Answer:
what are we rounding? like what is your question?
Step-by-step explanation:
**if you answer you have my promise and i'll go back and provide an actual answer**
Can someone plz help me with these United rates and explain step by step ?
f(x, y) = 2(xo-* yot) _40536 orgs 7.6 Otherwise covariance X 1 J = ?
Covariance is the measure of the joint variability of two random variables. It measures how much two random variables are linearly related to each other. It is denoted by Cov(X, Y).
f(x, y) = 2(xo-* yot) _40536 orgs 7.6, otherwise, covariance X1j = ?The value of covariance X1j can not be determined from the given equation f(x, y) = 2(xo-* yot) _40536 orgs 7.6. There is no relation mentioned between the variables of x and y in the equation. The equation f(x, y) = 2(xo-* yot) _40536 orgs 7.6 means that the function f(x, y) is the multiplication of the value of x to the value of y and then multiplying it by a constant value. The value of the constant is 2(xo-* yot) _40536 orgs 7.6 but the actual value of x and y is unknown. Therefore, we can not calculate the covariance X1j using the given equation. The covariance X1j needs at least two random variables to determine their interdependence, whereas the given equation is a single-variable function that has no relation between two variables .
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Solve the differential equation, 6 x dx + 4 x dy = 0, using separation of variables
The general solution to the differential equation is: y = -(3/2)x + C, where C is the constant of integration.
To solve the differential equation 6x dx + 4x dy = 0 using separation of variables, we need to rearrange the equation so that all the x terms are on one side and all the y terms are on the other side.
Let's start by dividing both sides of the equation by 4x:
(6x dx + 4x dy) / 4x = 0
(6x / 4x) dx + (4x / 4x) dy = 0
(3/2) dx + dy = 0
Now we can separate the variables by moving the dy term to the other side:
dy = -(3/2) dx
Integrating both sides with respect to their respective variables, we have:
∫ dy = ∫ -(3/2) dx
The integral of dy with respect to y is simply y, and the integral of -(3/2) dx with respect to x is -(3/2)x:
y = -(3/2)x + C
where C is the constant of integration. Thus, the general solution to the differential equation is:
y = -(3/2)x + C
This is the final solution using separation of variables.
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$12.60 for 3 boxes. Find the unit rate
Answer:
$4.20 / box
Step-by-step explanation:
12.60 / 3 = 4.20
I will give Brainliest
Emily rides her bike with a constant speed of 18 km/h. How long will she take to travel a distance of 18 kilometers?
Answer:
1 hour
Step-by-step explanation:
Travels 18 km/h
km/h = kilometers per hour
Answer:
1 hour
Step-by-step explanation:
Most quadratic equations have______roots?
Answer:
equal
Step-by-step explanation:
Most quadratic equations have equal roots.
25 points !! Please help me! What’s the volume to this question? Urgent !
Answer:
381 in^3
Step-by-step explanation:
The volume of the green block is V1 = length(width)(height), or
V1 = (16 in)(7 in)(3 in) = 336 in^3, and
the volume of the red triangular prism is V2 = (1/2)(3 in)(5 in)(6 in) =
V2 = 45 in^3
So the total volume is the sum of these two results: 381 in^3
what is the area of a semicircle that has a radius of 30 feet
Answer:
A = 1413.72 unit²
Step-by-step explanation:
area of a semicircle formula: A = (πr²)/2
To find the area of a semicircle that has a radius of 30 feet, apply r = 30 to the formula A = (πr²)/2:
A = (π30²)/2
A = 1413.72 unit²
Which ordered pair is a solution of the equation?
y+1=3(x-4)y+1=3(x−4)y, plus, 1, equals, 3, left parenthesis, x, minus, 4, right parenthesis
Choose 1 answer:
Choose 1 answer:
(Choice A)
A
Only (4,-1)(4,−1)left parenthesis, 4, comma, minus, 1, right parenthesis
(Choice B)
B
Only (5,2)(5,2)left parenthesis, 5, comma, 2, right parenthesis
(Choice C)
C
Both (4,-1)(4,−1)left parenthesis, 4, comma, minus, 1, right parenthesis and (5,2)(5,2)left parenthesis, 5, comma, 2, right parenthesis
(Choice D)
D
Neither
Both (4, 1) and (5, 2) is the solution to the given equation. Therefore, option C is the correct answer.
The given equation is y+1=3(x-4).
A) The given coordinate point is (4, -1).
Substitute (x, y)=(4, -1) in y+1=3(x-4), we get
-1+1=3(4-4)
0=0
So, (4, -1) is the solution to the given equation.
B) The given coordinate point is (5, 2).
Substitute (x, y)=(5, 2) in y+1=3(x-4), we get
2+1=3(5-4)
3=3
So, (5, 2) is the solution to the given equation.
C) Here, both (4, 1) and (5, 2) is the solution to the given equation.
Therefore, option C is the correct answer.
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Pls help it due soon
Answer:
I need help with math too. Can u help me and I’ll help u
Step-by-step explanation:
Kitchen chairs are purchased wholesale for $36 each by a discount furniture store. Then, the store marks up the chairs by 60 percent. The store has a special sale where all items are marked down by 20 percent. How much would two chairs cost during the sale? $46.08 $57.60 $92.16 $115.20Kitchen chairs are purchased wholesale for $36 each by a discount furniture store. Then, the store marks up the chairs by 60 percent. The store has a special sale where all items are marked down by 20 percent. How much would two chairs cost during the sale? $46.08 $57.60 $92.16 $115.20
Answer:
46.08
Step-by-step explanation:
you have to make your percentage a decimal, which 60% will be .60 and 20% will be .20. you then multiply your initial number which is 36 by .60 and add that on because youre adding 60%. After that you will multiply that given number by .20 and you subtract what that product is from your last product you received (36x.60) which if im not mistaken will give you $46.08.
Answer:
C
Step-by-step explanation:
I took the test
7. In multiplying 0.125 x 0.08, Dina knew the rule that says she should multiply 125 x 8 and then move the decimal point 5 places. When she did it on the calculator, she saw it was 0.01. She was confused. How would you help her?
The correct answer to 0.125 x 0.08 is 0.01. Dina's calculator provided the correct result.
The rule Dina mentioned about multiplying 125 x 8 and then moving the decimal point 5 places is not applicable in this case. When multiplying decimal numbers, we need to consider the number of decimal places in each factor.
Here's the correct approach to multiply 0.125 x 0.08:
Step 1: Ignore the decimal point and multiply the numbers as if they were whole numbers:
0.125 x 0.08 = 125 x 8
Step 2: Count the total number of decimal places in the original numbers:
0.125 has three decimal places
0.08 has two decimal places
Step 3: Add the number of decimal places from the original numbers:
Three decimal places + two decimal places = five decimal places
Step 4: Place the decimal point in the result by counting five places from the right:
125 x 8 = 1000
Adding the decimal point five places from the right gives us the final answer: 0.01000
Therefore, the correct answer to 0.125 x 0.08 is 0.01. Dina's calculator provided the correct result.
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If incomes increase by 58% and the quantity demanded of tennis balls drops by 87% as a result, what is the income elasticity of demand for tennis balls?
The income elasticity of demand for tennis balls is -1.5. The negative sign indicates an inverse relationship between income and the quantity demanded of tennis balls, suggesting that tennis balls are an inferior good.
To calculate the income elasticity of demand, we need to use the formula:
Income Elasticity of Demand = Percentage change in quantity demanded / Percentage change in income
Given that incomes increase by 58% and the quantity demanded of tennis balls drops by 87%, we can plug these values into the formula:
Income Elasticity of Demand = (-87%)/58%
Simplifying the calculation:
Income Elasticity of Demand = -1.5
The income elasticity of demand for tennis balls is -1.5. The negative sign indicates an inverse relationship between income and the quantity demanded of tennis balls, suggesting that tennis balls are an inferior good.
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The sale price for a jacket that regularly costs $85.00 is now $68.00. With sales tax, a customer pays $73.44. Which THREE statements are correct for this situation?
Answer:
The sales tax rate is 8%.
C. The sale price is 20% less than the regular price
Step-by-step explanation:
Here are the options to this question :
A) The sales tax rate is 6%
B) The sales tax rate is 8%.
C. The sale price is 20% less than the regular price
Sales tax rate = (tax paid / discounted selling price) x 100
tax paid = amount customer paid - discounted selling price
$73.44 - $68 = $5.44
($5.44 / $68) x 100 = 8%
The sales tax is 8%
Sales price = (discount / initial price) x 100
discount = $85 - $68 = $17
($17 / 85) x 100 = 20%
Answer:
B.) The Sales Tax Rate is 8%
C.) 85-68/85 = discount rate
E.) The Sale Price is 20% less than the regular price.
Step-by-step explanation:
i did the usatestprep :)
Out of her 5 gigabyte data plan, Debbie has used 37%. How much data does she have left?
jawkfjrnsidbjekwbxk2jw dlf 2kqbdkkebakdnrk8w Fflint
Answer:
740%
Step-by-step explanation:
A company prepares their shipments in two different-sized boxes.
In order to fit with new shipping regulations, the company needs to decrease the volume of the boxes and will do this by reducing each of the dimensions by at least x inches.
Which system of inequalities can be used to model V, the volume of each box, after each dimension has been reduced by at least x inches?
Answer:
Answer is A. -296x+960/ -636x+3,024
Step-by-step explanation:
Answer:
HEREEE besties
For each of these 10 samples, compute Statistics 1, 2, and 3. Enter your answers as 3 numbers separated by commas, such as 1,2,3. 2,3,3* 2,3,4 2, 3, 4* 2, 3+ 4 2, 3*, 4* 2, 4, 4* 3,3*, 4. 3, 3*, 4* 3, 4, 4* I 3*, 4,4* Which statistic would you recommend for estimating ? Average of smallest and largest values in the sample
Statistic 1 would you recommend for estimating average of smallest and largest values in the sample
To compute Statistics 1, 2, and 3 for the given samples, and determine which statistic would be recommended for estimating the average of the smallest and largest values, let's go through each sample one by one:
1. 2,3,3*
Statistics 1: Sum of the values = 2 + 3 + 3 = 8
Statistics 2: Average of the values = (2 + 3 + 3) / 3 = 2.67
Statistics 3: Median of the values = 3
2. 2,3,4
Statistics 1: Sum of the values = 2 + 3 + 4 = 9
Statistics 2: Average of the values = (2 + 3 + 4) / 3 = 3
Statistics 3: Median of the values = 3
3. 2,3,4*
Statistics 1: Sum of the values = 2 + 3 + 4 = 9
Statistics 2: Average of the values = (2 + 3 + 4) / 3 = 3
Statistics 3: Median of the values = 3
4. 2,3*,4*
Statistics 1: Sum of the values = 2 + 3 + 4 = 9
Statistics 2: Average of the values = (2 + 3 + 4) / 3 = 3
Statistics 3: Median of the values = 3
5. 2,4,4*
Statistics 1: Sum of the values = 2 + 4 + 4 = 10
Statistics 2: Average of the values = (2 + 4 + 4) / 3 = 3.33
Statistics 3: Median of the values = 4
6. 3,3*,4
Statistics 1: Sum of the values = 3 + 3 + 4 = 10
Statistics 2: Average of the values = (3 + 3 + 4) / 3 = 3.33
Statistics 3: Median of the values = 3
7. 3,3*,4*
Statistics 1: Sum of the values = 3 + 3 + 4 = 10
Statistics 2: Average of the values = (3 + 3 + 4) / 3 = 3.33
Statistics 3: Median of the values = 3
8. 3,4,4*
Statistics 1: Sum of the values = 3 + 4 + 4 = 11
Statistics 2: Average of the values = (3 + 4 + 4) / 3 = 3.67
Statistics 3: Median of the values = 4
9. 3*,4,4*
Statistics 1: Sum of the values = 3 + 4 + 4 = 11
Statistics 2: Average of the values = (3 + 4 + 4) / 3 = 3.67
Statistics 3: Median of the values = 4
For estimating the average of the smallest and largest values in the sample, the recommended statistic would be Statistics 1, which is the sum of the values. Taking the average of the smallest and largest values can be obtained by dividing the sum by the number of values, which in this case is 3.
Therefore, the recommended statistic for estimating the average of the smallest and largest values is Statistics 1 (sum of the values).
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To evaluate the performance of a new diagnostic test, the developer checks it out on 150 subjects with the disease for which the test was designed, and on 250 controls known to be free of the disease. Ninety of the
diseased yield positive tests, as do 30 of the controls.
What is the specificity of this test? (2 decimals)
The specificity of the diagnostic test is 88%, indicating its ability to accurately identify individuals without the disease as negative.
Specificity is a measure of the test's ability to correctly identify individuals without the disease as negative. To calculate the specificity, we need to consider the number of true negatives (controls who yield negative tests) and the total number of controls.
In this case, the number of controls tested is 250, and out of those, 30 yield positive tests. The number of true negatives can be calculated by subtracting the number of false positives (controls who yield positive tests) from the total number of controls:
Negatives which are true = Total Controls - False Positives
True Negatives = 250 - 30 = 220
The specificity is then calculated as the ratio of true negatives to the total number of controls:
Specificity = True Negatives / Total Controls
Specificity = 220 / 250 ≈ 0.88
Therefore, the specificity of this test is approximately 0.88 or 88%. This means that the test correctly identifies 88% of individuals without the disease as negative.
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Identify the end behavior of the function f(x) = 6x^4 - 12x^3 +8x -10
Answer:
Step-by-step explanation:
This is a quartic equation with a positive coefficient for x^4 so it is shaped like an M xo ir rises to both the left and the right.
please help me with this problem about growth and decay.
Answer:
The population of the town in Iowa after 13 years is 9,130
Step-by-step explanation:
The given parameters of the town are;
The population of the town in Iowa in 2007, a = 12,355
The rate at which the people of the town leave Iowa for Minnesota, r = 2.3% per year
We are required to find the population of the town after t = 13 years
The given population decay function is presented as follows;
[tex]f(t) = a \cdot (1 - r)^t[/tex]
Where;
a = The initial population of the town = 12,355
r = The annual percentage rate at which the people of the town leave Iowa for Minnesota = 2.3% per year = 0.023
t = The number of years over which the population changes = 13 years = 13
∴ f(13) = 12,355 × (1 - 0.023)¹³ = 9130.02734094
Therefore, the population of the town in Iowa after 13 years ≈ 9,130 (we round down to the nearest whole number).
100 points brainliest!!! plz answer all to the best of your ability no links they don't work for me
Answer the following questions using what you've learned from this unit. Write your responses in the space provided.
1. Part I: The degree of a polynomial is the (greatest / least) of the degrees of its terms. (Circle the term that correctly completes this definition.) (1 point)
Part II: In order to write a polynomial in descending order, you must write the terms with the exponents (decreasing / increasing) from left to right. (Circle the term that correctly completes this rule.) (1 point)
Part III: For each polynomial, determine the degree and write the polynomial in descending order. (4 points: 2 points each)
A. –4x2 – 12 + 11x4 B. 2x5 + 14 – 3x4 + 7x + 3x3
2. Use addition and subtraction to simplify the following polynomials.
A. Add polynomials: (3 – 4x + 8x2) + (–6 + 2x – 5x2)
Step 1: Rewrite the polynomials without the parentheses. (1 point)
Step 2: Write the polynomial in descending order and use parentheses around like terms. (1 point)
Step 3: Add the like terms identified in Step 2 to simplify the polynomial. (1 point)
B. Subtract polynomials: (3x – 5 – 7x2) – (–2 + 6x2 – 5x)
Step 1: Rewrite the polynomials without the parentheses. Remember to multiply each term in the second parentheses by –1. Show your work. (2 points)
Step 2: Write the polynomial in descending order and use parentheses around like terms. (1 point)
Step 3: Add the like terms identified in Step 2 to simplify the polynomial. (1 point)
3. Use the FOIL method to multiply binomials.
Part I: When multiplying binomials, the FOIL method helps you to organize the multiplication of each term of the first binomial by each term of the second. Fill in each blank with the word used to show how the binomials' terms are multiplied. (2 points: 0.5 point each)
F
O
I
L
Part II: Using the FOIL method, multiply the terms in the binomials below. Show your work in the blanks provided. Then, add any like terms and write the polynomial in standard form in the space provided. Show your work. (8 points: 2 points each)
A. (3x + 7)(2x – 5) B. (x2 + 2x)(5x2 – 3x)
______ + _______ + _______ + _______
______ + _______ + _______ + _______
______________________
______________________
C. (5x + 4)(5x – 4) D. (x2 – 7)(x2 – 4)
______ + _______ + _______ + _______
______ + _______ + _______ + _______
______________________
______________________
4. Use the distributive property to multiply the trinomial by the binomial.
Circle the first term in the trinomial, multiply it by each term in the binomial, and place each result in the blank spaces provided. Repeat this process for the second and third terms of the trinomial until all of the spaces are filled in. Finally, in the space provided beneath the blanks, simplify the expression by combining like terms and arranging the terms from highest to lowest order. Show your work. (4 points)
(3x2 – 2x + 7)(x2 + 2x)
______ + _______ + _______ + _______ + ______ + _______
5. Use the vertical method to multiply two trinomials.
Step 1: Multiply the top trinomial by the last term in the bottom trinomial. Place each result in the blank spaces provided. (2 points)
Step 2: Multiply the top trinomial by the middle term in the bottom trinomial. Place each result in the blank spaces provided. (2 points)
Step 3: Multiply the top trinomial by the first term in the bottom trinomial. Place each result in the blank spaces provided. (2 points)
Step 4: Add the three partial products to find the final answer. Place the result at the bottom. (1 point)
6. Sabrina is making an open box from a piece of cardboard that has a width of 12 inches and a length of 18 inches. She'll form the box by making cuts at the corners and folding up the sides. If she wants the box to have a volume of 224 in3, how long should she make the cuts?
Part I: Each dimension (width, length, and height) of the finished box can be represented by a polynomial expression. If the height is x inches, what is the length in terms of x? (2 points)
Height = x
Width = 12 – 2x
Length =
Part II: The formula for volume is v = l • w • h. Use the polynomial expressions from Part I to write an equation to represent the volume of the box. Simplify on the right side by multiplying the polynomials and writing the answer in descending order. Show your work.
HINT: First multiply the binomials representing length and width. Next, multiply the resulting trinomial by the expression representing height. (3 points)
v =
Part III: The length of each cut at the corners is represented by x. To make a box, x must fall in a certain range of values. For example, x cannot be 15 because that would make the flap longer than the width of the box. Identify one reasonable value of x. Why did you choose that value? (2 points)
Part IV: Use the polynomial equation you wrote in Part II to find the volume of the box for each value of x. Show your work. (5 points)
Answer:
Part I: The degree of a polynomial is the greatest of the degrees of its terms.
Part II: In order to write a polynomial in descending order, you must write the terms with the exponents decreasing from left to right.
Part III: A. -4x² + 11x - 12, degree = 2,
B. 5x³ + 4x + 14, degree = 3.
Step-by-step explanation:
Part I : Since the degree of a polynomial is the highest power of its monomials ( single term ),
e: degree of is 5.
Thus, in part I, the correct option is 'greatest'
Part II: When we write a polynomial then we write the terms of the polynomial in descending order of their degrees.
Thus, in part II the correct option is 'least'
Part III: A.
∵ -4x² has the highest power in the polynomial.
⇒ Degree = 2,
Also, in the polynomial descending order of degrees,
2 > 1 > 0
⇒ polynomial in descending order,
B.
Combining like terms,
∵ 5x³ has the highest degree,
⇒ Degree = 3,
Also, the order of the degrees in the polynomial is,
3 < 2 < 1 < 0
Thus, the polynomial in descending order,
Answer: I had trouble with this one too.
Step-by-step explanation:
find the 6th term of 6, 8, 32/3
Answer:
The 6th term of the sequence is 6144/243
Step-by-step explanation:
From what we have, we can see that the sequence might be geometric
to confirm this, we have to check if the common ratio is the same all through
To know this, we have to divide the succeeding term by the preceding term and check if the results for two sets are equal
thus, we have it that;
32/3 * 1/8 = 8/6
= 4/3 = 4/3
We can confirm that the sequence is thus geometric
Now, to find the nth term of a geometric sequence, we have it that;
Tn = ar^(n-1)
where a is the first term, given as 6
r is the common ratio given as 4/3
n is the term number given as 6
Thus, we have this as:
T6 = 6 * (4/3)^(6-1)
T6 = 6 * (4/3)^5
T6 = 6144/243
Mrs. Yellow is an email marketing manager, and she got the following characteristics: 20% open rate, 5% click-through rate, 5% conversion rate, and $1000 average order value. If she was sending the promotion email to a list of 20000 members, her company can achieve (number) of conversions, and $ (number) of revenue." Please write down your answers
The number of conversions Mrs. Yellow's company can achieve is 10, and the revenue they can earn is $10,000.
Mrs. Yellow is an email marketing manager and her company has the following characteristics:
20% open rate, 5% click-through rate, 5% conversion rate, and $1000 average order value. She is sending the promotion email to a list of 20000 members.
To calculate the number of conversions and revenue, we need to use the following formula:
Revenue = Number of conversions × Average order value
Number of conversions = Total number of emails × Conversion rate
Total number of emails = Total number of opened emails × Click-through rate
Total number of opened emails = Total number of sent emails × Open rate
Total number of sent emails = 20,000
Revenue = Number of conversions × Average order value
Total number of sent emails = 20,000
Total number of opened emails = 20,000 × 20/100 = 4,000
Total number of emails = 4,000 × 5/100 = 200
Number of conversions = 200 × 5/100 = 10
Revenue = 10 × 1,000 = $10,000
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A circus rents a rectangular building that has floor dimensions of 50 by 100 feet
Answer:
50 x 100=5,000
Step-by-step explanation:
50 x 100
I can use your help please
Answer:
he needs to play 24 games of basketball
PLEASEEEEEEEEEEEEEEE HELPPPPPPPPPPPPPP
Answer:
The equation is x-18 + 8x = 180 (because they form a linear pair of angles.
Each angle measure:-
x-18 + 8x = 180
x + 8x = 180 + 18 (-18 becomes +18)
9x = 198
x = 198/9
x = 22
Angle 1 = x-18 = 22-18 = 4 degrees
Angle 2 = 8x = 8 * 22 = 176 degrees
Hope it helps :')
5 6 7 8 9 + 10 11 12 13 14 15 16 17 18 data a Based on the boxplot above, identify the 5 number summary The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 44 ounces and a standard deviation of 11 ounces. Use the Standard Deviation Rule, also known as the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions. and a) 68% of the widget weights lie between b) What percentage of the widget weights lie between 11 and 55 ounces? S c) What percentage of the widget weights lie above 22?
a. The 68% of the widget weights lie between 33 ounces and 55 ounces.
b. The percentage of widget weights lying between 11 and 55 ounces is approximately 68%.
c. The percentage of widget weights lying above 22 ounces is approximately = 5%.
Based on the given data points: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, the five-number summary can be identified as follows:
Minimum: 5
First Quartile (Q1): 7
Median (Q2): 11
Third Quartile (Q3): 15
Maximum: 18
Now, let's answer the questions related to the distribution of widget weights using the Empirical Rule:
a) The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data lies within one standard deviation of the mean. Since the mean is 44 ounces and the standard deviation is 11 ounces, we can say that approximately 68% of the widget weights lie between 44 - 11 = 33 ounces and 44 + 11 = 55 ounces.
b) To determine the percentage of widget weights lying between 11 and 55 ounces, we need to calculate the z-scores for these values. The z-score is calculated using the formula: z = (x - mean) / standard deviation.
For 11 ounces: z1 = (11 - 44) / 11 = -33/11 = -3
For 55 ounces: z2 = (55 - 44) / 11 = 11/11 = 1
Using the Empirical Rule, we know that approximately 68% of the data lies within one standard deviation of the mean. Therefore, the percentage of widget weights lying between 11 and 55 ounces is approximately 68%.
c) To determine the percentage of widget weights lying above 22 ounces, we need to calculate the z-score for 22 ounces: z = (22 - 44) / 11 = -22/11 = -2.
Using the Empirical Rule, we know that approximately 95% of the data lies within two standard deviations of the mean. Therefore, the percentage of widget weights lying above 22 ounces is approximately 100% - 95% = 5%.
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A just answer pls I gotta write random words bc the question is a pic
Answer:
$84
Step-by-step explanation:
What is the value of y
when 8y/4y +5 = 5/6?
A: 25/7
B: 5/28
C: 15/8
D: 25/28
Answer:
D
Step-by-step explanation:
please mark brainliest thanks:)
Find a generalisation of Euler's Formula for graphs which are not necessarily connected. Be sure to prove that your formula always holds.
In Euler's Formula for graphs that are not necessarily connected states that the number of vertices minus the number of edges plus the number of connected components is equal to the Euler characteristic of the graph.
Euler's Formula, which states that the number of vertices minus the number of edges plus the number of faces is equal to 2 for planar graphs, can be extended to graphs that are not necessarily connected. In this generalization, we consider the number of connected components in the graph. A connected component is a subgraph where there is a path between any two vertices.
Let V be the number of vertices, E be the number of edges, C be the number of connected components, and X be the Euler characteristic of the graph. The generalization of Euler's Formula for non-connected graphs is given by V - E + C = X.
To prove this formula, we start with Euler's Formula for connected graphs, which states V - E + F = 2, where F is the number of faces. For a disconnected graph, the number of faces can be defined as the sum of the number of faces in each connected component minus the number of edges that belong to more than one connected component. This can be written as F = F1 + F2 + ... + FC - N, where Fi is the number of faces in the i-th connected component and N is the number of edges connecting different components.
By substituting F = F1 + F2 + ... + FC - N into Euler's Formula for connected graphs and rearranging terms, we get V - E + C = X, which is the generalization of Euler's Formula for non-connected graphs. Therefore, the formula holds true for any graph, whether connected or not.
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