Answer:
36
Step-by-step explanation:
3*12=36
Rewrite in simplest terms: -8(9r – 1) – 9(-85 + 2)
The Answer is:
−72 r + 755
Answer:
-72r + 755
Step-by-step explanation:
-8(9r – 1) – 9(-85 + 2)
-72r + 8 - 9(-83)
-72r + 8 + 747
-72r + 755
Test the claim about the population mean, h, at the given level of significance using the given sample statistics. Claim: h = 30; « = 0.08; 3 = 3.31. Sample statistics: x = 28.9, n = 65 Calculate the standardized test statistic. The standardized test statistic is -2.68 (Round to two decimal places as needed.) Determine the critical value
The standardized test statistic is -2.68 and the critical value is 1.44.
Given that, Claim: h = 30 Sample statistics: x = 28.9, n = 65 α = 0.08 σ = 3.31
We need to calculate the standardized test statistic and critical value.
Test statistic: The standardized test statistic can be calculated as follows:
Z = (x - μ) / (σ / √n)
Substitute the given values in the above formula to get
Z = (28.9 - 30) / (3.31 / √65) = -2.68
Critical value: Since the level of significance α is given, we can calculate the critical value using the z-score table for standard normal distribution.
Since α is given, the right-tailed test is conducted.
The critical value is obtained by subtracting the α-level from 1 and then find the z-score value.
Let's find the z-score value corresponding to the area 0.92 in the z-score table.
We get the corresponding z-score as 1.44, which is the critical value.
Hence, the standardized test statistic is -2.68 (Round to two decimal places as needed.) and the critical value is 1.44.
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Can someone pls help me ASAP I’ll give brainiest
Also show how u got the answer step by step pls
Using the example 2 2 4
3 3 4
•= •and a math drawing, explain why multiplying the numerator and
denominator of a fraction by the same number results in the same number (equivalent fraction).
In your explanation, discuss the following:
• what happens to the number of parts and the size of the parts;
• how your math drawing shows that the numerator and denominator are each multiplied by 4;
• how your math drawing shows why those two fractions are equal.
When you multiply the numerator and denominator of a fraction by the same number, you are effectively scaling up or scaling down the fraction without changing its value. The math drawing demonstrates how the number of parts and the size of the parts change, while still representing the same amount, thus showing why the two fractions are equal.
To explain why multiplying the numerator and denominator of a fraction by the same number results in an equivalent fraction, let's use the example you provided: 2/4.
First, let's understand the concept of a fraction. A fraction represents a part of a whole. The numerator represents the number of parts we have, and the denominator represents the total number of equal parts that make up the whole.
In the given example, 2/4, the numerator is 2, indicating that we have 2 parts out of a total of 4 equal parts. The denominator tells us that the whole is divided into 4 equal parts.
Now, let's say we want to multiply both the numerator and denominator by the same number, let's say 4. The new fraction becomes (2 * 4) / (4 * 4), which simplifies to 8/16.
Let's visualize this using a math drawing. Consider a rectangular shape representing the whole, divided into 16 equal parts, like a grid of squares, with 8 of those squares shaded. This represents the fraction 8/16.
Now, let's compare this to the original fraction, 2/4. If we draw a rectangle divided into 4 equal parts, and shade 2 of those parts, we can see that it represents the same amount as the fraction 8/16. By multiplying both the numerator and denominator by 4, we have essentially scaled up the size of each part and increased the number of parts.
Visually, the original fraction of 2/4 had fewer total parts (4) but larger-sized parts, while the equivalent fraction of 8/16 had more total parts (16) but smaller-sized parts. However, the total shaded area in both cases remains the same, which indicates that the fractions are equal.
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5potatos + 56 potatos
Answer:
61
Step-by-step explanation:
12 inches in this circle represents its
Answer:
I can't see the whole picture, but if it's half of the circle, it's the radius. if the line goes all the way through the circle, it's the diameter
Answer:
I think it’s the diameter
Step-by-step explanation:
The table shows the monthly salaries for employees at a company.
a. Find the mean, median, and mode of the data.
The mean is $
. The median is $
. The mode is $
.
Question 2
b. Each employee receives a 5% raise. Find the mean, median, and mode of the data with the raise. How does this increase affect the mean, median, and mode of the data? Round to the nearest cent.
The mean is $
The median is $
The mode is $
The raise increases the mean, median, and mode by
%.
Question 3
c. Use the original monthly salaries to calculate the annual salaries. Find the mean, median, and mode of the annual salaries. How are these values related to the mean, median, and mode of the monthly salaries?
The mean is $
. The median is $
. The mode is $
.
These values are ____ times the mean median, and mode of the monthly salaries.
(Also please explain how you got the answers.)
Answer:
30
Step-by-step explanation:
30
The mean, median, and mode of the given monthly salary is 1794, 1790, and 1940, respectively.
What is the measure of central tendency?Measures of central tendency assist in the discovery of a data set's middle, or average. The mode, median, and mean are the three most popular metrics of central tendency.
Mode: It is the most common value in a given data set.
Median: In an ordered data set, the median is the number in the middle.
Mean: It is the total number of values divided by the sum of all values.
For the given data, the mean, median and the mode of the data can be found as shown below,
Sum of the Observations
= 1940 + 1660 + 1860 + 2100 + 1720 + 1540 + 1760 + 1940 + 1820 + 1600
= 17940
Number of Observations = 10
The mean of the data is,
Mean = Sum of Observations / Number of Observations
= 17940 / 10
= 1794
The median of the data can be found by arranging the data in an order first. Therefore, the data can be arranged as,
1540160016601720176018201860194019402100The median of the data is,
Median = (Sum of the two mid values) / 2
= (1760 + 1820) / 2
= 1790
The mode of the data is 1940, since this is repeated twice.
b.
If the salary of each employee is increased by 5% that means that the value of each observation in the data will increase by 5%. Therefore, The raise increases the mean, median, and mode by 5% each.
c.
As it is now required to find the mean, median, and mode of the annual salary; there is a need to multiply the monthly salary of each employee by 12. Therefore, the value of each observation will be 12 times its original value. These values are 12 times the mean median, and mode of the monthly salaries.
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Suppose that instead of the model inc; = Be + B, ed; + B2 sex, + &i (sex equals 1 for males and 0 for females) we posited inc; = 20 + a, ed, + a2 sexi + ei = 365 where sex* equals 0 for males and 1 for females. What would you expect concerning the relationship between the parameter estimates and the interpretation of these estimates from OLS regressions of these alternative formulations?
The two alternative formulations of the OLS regressions have differences in their parameter estimates and interpretations. The first model can be represented as: inci = Be + B1edi + B2sexi + εi, where sexi equals 1 for males and 0 for females. The second model is given by: inci = a0 + a1edi + a2sexi + ei, where sexi equals 0 for males and 1 for females.
The first model, the intercept term B0 represents the average income earned by men (when sexi equals zero) who have no education. The coefficient B1 represents the increase in income for each additional year of education. B2 represents the average difference in income between men and women when education is the same. Thus, B2 is the coefficient of the dummy variable for sex.In the second model, a0 represents the average income earned by women (when sexi equals one) who have no education. a1 represents the increase in income for each additional year of education, and a2 represents the average difference in income between men and women when education is the same. Thus, a2 is the coefficient of the dummy variable for sex.The interpretation of the coefficients is different between the two models. In the first model, B2 represents the average difference in income between men and women. In the second model, a2 represents the average difference in income between women and men. Thus, the coefficients of the dummy variable for sex have opposite signs and different interpretations. Also, the intercept terms in the two models are different, which reflects the difference in the average income earned by men and women in the two models. The parameter estimates from the OLS regressions for these alternative formulations will differ as well.
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there is a line that includes the point 7,2 and has a slope of -1/3
Answer:
y= -1/3x + 13/3
Step-by-step explanation:
we can find the equation of the line thanks to this formula
y-y1 = m(x-x1) where y1 and x1 indicate the coordinates of the point, while m indicates the slope
y-2 = -1/3(x-7)
y = -1/3x+7/3+2
y= -1/3x + (7+6)/3
y = -1/3x+13/3
Fill in the blanks to complete the area model to solve 6.60 divided by 15 is the same as _ x ? = _ 6.60 = _ hundredths
Answer:
[tex]15 x = 6.60[/tex]
[tex]x = 44\ hundredths[/tex]
Step-by-step explanation:
Given
[tex]\frac{6.60}{15}[/tex]
Required
Complete the model
[ ] x = 6.60
x = _ hundredths
[tex]\frac{6.60}{15}[/tex]
Equate to x
[tex]x = \frac{6.60}{15}[/tex]
Multiply both sides by 15
[tex]15 * x = \frac{6.60}{15} * 15[/tex]
[tex]15 * x = 6.60[/tex]
[tex]15 x = 6.60[/tex]
So, we have:
[ ] x = 6.60 =====> [tex]15 x = 6.60[/tex]
Recall that:
[tex]x = \frac{6.60}{15}[/tex]
[tex]x = 0.44[/tex]
This implies that:
[tex]x = 44\ hundredths[/tex]
Hence:
x = _ hundredths =====> [tex]x = 44\ hundredths[/tex]
(Time complexity, 20pt) For the following languages, sketch (high-level, no need to mention states) a 1-tape TM program that solves the problem and include a time analysis of your program showing how much time it takes, e.g. O(n),O(nlog n), O(n^2), O(n^3), etc.; n is the length of the input.
a) [10pt] L = {u#v: U, v E {0,1}* and u is a substring of v}
b) [CSC 389, 10pt] L = {0^2^k : for k >= 0}. E.g. 0,00,0000,00000000 € L. Note that n = 2^k.
a) Language L = {u#v: u, v ∈ {0,1}* and u is a substring of v}
b) Language L = {0^2^k : for k ≥ 0}
Time Analysis: The time complexity of this program is O(n), where n is the length of the input. This is because we only need to scan through the input string once, counting the consecutive '0' symbols, and check if the count is a power of 2.
a) Language L = {u#v: u, v ∈ {0,1}* and u is a substring of v}
To sketch a 1-tape Turing machine program for this language, we can follow these steps:
1. Start at the leftmost symbol of the input.
2. Scan to the right until we encounter the '#' symbol, marking all the scanned symbols.
3. Once we encounter the '#' symbol, start scanning to the right again, comparing each scanned symbol with the marked symbols.
4. If a match is found, continue scanning until the end of the input, ensuring that the remaining symbols match the unmarked symbols.
5. If a mismatch is found at any point, reject the input.
6. If the end of the input is reached and all symbols match, accept the input.
Time Analysis: The time complexity of this program is O(n^2), where n is the length of the input. This is because, in the worst case, we need to compare each symbol in the input string with the marked symbols, and we may need to scan through the input string twice.
b) Language L = {0^2^k : for k ≥ 0}
To sketch a 1-tape Turing machine program for this language, we can follow these steps:
1. Start at the leftmost symbol of the input.
2. Scan to the right, counting the number of consecutive '0' symbols until a different symbol is encountered.
3. If the count of '0' symbols is not a power of 2, reject the input.
4. If a different symbol is encountered, check if it is the end of the input. If it is, accept the input.
5. If a different symbol is encountered and it is not the end of the input, reject the input.
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The aggregate claim amount is defined as S X₁, where Xį are independent and i=1 identically distributed claim amounts and N is the number of claims. (a) Suppose N follows a negative-binomial distribution with E(N) = k(1 − p)/p and variance Var(N) = k(1 − p)/p². Show that: k(1 − p) (i) E(S) = -m₁ and р k(1 − p) k(1-P)² m², (ii) Var (S) = -M2 + p² - (iii) Skew k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P
The required results are:
(i) E(S) = -m₁
(ii) [tex]Var(S) = k(1 − p)(m₂ - m₁²)/p + [k(1-p)/p]² (m₁² + m₂ - m₁²)[/tex]
(iii) Skew = 3(1 − 2p)/sqrt{k(1 − p)}.
The aggregate claim amount is defined as S X₁, where Xį are independent and i=1 identically distributed claim amounts and N is the number of claims.
And also suppose N follows a negative-binomial distribution with
E(N) = k(1 − p)/p
and variance
Var(N) = k(1 − p)/p².
Now we have to show that:
k(1 − p)
(i) [tex]E(S) = -m₁ and р k(1 − p) k(1-P)² m²,[/tex]
(ii) Var (S) = -M2 + p² -
(iii) [tex]Skew k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P(i) E(S) = -m₁[/tex]
Where S = X1 + X2 + ...... + XN.
Also, N and Xi's are independent.
So, E(S) = E(E(S|N))
From the law of total expectation,
E(S|N) = E(X1 + X2 +......+XN|N)
Using linearity of expectation
,E(X1 + X2 +......+XN|N) = E(X1|N) + E(X2|N) +......+ E(XN|N) = NE(X1)Now, E(S) = E(E(S|N)) = E(NE(X1))= NE(X1) = m₁
Therefore,
E(S) = -m₁.(ii) Var (S) = -M2 + p² -Where Var(S|N) = Var(X1 + X2 +......+XN|N) = Var(X1|N) + Var(X2|N) +......+ Var(XN|N)Also, E(S|N) = NE(X1)Thus, E(S) = E(E(S|N)) = E(NE(X1)) = NE(X1) = m₁ And Var(S) = E(Var(S|N)) + Var(E(S|N))
Using independence,
[tex]Var(S|N) = Var(X1 + X2 +......+XN|N) = Var(X1|N) + Var(X2|N) +......+ Var(XN|N)Var(S|N) = N Var(X1)Using independence, E(S|N) = NE(X1)Var(E(S|N)) = Var(NE(X1)) = N²Var(X1)Hence, Var(S) = E(Var(S|N)) + Var(E(S|N))= E(N Var(X1)) + Var(NE(X1))= (k(1 − p)/p) Var(X1) + N²Var(X1)= k(1 − p)Var(X1)/p + N²Var(X1)[/tex]
Also, [tex]Var(X1) = E(X1²) - (E(X1))² = m₂ - m₁²[/tex]
Therefore,
Var(S) = k(1 − p)(m₂ - m₁²)/p + N²(m₂ - m₁²)
And N follows a negative-binomial distribution with
E(N) = k(1 − p)/p and variance
[tex]Var(N) = k(1 − p)/p². Thus, E(N²) = Var(N) + (E(N))²E(N²) = k(1-p)/p² + [k(1-p)/p]²[/tex]
Therefore,
[tex]Var(S) = k(1 − p)(m₂ - m₁²)/p + [k(1-p)/p]² (m₁² + m₂ - m₁²)(iii) Skew = 3(1 − 2p)/sqrt{k(1 − p)}[/tex]
Therefore,
[tex]k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P = 3(1 − 2p)/sqrt{k(1 − p)}.[/tex]To know more about binomial distribution, visit:
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whats 40/8 but not 5? if u get it right, u get branliest cuz u big brain
Answer:
it could also be [tex]\frac{20}{4}[/tex] or [tex]\frac{10}{2}[/tex]
Step-by-step explanation:
Simplify.
[tex]\frac{40}{8}[/tex] ÷ 2 = [tex]\frac{20}{4}[/tex]
[tex]\frac{20}{4}[/tex] ÷ 2 = [tex]\frac{10}{2}[/tex]
one card is selected at random from a deck of cards. determine the probability of selecting a card that is less than 8 or a club. note that the ace is considered a low card.
To determine the probability of selecting a card that is less than 8 or a club from a standard deck of cards, we need to consider the number of favorable outcomes and the total number of possible outcomes.
First, let's calculate the number of cards that are less than 8. There are four suits (hearts, diamonds, clubs, and spades), and each suit has cards numbered 2 through 7. So, there are 4 suits * 6 cards per suit = 24 cards that are less than 8.
Next, let's calculate the number of clubs in the deck. There are 13 cards in each suit, and one of those suits is clubs. Therefore, there are 13 clubs in the deck.
To find the probability, we add the number of favorable outcomes (cards less than 8 or clubs) and divide it by the total number of possible outcomes (52 cards in a deck).
Probability = (24 + 13) / 52 = 37 / 52 ≈ 0.7115
Therefore, the probability of selecting a card that is less than 8 or a club is approximately 0.7115 or 71.15%.
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Pi is: (click all that are correct) *
always the same, and is 3.14 or 22/7
delicious
used to help find the circumference of a circle
always a different number pls help lol
yes,Pi is 3.142 or 22/7
Step-by-step explanation:
22/7=3.14285714
ANSWER ASAP! DUE TODAY
5. Find the measure of arc AC.
Find mAC
A) 52
B) 72
C) 92
D) 112
Answer:
A. 52°
Step-by-step explanation:
The measure of an inscribed angle is 1/2 the measure of the intercepted arc.
m∠B = 1/2 (m of arc AC)
26 = 1/2 (m of arc AC)
52 = (m of arc AC)
Which of the following is a sine of a in the right triangle below
Answer:
[tex]\frac{5}{13}[/tex]
Step-by-step explanation:
sine of A = [tex]\frac{opposite}{hypotenuse}[/tex]
opposite = 5
adjacent = 12
hypotenuse = 13
Answer:
C.) 12/13
Explanation:
trust me
A point P(x, y) moves along the graph of the equation y = x3 + x2 + 6. The x-values are changing at the rate of 2 units per second. How fast are the y-values changing (in units per second) at the point Q(1, 8)?
The y-values are changing at a rate of 24 units per second at the point Q(1, 8).
To find how fast the y-values are changing at the point Q(1, 8), we need to calculate the derivative of the given equation with respect to x. The derivative of y = x^3 + x^2 + 6 is dy/dx = 3x^2 + 2x.
At the point Q(1, 8), we substitute x = 1 into the derivative to find the rate of change.
dy/dx = 3(1)^2 + 2(1) = 3 + 2 = 5.
Therefore, the y-values are changing at a rate of 5 units per second at the point Q(1, 8). However, the rate at which the x-values are changing is given as 2 units per second.
Thus, to determine how fast the y-values are changing with respect to time, we multiply the rate of change in y by the rate of change in x: 5 units/second * 2 units/second = 10 units/second. Hence, the y-values are changing at a rate of 10 units per second at the point Q(1, 8).
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Each of the following sentences has three errors in grammar, capitalization, usage, or spelling. Write a correct version for each sentence. Avoid adding new phrases, starting new sentences, or rewriting in your own words.
1) Among the oddly named towns in the United States are: What Cheer, Iowa, Peculiar, Missouri, and Cheesecake, New Jersey.
2) After our supervisor and her returned from their meeting at 2:00 p.m., we were able to sort the customers names more quickly.
3) 6 of the 18 workers in my department were fired, as a result we had to work harder to achieve our goals.
4) Jeffreys presentation to a nonprofit group netted him only three hundred dollars, which is a tenth of his usual honorarium but he believes in pro bono work.
5) Of all the discoveries and inventions in human history the 4 greatest are said to be these speech, fire, agriculture, and the wheel.
6) Our latest press release written by our corporate communication department announces the opening of three Asian offices.
7) In his justification report dated September first, Justin argued that expansion to twelve branch offices could boost annual revenue to 22 million dollars.
8) The practicality and advisability of opening 12 branch offices are what will be discussed in the consultants feasibility report.
9) 3 categories of Apps (social media, productivity, and entertainment—are most popular for installing on smartphones.
10) Because some organizations prefer single spaced reports be sure to check with your organization to learn it’s preference.
Below are the corrected versions of the sentences with grammar, capitalization, usage, and spelling errors addressed.
Oddly named towns in the United States include What Cheer, Iowa; Peculiar, Missouri; and Cheesecake, New Jersey.After our supervisor and she returned from their meeting at 2:00 p.m., we were able to sort the customers' names more quickly.Six of the eighteen workers in my department were fired. As a result, we had to work harder to achieve our goals.Jeffrey's presentation to a nonprofit group netted him only three hundred dollars, which is a tenth of his usual honorarium. But he believes in pro bono work.Of all the discoveries and inventions in human history, the four greatest are said to be speech, fire, agriculture, and the wheel.Our corporate communication department has written the latest press release, which announces the opening of three Asian offices.In his justification report dated September 1st, Justin argued that expanding to twelve branch offices could boost annual revenue to 22 million dollars.The feasibility report from the consultants will discuss the practicality and advisability of opening twelve branch offices.Three categories of apps (social media, productivity, and entertainment) are most popular for installing on smartphones.Because some organizations prefer single-spaced reports, be sure to check with your organization to learn its preference.The sentence lists the oddly named towns correctly, but the capitalization of "Iowa" and "New Jersey" should be in lowercase, and colons are not needed in this context.The pronoun "her" should be replaced with "she" to be grammatically correct. "Customers names" should be "customers' names" to show possession.The numbers "6" and "18" should be written as words, and the sentence should be divided into two sentences for clarity."Jeffreys" should be corrected to "Jeffrey's" to indicate possession. "Pro bono work" should be in italics or quotation marks for correct formatting.The sentence lacks proper punctuation and capitalization. The four greatest discoveries and inventions should be separated by commas.The sentence is grammatically correct, but the use of "our" twice can be avoided for better flow.The date should be written as "September 1st" instead of "September first" for proper formatting.The subject-verb agreement is incorrect. "The practicality and advisability" should be followed by "is," not "are."The number "3" should be written as "Three." The parentheses should be placed correctly, and a comma is needed after "entertainment.""It's" should be corrected to "its" as it denotes possession, not a contraction. The sentence should be divided into two sentences for clarity.Learn more about spelling errors here:
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Will takes out a loan of $1400 at 4.5% interest. The loan term is 3 years, with a monthly payment of $101.89. How much interest does he
pay?
Answer:
+100 nasan yung tanong ah
Complete the equation of the line through (-8,-2) and (-4,6)
Use exact numbers.
y=
Answer:
round 55666777 for 5th 50th is not a big game for a player who will give the 8th or 3rd season to 466
Help help pretty please
Answer:
Yes it is
Step-by-step explanation:
It passes the vertical line test.
Let : = (a + ai)(b + b/3i) where a and b are positive real numbers. Without using a calculator, determine arg 2.
The answer is arg 2 = tan^-1(3/2) where : = (a + ai)(b + b/3i) where a and b are positive real numbers.
To determine arg 2, we need to first find the value of :.
Expanding the given expression, we get:
: = (a + ai)(b + b/3i)
: = ab + ab/3i^2 + abi + ab/3i
: = ab - ab/3 + abi + ab/3i
: = (2ab/3) + (ab)i
Now, we can find the modulus of : as:
|:| = sqrt((2ab/3)^2 + (ab)^2)
|:| = sqrt(4a^2b^2/9 + a^2b^2)
|:| = sqrt(13a^2b^2/9)
And, we can find the argument of : as:
arg(:) = tan^-1((ab)/(2ab/3))
arg(:) = tan^-1(3/2)
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If a man invests $1000 in a savings account and another $1000 in a fixed deposit. If the savings account pays 4.5% interest per annum and the fixed deposit pays 6% interest per annum, find the total amount after 1 year of investment. Please I need a quick answer to this!
Answer:
Results are below.
Step-by-step explanation:
Giving the following information:
Savings account:
PV= $1,000
n= 1
i= 0.045
Fixed Deposit:
PV= $1,000
n= 1
i= 0.06
To calculate the future value, we need to use the following formula:
FV= PV*(1+i)^n
Savings account:
FV= 1,000*1.045^1
FV= $1,045
Fixed deposit:
FV= 1,000*1.06^1
FV= $1,060
Pls help me I don’t know how to do it!
Answer:
864
Step-by-step explanation:
To find the area of a shape, you have to multiply length times width.
Answer:
72 ft. sq.
Step-by-step explanation:
The way to look at it it to multiply length times width ( l x w )
there are 2 obvious numbers to get the answer, which is 8 and 9 so 8 x 9 = 72
the factors. 2. Fill in the blanks with appropriate word(s) or numbers. a) 6x 10³ +3× 10 + 4 x 10-2is written in standard form as b) Four hundred seven and thirty-four thousandths in standard form is written as c) 86.00405 is written in words as d) (3x*y) written without negative exponents is e) 0.0675= % f) 44% is the same as the fraction in simplest form. g) When you multiply a natural number by a decimal, when do you get a result more than that number? h) 13.982 rounded to the nearest tenths is i) 4 tens minus 4 tenths = (write the answer in standard form)
Answer:
answer is a
Step-by-step explanation:
first solve
solve for gcf
flip
divide
then you add
and you get 0.928838
Let f : R → R be a function. Define a sequence fn : R → R by 1 fn(x) = f ( x + = = (₁ + ₁/ ). n (a) Prove that if f is uniformly continuous, then fn → f uniformly. (b) Prove that uniform continuity of f is a necessary assumption to conclude that fn → ƒ uniformly. That is, give an example of a function ƒ : R → R such that f is continuous, but that fn does not converge to f uniformly.
(a) If f is uniformly continuous, then fn converges to f uniformly.
(b) Uniform continuity of f is necessary for the uniform convergence of fn to f; a counterexample is provided with a continuous f but non-uniform convergence of fn.
(a)If f is uniformly continuous, then we want to show that fn converges to f uniformly.
Given ε > 0, since f is uniformly continuous, there exists δ > 0 such that for any x, y in R, if |x - y| < δ, then |f(x) - f(y)| < ε.
Now, consider fn(x) = f(x + 1/n) for each n. We need to show that for any ε > 0, there exists N such that for all n ≥ N, |fn(x) - f(x)| < ε for all x in R.
Fix ε > 0. By the uniform continuity of f, there exists δ > 0 such that for any x, y in R, if |x - y| < δ, then |f(x) - f(y)| < ε.
Now, for any x in R, let N > 1/δ. Then for all n ≥ N and for any y = x + 1/n, we have |x - y| = |x - (x + 1/n)| = 1/n < δ.
By choosing y = x + 1/n, we have |fn(x) - f(x)| = |f(x + 1/n) - f(x)| < ε.
Since ε was chosen arbitrarily, this shows that fn converges to f uniformly.
(b) Counterexample: To show that uniform continuity of f is necessary for the uniform convergence of fn to f, we need to find an example where f is continuous but fn does not converge to f uniformly.
Consider the function f(x) = x on R. This function is continuous.
Now, let's examine the sequence fn(x) = f(x + 1/n). For any x in R, we have:
|fn(x) - f(x)| = |(x + 1/n) - x| = 1/n.
Notice that for any fixed x, as n approaches infinity, 1/n approaches 0. However, the convergence is not uniform since the choice of x affects the convergence rate.
For any ε > 0, no matter how large N is chosen, there will always exist x such that |fn(x) - f(x)| = 1/n ≥ ε, for n > 1/ε.
Therefore, fn does not converge uniformly to f(x) = x, despite f(x) being continuous.
This counterexample demonstrates that the assumption of uniform continuity of f is necessary for the uniform convergence of fn to f.
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A dump truck brought 1⁄3 of a ton of rock on the first trip, 1⁄2 of a ton on the second trip, but on the third trip, had to take back 4⁄5 of a ton. What was the total weight of the rock left at the construction site?
Answer:
1/30 ton of rock left
Step-by-step explanation:
1/3+ 1/2= 5/6
5/6 - 4/5= 1/30 ton of rock left at the construction site.
log(x)-log(x+6)=log(k)
Find x if k=1/2
Answer:
x = 6
Step-by-step explanation:
Using the rules of logarithms
logx - logy = log ([tex]\frac{x}{y}[/tex] )
logx = logy , then x = y
Given
log(x) - log(x + 6)
= log([tex]\frac{x}{x+6}[/tex] )
Then
log )[tex]\frac{x}{x+6}[/tex] ) = log([tex]\frac{1}{2}[/tex] ) , thus
[tex]\frac{x}{x+6}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
2x = x + 6 ( subtract x from both sides )
x = 6
Which groups of numbers could 1/3 fall under. Natural,whole,integer,or rational
Answer:
The number 1/3 is a rational number. A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
1/3 is not a natural number, whole number, or integer. Natural numbers are the set of positive integers (1, 2, 3, …). Whole numbers are the set of non-negative integers (0, 1, 2, 3, …). Integers are the set of whole numbers and their additive inverses (-3, -2, -1, 0, 1, 2, 3, …).
Step-by-step explanation: