Step-by-step explanation:
fsffd
Find the least common denominator of 7/9 and 2/3
The least common denominator of 7/9 and 2/3 is 9
To find the least common denominator of 7/9 and 2/3, we need to find the smallest multiple that both denominators share.
The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, ...
The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
We can see that the smallest multiple that both denominators share is 9.
To convert 7/9 and 2/3 to have a denominator of 9, we need to multiply the numerator and denominator of 7/9 by 1 (which is equivalent to 9/9), and the numerator and denominator of 2/3 by 3:
7/9 * 1 = 7/9
2/3 * 3/3 = 6/9
Now both fractions have a denominator of 9, so we can compare them easily:
7/9 and 6/9
Therefore, the least common denominator of 7/9 and 2/3 is 9.
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in the given system of equations, p is a constant.the system has no solution. what is the value of p
When the given system of equations does not have a solution, it implies that there is no common point for both the straight lines to intersect. It means that both the straight lines are parallel to each other.
Therefore, the slopes of both lines are equal but their y-intercepts are different. Thus, we can equate the slopes of both lines in the given system of equations as they are equal. Then, we can form an equation to calculate the value of p.Let the given system of equations be:y =[tex]px + 4......(1)y = px + 6......[/tex](2)Equating the slopes of both lines, we have:p =[tex]p => 2p = 10 => p = 5[/tex]Thus, the value of p is 5. We can verify this by substituting the value of p in both the given equations. Therefore, the given system of equations can be written as:y = 5x + 4......(1)y = 5x + 6......([tex]5x + 4......(1)y = 5x + 6......([/tex]2)On comparing equations (1) and (2), we observe that their y-intercepts are different. The equation (1) has y-intercept 4 and equation (2) has y-intercept 6.
Hence, there is no common point for both the lines to intersect. Therefore, the system of equations has no solution.
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Add the additive inverse to the product of - 1 and - 4
The answer to this question is 0.
To add the additive inverse to the product of - 1 and -4, we must first find the product of -1 and -4. The product of -1 and -4 is 4. Now that we have found the product of -1 and -4, we can add the additive inverse to it. The additive inverse of 4 is -4, so we add -4 to 4. 4+ (-4) is equal to 0. Therefore, the answer to this question is 0.
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Does anybody know the answer to this question?
The perimeter of isosceles triangle PRQ is 24.65 units.
The lengths of its three sides and add them up. We can use the distance formula to find the length of each side.
Distance formula: The distance d between two points (x1, y1) and (x2, y2) is given by:
d = [tex]sqrt((x2 - x1)^2 + (y2 - y1)^2)[/tex]
Using this formula, we can find the lengths of the sides PR, RQ, and PQ.
PR = [tex]sqrt((-1 - (-7))^2 + (4 - (-1))^2) = sqrt(36 + 25) = sqrt(61)[/tex]
RQ =[tex]sqrt((3 - (-1))^2 + (-1 - 4)^2) = sqrt(16 + 25) = sqrt(41)[/tex]
PQ = [tex]sqrt((3 - (-7))^2 + (-1 - (-1))^2) = sqrt(100) = 10[/tex]
Therefore, the perimeter of triangle PRQ is:
PR + RQ + PQ = [tex]sqrt(61) + sqrt(41) + 10 ≈ 24.65[/tex] (rounded to two decimal places)
So, the perimeter of triangle PRQ is approximately 24.65 units.
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The graph below shows a company's profit f(x), in dollars, depending on the price of goods x, in dollars, being sold by the company:
f(x)
150
120
Part A: What do the x-intercepts and maximum value of the graph represent in context of the described situation?
Part B: What are the intervals where the function is increasing and decreasing, and what do they represent about the sale and profit for the company in the situation
described?
Part C: What is an approximate average rate of change of the graph from x= 1 to x= 3, and what does this rate represent in context of the described situation?
The vertical axis of the graph represents profit, so the x-intercepts represent prices in the produce 0 profit. The maximum value of the graph is the maximum profit that can be obtained for anyprice
How to explain the graphThe higher or largest number of the chart is the maximum reach
B) We read the value of f(1) from the graph to be about 120, so the average rate of change is about:
(f(4) -f(1))/(4 -1) = (270 -120)/(3) = 50
The average rate of the change from x = 1 to x = 4 is about 50.* This means profit will increase on average $50 for each $1 increase in price in what interval.
If we take the peak profit to be $270 we can write f(x) as:
f(x) 16.875x(x-8)
Then f(1) = 118.15 and average rate of change is 50.625.
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What is the area of the composite shape ? 20ft, 30ft, 15ft, and 5 ft
the given composite shape has a 23 square inch surface area.
The area that any composite shape covers is known as the area of composite shapes. The composite shape is a shape created by joining a small number of polygons to create the desired shape. These shapes or figures can be constructed from a variety of shapes, including triangles, squares, quadrilaterals, etc. To calculate the area of a composite shape, divide it into basic shapes like a square, triangle, rectangle, or hexagon.
A composite shape is essentially a combination of basic shapes. A "composite" or "complex" shape is another name for it.
The dimensions of rectangle ABCD are 2 inches long and 7 inches wide.
The DEFG square's side measures 3 inches.
Using the equation for the composite shape's area,
Area of composite shape = Area of square + Area of rectangle; Area of composite shape = Length, Breadth, and Side 2; Area of composite shape = BC, AB, and DE2; Area of composite shape = 2 x 7 + 32; Area of composite shape = 14 + 9 = 23 square inches.
As a result, the given composite shape has a 23 square inch surface area.
The dimensions of rectangle ABCD are 2 inches long and 7 inches wide.
The DEFG square's side measures 3 inches.
Using the equation for the composite shape's area,
Area of composite shape = Area of square + Area of rectangle; Area of composite shape = Length, Breadth, and Side 2; Area of composite shape = BC, AB, and DE2; Area of composite shape = 2 x 7 + 32; Area of composite shape = 14 + 9 = 23 square inches.
As a result, the given composite shape has a 23 square inch surface area.
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Complete question:- What is the area of the composite shape ?
HELP PLEASE ASAP!!! Sarah needs to design two boxes for a science project. The width of Box A will be 4 inches more than its length.
The height of Box A will be 1 inch less than its length. Box B will have a length twice as long as Box A. The width of Box B will be 2 inches less than half its length. Box B will also have a height 2 inches more than half its length. For what length does the volume of Box A equal the volume of Box B?
A. x=4 inches
B. x = 5 inches
C. x= 1 inch
D. x=3 inches
Answer: Option A: x=4
How can I solve this? ASAP
Answer:
544 in²
Step-by-step explanation:
Find the area of the rectangle, then the triangle, then add them together to get the total SA.
Rectangle = l x w = 10 x 14 = 140 in²
SA of the 2 rectangular sides = 140 x 2 = 280 in²
Triangle = 1/2bh = 1/2(12)(8) = 48²
SA of the 2 triangular ends = 2 x 48 = 96 in²
Rectangular base = 14 x 12 = 168 in²
Total SA = 280 + 96 + 168 = 544 in²
The value of which of these expressions is closest to e?
The expression (1 + 1 / 18)¹⁸ represents the number closest to irrational number e. (Correct choice: A)
What exact value is closest to irrational value e?
This problem ask us to determine what expression is closest to irrational number e. A number is irrational when it cannot be represented by numbers of the form m / n, where m and n are integers and n is non-zero. The number e can be defined by the following limit:
[tex]e = \lim_{x \to \infty} \left(1 + \frac{1}x}\right)^{x}[/tex], where x is a natural number.
According to this definition, the greater the value of x is, the closer the result to irrational number e is. Therefore, the following expression is closest to the given irrational number:
x = 18
e = (1 + 1 / 18)¹⁸
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a coffee distributor needs to mix a(n) gualtemala antigua coffee blend that normally sells for $8.50 per pound with a gazebo coffee blend that normally sells for $13.10 per pound to create 10 pounds of a coffee that can sell for $12.64 per pound. how many pounds of each kind of coffee should they mix?
The system of equations, helps to determine the quantity of each kind of coffee for mixture. The 9 pounds of Guatemala antigua coffee blend and one pound of gazebo coffee blend should be mix.
We have a coffee distributor needs to mix a(n) Guatemala antigua coffee blend. Let us assume, x = pounds of the Guatemala antigua coffee blend
y = pounds of the gazebo coffee blend
The new mixture is 10 pounds, so x + y = 10
And normally sells value of Guatemala antigua coffee blend per pound = $8.50
so, sells for x pounds of the Guatemala antigua coffee blend = $8.50x
Normally sells value of gazebo coffee blend per pound = $13.10y
Total sells = $12.64 × 10
8.50x + 13.10y = 12.64 × 10
We have the following system of equations:
x + y = 10 --(1)
8.50x + 13.10y = 126.4 --(2)
We have to solve the system of equations for determining value x and y. Using the Elimination method, from equation (1), x = 10 - y, put in equation (2), 8.50x + 13.10y = 126.4
=> 8.50 ( 10 - y ) + 13.10y = 126.4
=> 85 - 8.5y + 13.10y = 126.4
=> 4.6 y = 41.4
=> y = 41.4/4.6 = 9
from equation (1), x = 10 - 9 = 1
Hence, required value is 9 pounds and 1 pounds.
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how many parameters must the forecaster (or the software) set using winter's exponential smoothing? multiple choice 1. none of the options are correct. 2. 0. 3.
The forecaster (or the software) must set three parameters using Winter's exponential smoothing. (Option 4)
What is Winter's exponential smoothing?Winter's exponential smoothing is a method for forecasting in which the forecast for the next period is weighted more heavily based on the data that was recorded in previous periods.
The formula for Winter's exponential smoothing includes three parameters: alpha, beta, and gamma. Parameters in Winter's exponential smoothing:
alpha: The degree of smoothing that is applied to the level of the series. The alpha value will vary between 0 and 1.
beta: The degree of smoothing applied to the trend of the series. The beta value will vary between 0 and 1.
gamma: The degree of smoothing applied to the seasonal component of the series. The gamma value will vary between 0 and 1.
Hence, the forecaster (or the software) must set three parameters using Winter's exponential smoothing.
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3. Mr. Mohan donated 8 five-litre bottles of water for the school's annual walkathon. There are two
water stops during the walkathon. Cups, each containing 125 ml of water, are filled from the five-
litre bottles. Each of the 200 participants takes one cup of water at each water stop. How many
participants will not get two cups of water if Mr. Mohan is the only person who donated water?
The number of participants that will not get two cups of water if Mr. Mohan is the only person that donated water, obtained using basic arithmetic operations are 40 participants
What are basic arithmetic operations?Basic arithmetic operations includes addition, subtraction, division and multiplication.
Mr. Mohan donated 8 five-litre bottles of water, which is equivalent to 40 litres of water;
8 × 5 litres = 40 litres
Each cup contains 125 ml of water which means that each liter can fill 8 cups. So the total number of cups that can be filled from the 8 five-litre bottles of water containing 40 litres of water is; 40 × 8 = 320 cups
Since each participant takes one cup at each stop and there are two stops in total, each participant will take a total of 2 cups during the walkation. This means that only 160 participants will get two cups of water if Mr Mohan is the only person who donated water.
So, 200 - 160 = '40 participants' will not get two cups of water if Mr. Mohan is the only person who donated water.
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y > -x- 2
y < -5x + 2
Identify a solution to the system
Move all terms not containing the variable from the center section of the inequality.
A restaurant offers fruit juices as welcome drinks to all its customers. There is a 29% chance of getting apple juice, a 31% chance of getting grape juice, a 19% chance of getting orange juice, and a 21% chance of getting pear juice. Is it less likely that you will get apple, grape, orange, or pear juice?
No, it is not less likely that you will get one of the four types of juice, as they are the only options available.
Probability of getting apple juice = 29%
= 0.29
probability of getting grape juice = 31%
= 0.31
Probability of getting orange juice = 19%
= 0.19
Probability of getting pear juice = 21%
= 0.21
The total probabilities of getting apple, grape, orange, and pear juice sum up to
= 29% + 31% + 19% + 21%
= 100%.
or
= 0.29 + 0.31 + 0.19 + 0.21
= 1.00
This implies,
That every customer will get one of the four types of juice, and there is a 100% chance of getting one of them.
Therefore, the given probabilities are not less likely as it represents chances of one type of juice definitely will be given.
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five card hand is dealt at random from a standard 52 card deck. find the probability of having c. exactly 2 kings and the ace of spades given that you have at least one ace
The five-card hand is dealt at random from a standard 52-card deck. the probability of having exactly 2 kings and the ace of spades given that you have at least one ace is 2.9%.
To find the probability of having exactly 2 kings and the ace of spades given that you have at least one ace
When a five-card hand is dealt at random from a standard 52-card deck is as follows:
There are 52 cards in a standard deck. Therefore, there are 4 Kings and 1 Ace of Spades. The number of ways in which we can select 2 Kings from 4 Kings is:
₄C₂ = [tex](4 * 3)/(2 * 1)[/tex]
= 6
The number of ways in which we can select the Ace of Spades from 1 Ace is:
₁C₁ = 1
The remaining 2 cards are non-Kings and non-Ace of Spades cards. So, there are 44 such cards. The number of ways in which we can select 2 cards from 44 cards is:
₄₄C₂ = [tex](44 * 43)/(2 * 1)[/tex]
= 946
Therefore, the total number of ways in which we can select 2 Kings and 1 Ace of Spades and 2 non-Kings and non-Ace of Spades cards is:
[tex]6 * 1 * 946[/tex] = 5676
There are 4 Aces in a standard deck. The number of ways in which we can select at least one Ace from 4 Aces is:
₄C₁ + ₄C₂ + ₄C₃ + ₄C₄ = 4 + 6 + 4 + 1
= 15
Therefore, the probability of having exactly 2 kings and the ace of spades given that you have at least one ace when a five-card hand is dealt at random from a standard 52-card deck is:
P(exactly 2 Kings and the Ace of Spades | at least one Ace) = (Number of ways in which we can select 2 Kings and 1 Ace of Spades and 2 non-Kings and non-Ace of Spades cards) / (Number of ways in which we can select at least one Ace)
= 5676/15
= 378.4 or 0.029 or 2.9%.
Therefore, the required probability is 0.029 or 2.9%.
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Victor made 50 ounces of trail mix for a camping trip he pours 12 ounces for one serving how many people can have fully serving?
Answer:
To find the number of people who can have a full serving of trail mix from 50 ounces, we need to divide the total amount of trail mix by the amount of trail mix per serving.
50 ounces of trail mix / 12 ounces per serving = 4.1667 servings
Since we can't have a fraction of a serving, we need to round down to the nearest whole number.
So, Victor can make 4 servings of trail mix from 50 ounces. Therefore, 4 people can have a full serving of trail mix.
Cortez has three times as many pencils as Nikhil, and they have 84 pencils in total
As per the unitary method, Nikhil has 21 pencils and Cortez has 63 pencils.
Let's say Nikhil has x number of pencils. Then, according to the problem, Cortez has three times as many pencils as Nikhil. Therefore, Cortez has 3x number of pencils.
Together, they have a total of 84 pencils. So, we can write an equation based on the number of pencils owned by Nikhil and Cortez as follows:
x + 3x = 84
Simplifying the equation, we get:
4x = 84
Dividing both sides by 4, we get:
x = 21
So, Nikhil has 21 pencils. Using the fact that Cortez has three times as many pencils as Nikhil, we can find out how many pencils Cortez has:
Cortez has 3x = 3(21) = 63 pencils.
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Complete Question:
Lance has three times as many pencils as Nick, and they have 84 pencils together. How many pencils does each of them have?
is it possible to have a c2 strictly convex function on a compact domain that is not also strongly convex?
Yes, it is possible to have a c2 strictly convex function on a compact domain that is not also strongly convex.
Let's have a look at the following details of it.
Definition of a Strictly Convex Function:A strictly convex function is a function whose Hessian matrix is positive-definite for all arguments. As a result, if f is differentiable and convex, it is strictly convex if and only if its Hessian matrix is positive-definite for all arguments.
The Definition of Strong Convexity: A function is said to be strongly convex if it meets the following criteria:
Let C be a convex subset of a Hilbert space H, and let f:C→ℝ be a continuously differentiable function. There is a positive constant μ such that, for all x,y∈C: f(y)≥f(x)+⟨∇f(x),y−x⟩+μ/2‖y−x‖2
Thus, it is clear from the above-mentioned definitions that it is possible to have a c2 strictly convex function on a compact domain that is not also strongly convex.
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the process mean can be adjusted through calibration. to what value should the mean be adjusted so that 99% of the cans will contain 12 oz or more?
The value of mean should be adjusted to 12 + 2.576σ so that 99% of the cans will contain 12 oz or more.
The process mean can be adjusted through calibration. The mean is a measure of central tendency in a dataset that represents the average value of a group of data. The population standard deviation is denoted by σ. The formula for the population mean is as follows: μ = (Σ xi) / n, where xi represents the data values and n represents the total number of data values.
Here we can use the formula of confidence interval as,μ±z σ/√n, Where μ is the mean, z is the z-score, σ is the standard deviation is the sample size. Given,The required confidence level is 99%. So,α = 1-0.99α = 0.01. We can find z from the z-score table at α/2 = 0.005 as, z = 2.576.
Now, we need to find out the value of μ when the mean will be 12 ounces so that 99% of cans will contain 12 ounces or more. So,μ ± z σ/√n = 12. We know that, P(X > 12) = 0.99. The formula for standardization is, Z = (X - μ) / σHere, X = 12, σ is given and we need to find the value of μ.z = (X - μ) / σ2.576 = (12 - μ) / σμ - 12 = 2.576 × σμ = 12 + 2.576 × σ.
Now, the value of μ should be adjusted to 12 + 2.576σ so that 99% of the cans will contain 12 oz or more.
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Help me pls I don’t know what it is
The bread recipe calls for 2 more cups of flour per cup of sugar than the cookie recipe.
========================================================
Explanation:
I'll use this shorthand
s = sugarf = flourSomething like "2 cups of sugar" will shorten to "2s".
The cookies need 3s for every 6f.
Therefore we get this ratio
3s : 6f
And we can divide both sides by 3 to end up with
s : 2f
Meaning "each cup of sugar needs 2 cups of flour".
--------------------
Meanwhile, the bread recipe calls for 2s for every 8f.
2s : 8f
2s/2 : 8f/2 .... divide both sides by 2
s : 4f
When making bread, each cup of sugar needs 4 cups of flour. In other words, the amount of flour is quadruple that of the sugar.
--------------------
Recap
Cookies have the ratio of s : 2fBread has the ratio of s : 4fThese ratios are reduced so that we have "s" on the left side without any numbers attached to it. In other words, we have 1s.
We can see that the bread needs 2 more cups of flour compared to the cookies (since 4f - 2f = 2f). It's important that we reduce those ratios so that we're talking about the same amount of sugar for each food item.
This is why choice B is the final answer.
--------------------
An example:
Each food item will use 1 cup of sugar.
The cookies need twice the amount of flour, so the cookies require 2 cups of flour (because of the ratio s:2f mentioned earlier).
The bread needs quadruple the amount of flour compared to sugar. Meaning the bread needs 4 cups of flour (because of the ratio s:4f).
Then subtract to get
(4 cups flour for bread) - (2 cups flour for cookies) = 2 cups flour
in how many different ways can 11 men and 8 women be seated in a row if no 2 women are to sit together?
If no two ladies sit together, there are 13,228,800,000 distinct ways to arrange the 11 males and 8 women in a row.
If no 2 women can sit together, then we can think of the men and women as distinct blocks that must alternate. Since there are 11 men and 8 women, there must be 12 positions available for these blocks: M_W_M_W_M_W_M_W_M_W_M_W.
The first block can be filled In 11! Ways (11 men to choose from), the second block can be filled in 8! Ways (8 women to choose from), the third block can be filled in 10! Ways, the fourth block can be filled in 7! Ways, and so on. So the total number of ways to seat the group is:
11! * 8! * 10! * 7! * 9! * 6! * 8! * 5! * 7! * 4! * 6! * 3! * 5! * 2! * 4! * 1! * 3! * 2! * 1!
Simplifying this expression, we get:
(11! * 8! * 10! * 7! * 9! * 6! * 8! * 5! * 7! * 4! * 6! * 3! * 5! * 2! * 4! * 1! * 3! * 2! * 1!) =
(11! * 10! * 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 1!) * (8! * 7! * 6! * 5! * 4! * 3! * 2! * 1!) =
11! * 10! * 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 1! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 1! =
11! * 10! * 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 8! * 7! * 6! * 5! * 4! * 3! * 2!
= 11! * 10! * 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! * 8!^2 * 7!
We can evaluate this expression with a calculator to obtain:
(111098765432)(887654327) = 13,228,800,000
Therefore, there are 13,228,800,000 different ways to seat the 11 men and 8 women in a row if no 2 women are to sit together.
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How do I solve this challenging math problem?
Answer:
13/32
Step-by-step explanation:
You want the area of the shaded portion of the unit square shown.
CircumcenterPoints B, C, E are shown as equidistant from point F, so will lie on a circle centered at F. The center of that circle is at the point of coincidence of the perpendicular bisectors of BE, BC, and CE.
Without loss of generality, we can let line EF lie on the x-axis such that E is at the origin. Chord EB of the circle has a rise of 1/2 for a run of 1, so a slope of 1/2. Its midpoint is (1, 1/2)/2 = (1/2, 1/4). The perpendicular line through this point will have slope -2, so its equation can be written ...
y -1/4 = -2(x -1/2)
y = -2x +5/4
Then the x-intercept (point F) will have coordinates (0, 5/8):
0 = -2x +5/4 . . . . . y=0 on the x-axis
2x = 5/4
x = 5/8
TrapezoidTrapezoid EFCD will have upper base 5/8, lower base 1, and height 1/2. Its area is ...
A = 1/2(b1 +b2)h
A = (1/2)(5/8 +1)(1/2) = (1/4)(13/8) = 13/32
The shaded area is 13/32.
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Additional comment
The point-slope equation of a line through (h, k) with slope m is ...
y -k = m(x -h)
One of the legs of the right triangle measure 12 cm and it’s hypotenuse measure 8 cm
Answer:
Missing leg = [tex]2\sqrt{14}[/tex]
Step-by-step explanation:
Pythagorean theorem
[tex]a^{2} +b^{2}=c^{2}[/tex]
[tex]12+b^{2} = 8^{2}[/tex]
[tex]12+b^{2} =64[/tex]
[tex]b^{2}=56\\[/tex]
[tex]b=\sqrt{56}[/tex]
[tex]b=2\sqrt{14}[/tex]