Answer:
The height of the flagpole is 36.363 feet.
Step-by-step explanation:
This problem can be represented geometrically by two similar right triangles, represented in the image attached below. By definition of similarity, we construct the following relationship:
\(\frac{4\,ft}{5.5\,ft} = \frac{x}{50\,ft}\) (1)
Where \(x\) is the height of the flagpole, measured in feet.
Lastly, we find the required value:
\(x = \left(\frac{4\,ft}{5.5\,ft} \right)\cdot (50\,ft)\)
\(x = 36.363\,ft\)
The height of the flagpole is 36.363 feet.
HELP ME WITH THIS PLEASE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
The answer is 5
hope it helps
Answer:
n = 5 units
Step-by-step explanation:
We can solve this using 2 different equations which lead to the same answer:
Equation #1: Sin(30) = n/10
=>Sin(30) * 10 = n
=> 5 = n
Equation #2: Cos(60) = n/10
=> Cos(60) * 10 = n
=> 5 = n
If a number is a whole number, it is in it's simplest radical form, therefore; n = 5 units is the correct answer
Hope this helps!
Number 14 please help I’m feolifn
Answer:
see explanation
Step-by-step explanation:
∠ R = 2x + 5 , ∠ S = x + 1 , ∠ T = 7x - 16
the sum of the 3 angles in a triangle = 180°
sum the 3 angles and equate to 180
2x + 5 + x + 1 + 7x - 16 = 180
10x - 10 = 180 ( add 10 to both sides )
10x = 190 ( divide both sides by 10 )
x = 19
Then
∠ R = 2x + 5 = 2(19) + 5 = 38 + 5 = 43°
∠ S = x + 1 = 19 + 1 = 20°
∠ T = 7x - 16 = 7(19) - 16 = 133 - 16 = 117°
A and B together can complete a work in 3 days. They start together but, after 2 days, B left the work. If the work is completed after 2 more days, B alone could do the work inA. 8 days
B. 3 days
C. 2 days
D. 6 days
B alone could do the work in D) 6 days.
Let's assume that A and B together can complete the work in x days. According to the question, we know that:
1/x = 1/3 (since they can complete the work in 3 days)
Solving for x, we get x = 3.
Now, we know that A and B worked together for 2 days. So, the amount of work done in those 2 days is:
2/x = 2/3
This means that the remaining work that needs to be done is:
1 - 2/3 = 1/3
We also know that B alone can complete the work in y days. So, the amount of work B can do in 2 days is:
2/y
Therefore, the remaining work that A needs to do is:
1/3 - 2/y
We also know that A and B together can complete the work in 5 days (after B left). So, we have:
1/x + 1/y = 1/5
Substituting the value of x, we get:
1/3 + 1/y = 1/5
Solving for y, we get y = 6.
Therefore, B alone could do the work in 6 days.
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Today Sean read the first 14 pages of his
book. He wants to read at least 80 pages
the end of the week. Write and solve an
inequality to find how many more pages
Sean must read.
Sean must read at least 66 more pages to reach his goal of reading at least 80 pages by the end of the week.
Let's say the number of pages Sean needs to read is x.
he has already read 14 pages and wants to read at least 80 pages by the end of the week.
We can write this as an inequality:
x + 14 ≥ 80
To solve this inequality, we need to isolate x.
First, we subtract 14 from both sides of the inequality:
x + 14 - 14 ≥ 80 - 14
This simplifies to:
x ≥ 66
So, Sean must read at least 66 more pages to reach his goal of reading at least 80 pages by the end of the week.
Sean must read at least 66 more pages to reach his goal of reading at least 80 pages by the end of the week.
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A home improvement store advertises 60 square feet of flooring for $253. 00, plus an additional $80. 00 installation fee. What is the cost per square foot for the flooring
If home improvement store advertises 60 square feet of flooring for $253. 00, plus an additional $80. 00 installation fee, the cost per square foot for the flooring is $5.55.
To find the cost per square foot of the flooring, we need to divide the total cost (including installation) by the total square footage.
First, we need to determine the total square footage that $253.00 covers. We know that 60 square feet of flooring cost $253.00, so we can set up a proportion:
60 sq. ft. / $253.00 = x sq. ft. / $1.00
Solving for x, we get:
x = (60 sq. ft. x $1.00) / $253.00
x ≈ 0.24 sq. ft.
So, $253.00 covers 60 square feet of flooring, which gives us a cost of:
$253.00 + $80.00 = $333.00 total cost for 60 sq. ft. of flooring and installation
To find the cost per square foot, we divide the total cost by the total square footage:
$333.00 / 60 sq. ft. = $5.55 per square foot
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select the appropriate reagents for the transformation at −78 °c.
For the transformation at -78 °C, appropriate reagents include lithium aluminum hydride (LiAlH4) and diethyl ether.
What reagents are suitable for -78 °C transformations?At -78 °C, certain chemical reactions require the use of specific reagents to achieve the desired transformation. One commonly used reagent is lithium aluminum hydride (LiAlH4), which acts as a strong reducing agent. It is capable of reducing various functional groups, such as carbonyl compounds, to their corresponding alcohols.
Diethyl ether is typically employed as a solvent to facilitate the reaction and ensure efficient mixing of the reactants. Researchers often utilize this low temperature for reactions involving sensitive or reactive intermediates, as it helps control the reaction and prevent unwanted side reactions.
The use of LiAlH4 and diethyl ether provides a reliable combination for achieving the desired transformation at this temperature, enabling chemists to manipulate and modify compounds in a controlled manner.
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Solve for c.
c= -9 - 8c
c=-9-8c
+8c
9c=-9
c=-1
Hope this helps:D!!
The letter C is equal to negative one.
c = -1
Hope this helps!
Use power series operations to find the Taylor series at x=0 for the following function. xcos 2
3πx
The Taylor series for cosx is a commonly known series. What is the Taylor series at x=0 for cosx ? ∑ n=0
[infinity]
(Type an exact answer.) Use power series operations and the Taylor series at x=0 for cosx to find the Taylor series at x=0 for the given function. ∑ n=0
[infinity]
(Type an exact answer.)
The Taylor series at x=0 for the given function x * cos²((3πx)/2) is:
∑(n=0 to infinity) ∑(m=0 to infinity) (-1)^(n+m) * (x²ⁿ⁺¹) * ((((3π)/2)^(2n+2m)) / ((2n)!(2m)!))
Here, we have,
The Taylor series at x=0 for cos(x) is given by:
cos(x) = ∑(n=0 to infinity) (-1)ⁿ * (x²ⁿ)) / (2n)!
Now, let's find the Taylor series at x=0 for the given function x * cos²((3πx)/2):
To find the Taylor series for the given function, we'll use power series operations.
We'll substitute the Taylor series expansion for cos(x) into the given function and then perform the necessary operations.
Let's start with cos²((3πx)/2):
cos²((3πx)/2) = (cos((3πx)/2))²
= (∑(n=0 to infinity) (-1)ⁿ * (((3πx)/2)²ⁿ) / (2n)!)²
Expanding the square of the series, we get:
cos²((3πx)/2)
= (∑(n=0 to infinity) (-1)ⁿ * (((3πx)/2)²ⁿ) / (2n)!) * (∑(m=0 to infinity) (-1)^m * (((3πx)/2)^(2m)) / (2m)!)
Now, we'll multiply the x term to obtain the Taylor series for the given function:
x * cos²((3πx)/2) = x * (∑(n=0 to infinity) (-1)ⁿ * (((3πx)/2)²ⁿ) / (2n)!) * (∑(m=0 to infinity) (-1)^m * (((3πx)/2)^(2m)) / (2m)!)
Expanding the multiplication and rearranging the terms, we have:
x * cos²((3πx)/2) = ∑(n=0 to infinity) ∑(m=0 to infinity) (-1)^(n+m) * (x²ⁿ⁺¹) * ((((3π)/2)^(2n+2m)) / ((2n)!(2m)!))
Therefore, the Taylor series at x=0 for the given function x * cos²((3πx)/2) is:
∑(n=0 to infinity) ∑(m=0 to infinity) (-1)^(n+m) * (x²ⁿ⁺¹) * ((((3π)/2)^(2n+2m)) / ((2n)!(2m)!))
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your sketching on a standard sheet of paper 8.5x11 in. if you draw a straight vertical line down half the width of the paper your line will be approximately
The line draw down as per given dimensions of standard sheet of paper 8.5x11 in then half the width of paper is 4.25 inches long approximately.
Let's consider a standard sheet of paper, which typically has dimensions of 8.5 inches by 11 inches (width by height).
If you draw a straight vertical line down half the width of the paper,
It is essentially drawing a line from the top edge to the bottom edge, dividing the paper into two equal halves.
The width of the paper is 8.5 inches.
When you draw a line down half the width, you are going from one edge to the midpoint.
Since you are going halfway, the length of the line will be half of the width,
which is 8.5 inches divided by 2, resulting in approximately 4.25 inches.
Therefore, the line you draw down half the width of the paper will be approximately 4.25 inches long.
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Francis surveyed a random sample of 70 7070 students at Franklin High School about their favorite season. Of the students surveyed, 18 1818 chose fall as their favorite season. There are 1816 18161816 students at Franklin High School. Based on the data, what is the most reasonable estimate for the number of students at Franklin High School whose favorite season is fall? Choose 1 answer: Choose 1 answer:
Answer:
467
Step-by-step explanation:
Determine the percentage of those sampled whose favourite season is fall
(18/70) x 100 = 25.7%
the sample is representative of the population of students. So, 25.7% of the students would like fall
number of students at Franklin High School whose favourite season is fall = 25.7% x 1816 = 466.7 students
rounding off to the nearest whole number gives 467
Answer:
467
Step-by-step explanation:
1. a certain college wants to estimate the amount of time students, who live off campus, spend commuting to classes each week. a random sample of 45 off-campus students is surveyed. a) identify the population and sample for this study. b) what data is being collected? what type of data is this? c) what type of study is this?
The objective of this research is to collect data on a specific component of the off-campus student population.The amount of time they spend travelling to courses each week.
a) The population for this study is all off-campus students at this college, and the sample is a random sample of 45 off-campus students who are polled. The sample is chosen at random to be representative of the entire population.
b) The data being gathered is the amount of time the surveyed students spend each week travelling to school. Because it reflects a numerical measurement, this is quantitative data. This information will be used to calculate the average commuting time for off-campus students and to identify any possible concerns or areas for improvement in commuting.
c) This is a descriptive research, since it seeks to characterise and summarise the features of the off-campus student population in terms of commuting time. The study's purpose is to better understand off-campus students' commuting patterns and gather ideas into how to enhance their commuting experience.
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What is the value of the expression below when z =
6?
2z2 + 22 - 3
Answer:
43
Step-by-step explanation:
2z2+22-3
2 x 6 x 2+22-3
12 x 2+22-3
24+22-3
46-3
43
Which of the following is true regarding a regression model with multicollinearity, a high r2 value, and a low F-test significance level? a.The model is not a good prediction model. b.The high value of 2 is due to the multicollinearity. c.The interpretation of the coefficients is valuable. d.The significance level tests for the coefficients are not valid. e.The significance level for the F-test is not valid.
The correct answer is d. The significance level tests for the coefficients are not valid.
Multicollinearity is a statistical term that refers to the presence of high correlation among predictor variables in a regression model. This can cause issues in the model, such as unstable or unreliable coefficients, and can lead to incorrect conclusions about the relationships between the predictors and the response variable.
When multicollinearity is present, the R-squared value of the model can become inflated because the model is able to explain more of the variation in the response variable due to the high correlation among the predictor variables. This can give the impression that the model is a good predictor when in fact it may not be. Additionally, multicollinearity can cause the F-test significance level to be low, indicating that the model is a good fit, even though the individual coefficients may not be statistically significant.
Multicollinearity can cause inflated R-squared values and low F-test significance levels. However, it does not necessarily mean that the model is a poor predictor. The interpretation of coefficients may also be affected by multicollinearity.
However, the most significant issue with multicollinearity is that it can lead to unreliable significance tests for individual coefficients, making it difficult to determine which predictors are contributing significantly to the model.
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An electrician charges a set fee of $50 for every house call and then charges an additional hourly rate. One day, the electrician earned a total of $400 for a 5 hour job. Write an equation for C,C, in terms of t,t, representing the total cost of the electrician's services if the electrician spends tt hours at the house working.
Answer:
\(50+5t=C\)
Step-by-step explanation:
Suppose the electrician spends t hours at the house working.
Amount charged for every house call by electrician = $50
The electrician earned a total of $400 for a 5 hour job.
Also, C denotes total cost of the electrician's services.
So, the equation for C in terms of t is given below.
\(50+5t=C\)
Here, \(C=400\)
8 in. 8 in. 8 in Find the area of the figure above, [?] in
Answer:
96
Step-by-step explanation:
Total area = area of cube + area of triangle
A = 8² + \(\frac{8*8}{2}\) = 64 + 32 = 96 in²
Answer:
96 square feet
to find the area of a triangle the formula is
A=1/2×b×h
8×8=64
1/2×8×8=32
32+64= 96
carla wants to investigate whether a person’s political party affiliation causes the person to be more vocal about political issues. she plans to administer a survey to a large sample of people. which of the following describes why the method of data collection used will prevent carla from achieving her goal?
Answer:
causation cannot be determined from a survey
Pls provide clearer calculation. the answer has been given
For Q4 and Q5 evaluate Q4 the determinant, given that 3a 36 3c -d - e - f 4g 4h 4i Ans: 72 Q5 a 2d g+3a Ans: -12 b с 2e 2f h+3b i+ 3c] a b c def-6 gh i
The values of determinant ,
\(\left|\begin{array}{ccc}3a&3b&3c\\-d&-e&-f\\4g&4h&4i\end{array}\right| = 72\)
\(\left|\begin{array}{ccc}a&b&c\\2d&2e&2f\\g+3a&h+3b&+3ci\end{array}\right| = - 12\)
The given determinant is,
\(\left|\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right| = -6\)
This is possible when we take,
a = -1 , e = 1, i = 6, b = c = 1
And d = g = h = 0
The the determinant becomes,
\(\left|\begin{array}{ccc}-1&1&1\\0&1&1\\0&0&6\end{array}\right|\)
Since this is an upper triangular matrix,
Therefore,
Determinant be the product of diagonal element,
Thus,
⇒ -1x1x6 = -6
Now using these values we can calculate the further given determinants,
Q4: Given that,
\(\left|\begin{array}{ccc}3a&3b&3c\\-d&-e&-f\\4g&4h&4i\end{array}\right|\)
Therefore put the values of a, b, c, d, e, f, g, h, and i defined above,
\(\left|\begin{array}{ccc}3(-1)&3(1)&3(1)\\0&-1&-1\\0&0&4(6)\end{array}\right|\)
Therefore, Determinant be,
⇒ -3 x (-1) x 24 = 72
Hence,
\(\left|\begin{array}{ccc}3a&3b&3c\\-d&-e&-f\\4g&4h&4i\end{array}\right| = 72\)
Q5 : Given determinant is,
\(\left|\begin{array}{ccc}a&b&c\\2d&2e&2f\\g+3a&h+3b&+3ci\end{array}\right|\)
put the values of a, b, c, d, e, f, g, h, and i defined above,
we get,
\(\left|\begin{array}{ccc}-1&1&1\\0&2&2\\-3&3&9\end{array}\right|\)
= -1(12) -1(6) + 1(6)
= -12
Hence, The determinant be
\(\left|\begin{array}{ccc}a&b&c\\2d&2e&2f\\g+3a&h+3b&+3ci\end{array}\right| = - 12\)
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The complete question is attached below:
12) (-4,4), (-2, 2)
what is the midpoint?
Answer:
-3,3
Step-by-step explanation:
Answer:
_3,3 this is the right answer
In a randomly selected 100 students in a large college, 20 of them had at least one sibling. Does this provide strong evidence that more than 15% of college students in America have at least one sibling? Find the p-value (round off to second decimal place).
The p-value is approximately 0.027 (rounded off to two decimal places).
To determine whether there is strong evidence that more than 15% of college students in America have at least one sibling, we can use a hypothesis test with a significance level of 0.05.
Let p be the proportion of college students in America who have at least one sibling. The null hypothesis is that p ≤ 0.15, and the alternative hypothesis is that p > 0.15.
Under the null hypothesis, the number of students in a sample of 100 who have at least one sibling follows a binomial distribution with parameters n = 100 and p = 0.15. We can use this distribution to calculate the probability of observing 20 or more students with at least one sibling in a sample of 100 if the null hypothesis is true.
The p-value is the probability of observing a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. In this case, the test statistic is the number of students with at least one sibling in a sample of 100.
Using a binomial distribution calculator or software, we find that the probability of observing 20 or more students with at least one sibling in a sample of 100 if p = 0.15 is approximately 0.027.
Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is strong evidence that more than 15% of college students in America have at least one sibling.
Therefore, the p-value is approximately 0.027 (rounded off to two decimal places).
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Enter a decimal that equivalent to 6/100
Answer:
0.06
Step-by-step explanation:
Lucy and Greg each want to buy a bicycle. Lucy has already saved $35 and plans to save $10 per week. Greg has $26 and plans to save $13 per week. How many weeks will it take for Lucy and
Gregory to have the same amount of money saved? How much money will they have saved?
Answer:
3 weeks
65 dollars
Step-by-step explanation:
Alright to answer this question we have to set up an equation. For greg the equation is y=13x+26 and for lucy the equation is y=10x+35. y is total amount of money saved and x is number of weeks. Since we want them to each of the same amount of money we set up the equations to be 10x+35=13x+26
then we want to get the constant and the variables alone on different sides of the equal sign so we first subtract 26 from each side to get 10x+9=13x. next we want to subtract 10x from each side and then we end up with 3x=9. Finally we divide 3 from each side to get x=3, We have answered the question to how many week it will take them to save the same amount of money but not the other part. To find out how much money they will both have in 3 weeks we plug in 3 to x. 10(3)+35 which is equal to 65. They will both have 65 dollars in three weeks
Low Carb Diet Supplement, Inc., has two divisions. Division A has a profit of $230,000 on sales of $2,120,000. Division B is able to make only $34,700 on sales of $381,000.
Compute the profit margins (return on sales) for each division. (Input your answers as a percent rounded to 2 decimal places.)
Division A= ______%
Division B= ______%
___________________________________________________________________________________________________________________________________________________
Polly Esther Dress Shops Inc. can open a new store that will do an annual sales volume of $1,220,400. It will turn over its assets 2.7 times per year. The profit margin on sales will be 7 percent.
What would net income and return on assets (investment) be for the year? (Input your return on assets answer as a percent rounded to 2 decimal places.)
Net Income=
Return on Assets= __________ %
The profit margins (return on sales) for each division are approximately :Division A = 10.85%,Division B = 9.11% and The calculations for the year would be:Net Income = $85,428,Return on Assets = 18.9%.
To compute the profit margins (return on sales) for each division, we divide the profit by the sales and multiply by 100 to express the result as a percentage.
For Division A:
Profit Margin = (Profit / Sales) * 100
Profit Margin = ($230,000 / $2,120,000) * 100
Profit Margin ≈ 10.85%
For Division B:
Profit Margin = (Profit / Sales) * 100
Profit Margin = ($34,700 / $381,000) * 100
Profit Margin ≈ 9.11%
To calculate the net income and return on assets for Polly Esther Dress Shops Inc., we use the given information.
Net Income = Profit Margin * Sales
Net Income = 7% * $1,220,400
Net Income = $85,428
Return on Assets = Profit Margin * Asset Turnover
Return on Assets = 7% * 2.7
Return on Assets = 18.9%
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Jose swims 3 laps in a pool every 2 minutes. How long would it take him to swim 27 laps?
A.
8
B.
9
O c.
16
o
D.
18
Answer:
D
Step-by-step explanation:
time = 27 ÷ 3 × 2 = 9 × 2 = 18 minutes → D
Answer:
The answer is below
Step-by-step explanation:
You first divide 27/3
= 9
you divide it cause you know how many times you have done a lap every 2 minutes.
Now to find how long you took you times 9 by 2
= 18
Therefore Jose took 18 minutes to swim 27 laps
there are six balls in a box if we select three balls what is the probability of having one white ball
The probability of having one white ball is 1/20 or approximately 0.05.
To find the probability of selecting one white ball out of three balls, you need to know the number of white balls in the box and the total number of balls in the box. You also need to know whether the balls are being selected randomly or not.
Assuming that there is only one white ball in the box and the rest are some other color, and that the balls are being selected randomly.
The probability of selecting one white ball out of three would be:
Probability = (number of ways to select one white ball out of three) / (total number of ways to select three balls)
Since there is only one white ball, there is only one way to select one white ball out of three. The total number of ways to select three balls out of six is 6 choose 3, which is equal to 20.
Therefore, the probability of selecting one white ball out of three would be
Probability = (number of ways to select one white ball out of three) / (total number of ways to select three balls)
Probability = 1/20
Probability = approximately 0.05.
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Consider the following data for MRP. Develop the material requirements plan for item B and C, and answer questions below
MPS start:
End item Week 1 Week 2 Week 3 Week 4 Week 5 Week 6
A 120 80
Bill of materials:
End item A is made of 1 unit of item B.
Item B is made of 2 units of item C.
MRP-related information for each item:
Item Leadtime (weeks) Lot sizing Safety stock On-hand inventory Scheduled receiptsB 1 L4L 0 30 60 in
week 1
C 2 FOQ (50) 20 80 None
What is the gross requirement for item B in Week 3?
i. 0 unit
ii. 80 units
iii. 120 units
iv. 200 units
v. 240 units
The gross requirement for item B in Week 3 is 80 units.
The gross requirement for item B in Week 3, we need to calculate the demand for item B based on the bill of materials and the requirements for item A.
From the MPS (Master Production Schedule) provided, we can see that the gross requirement for item A in Week 3 is 80 units.
According to the bill of materials, item A requires 1 unit of item B. Therefore, the gross requirement for item B can be calculated by multiplying the gross requirement for item A by the quantity needed for item B in the bill of materials.
Gross requirement for item B = Gross requirement for item A × Quantity needed for item B in bill of materials
Gross requirement for item B = 80 units × 1 unit
Gross requirement for item B = 80 units
Therefore, the gross requirement for item B in Week 3 is 80 units.
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I really need help with this question as fast as possible
Answer:
-1625
Step-by-step explanation:
→ (-125) × 5 + (-125) × 8
Taking (-125) as in common,
→ (-125) × (8 + 5)
→ (-125) × 13
→ -1625
For each expression, use the distributive property to write an equivalent expression.4(2x + 3) =6(x + y + z) =
The distributive property of multiplications states allows you to multiply each term on the parentheses term by the constant outside the parentheses, for example:
\(a(b+c)=ab+ac\)To solve
\(4(2x+3)\)You have to multiply each term on the parentheses by 4, that is, multiply 2x by 4 and 3 by 4
\(\begin{gathered} 4\cdot2x+4\cdot3 \\ 8x+12 \end{gathered}\)And you get that the equivalent expression is 8x+12
To solve
\(6(x+y+z)\)You have to multiply the three terms inside the parentheses by 6, so that
\(6\cdot x+6\cdot y+6\cdot z=6x+6y+6z\)The equivalent expression is 6x+6y+6z
Lesson Performance Task
The image shows the angles formed by a pair of scissors. When the
scissors are closed, m<1= 0°. As the scissors are opened, the measures
of all four angles change in relation to each other. Describe how the
measures change as m<1 increases from 0° to 180°.
Answer: Angle 3 is a vertical angle of angle 1 and will always be identical to what angle 1 is.
Angle 2 and angle 4 are supplementary angles to angle 1 and will always be the difference between 180 and what angle 1 is.
For example, if angle 1 were 45 degrees, angle 3 would also be 45 degrees, and angles 2 and 4 would be 180-45 = 135 degrees.
Step-by-step explanation:
Let R be a ring and r1,...,rn ∈ R. Prove that the subset ={λ1r1 +···+ λnrn | λ1,...,λn ∈ R} is an ideal in R.
Since S satisfies both defining properties of an ideal in R, we can conclude that S is indeed an ideal in R.
To prove that the subset S = {λ1r1 +···+ λnrn | λ1,...,λn ∈ R} is an ideal in R, we need to show that it satisfies the two defining properties of an ideal:
1. S is a subgroup of R under addition.
2. S is closed under multiplication by elements of R.
First, let's show that S is a subgroup of R under addition.
- Closure under addition: Let x,y ∈ S, so that x = λ1r1 + ··· + λnrn and y = μ1r1 + ··· + μnrn for some λi,μi ∈ R. Then their sum is x + y = (λ1 + μ1)r1 + ··· + (λn + μn)rn, which is in S since each coefficient is still in R.
- Additive inverse: Let x ∈ S, so that x = λ1r1 + ··· + λnrn for some λi ∈ R. Then its additive inverse is -x = (-λ1)r1 + ··· + (-λn)rn, which is also in S since each coefficient is still in R.
Therefore, S is a subgroup of R under addition.
Next, let's show that S is closed under multiplication by elements of R.
- Closure under left multiplication: Let r ∈ R and x ∈ S, so that x = λ1r1 + ··· + λnrn for some λi ∈ R. Then their product is rx = (rλ1)r1 + ··· + (rλn)rn, which is in S since each coefficient is still in R.
- Closure under right multiplication: This follows from the distributive property of multiplication over addition.
Therefore, S is closed under multiplication by elements of R.
Since S satisfies both defining properties of an ideal in R, we can conclude that S is indeed an ideal in R.
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Jaylan makes limeade using cup water per
cup lime juice. Wanchen makes limeade
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using cup water per cup lime juice. Find the unit rates of water (cups) per lime
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3
juice (cups). Whose limeade has a weaker lime flavor? Enter your answer as a proper
fraction or a mixed number.
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Part 1 out of 2
Jaylan's limeade contains
Wenchan's limeade contains
cups water per cup of lime juice.
cups water per cup of lime juice.
Answer:
limeade has a weaker lime flavor? Enter your answer as a proper
fraction or a mixed number.