Show that the following sequences of functions converge uniformly to 0 on the given ser sin nx nx (a) on [0, 00 ) where a > 0. (b) {xe n} on [0, 0). их х ln(1 + nx) In1 (c) on (0,1). (d) 1 + nx *} on [0, M] n
Answers
(a) Converges uniformly to 0 on [0, ∞).
(b) Converges uniformly to 0 on [0, 0).
(c) Converges uniformly to 0 on (0, 1).
(d) Does not converge uniformly to 0 on [0, M].
To show that the sequences of functions converge uniformly to 0 on the given intervals, we need to show that for any ε > 0, there exists an N such that |f_n(x) - 0| < ε for all x in the given interval and for all n ≥ N.
(a) For the sequence {sin(nx)/nx} on [0, ∞) where a > 0:
We know that |sin(nx)/nx| ≤ 1/n for all x in [0, ∞).
Given ε > 0, we can choose N such that 1/N < ε.
Then, for all x in [0, ∞) and for all n ≥ N, we have |sin(nx)/nx| ≤ 1/n < ε.
Thus, the sequence {sin(nx)/nx} converges uniformly to 0 on [0, ∞).
(b) For the sequence {xe^n} on [0, 0):
We know that xe^n → 0 as x → 0.
Given ε > 0, we can choose N such that e^(-N) < ε.
Then, for all x in [0, 0) and for all n ≥ N, we have |xe^n - 0| = xe^n ≤ e^(-N) < ε.
Thus, the sequence {xe^n} converges uniformly to 0 on [0, 0).
(c) For the sequence {xln(1 + nx)} on (0, 1):
We know that xln(1 + nx) → 0 as x → 0.
Given ε > 0, we can choose N such that 1/N < ε.
Then, for all x in (0, 1) and for all n ≥ N, we have |xln(1 + nx) - 0| = xln(1 + nx) ≤ x ≤ 1 < ε.
Thus, the sequence {xln(1 + nx)} converges uniformly to 0 on (0, 1).
(d) For the sequence {1 + nx*} on [0, M]:
We know that 1 + nx* → 0 as x* → -∞ and as x* → ∞, but it does not converge uniformly to 0 on [0, M] for any finite M.
Thus, the sequence {1 + nx*} does not converge uniformly to 0 on [0, M].
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Related Questions
The inside diameter of a randomly selected piston ring is a random variable with mean value 10 cm and standard deviation 0.07 cm.
(a) If
X
is the sample mean diameter for a random sample of n = 16 rings, where is the sampling distribution of
X
centered and what is the standard deviation of the
X
distribution? (Enter your standard deviation to five decimal places.)
center cm
standard deviation cm
(b) Answer the questions posed in part (a) for a sample size of n = 64 rings. (Enter your standard deviation to five decimal places.)
center cm
standard deviation cm
(c) For which of the two random samples, the one of part (a) or the one of part (b), is
X
more likely to be within 0.01 cm of 10 cm? Explain your reasoning.
X
is more likely to be within 0.01 cm of 10 cm in sample (a) because of the increased variability with a smaller sample size.
X
is more likely to be within 0.01 cm of 10 cm in sample (b) because of the increased variability with a larger sample size.
X
is more likely to be within 0.01 cm of 10 cm in sample (b) because of the decreased variability with a larger sample size.
X
is more likely to be within 0.01 cm of 10 cm in sample (a) because of the decreased variability with a smaller sample size.
Answers
We are given that the inside diameter of a piston ring is a random variable with mean value 10 cm and standard deviation 0.07 cm.
We are asked to find the sampling distribution of the sample mean diameter for two different random samples of n=16 and n=64 piston rings, respectively, and to calculate the standard deviation of each sampling distribution.
The sampling distribution of the sample mean diameter is the distribution of all possible sample means that could be obtained from a given sample size n.
The central limit theorem tells us that for large enough sample sizes (typically n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the shape of the underlying population distribution.
The mean of the sampling distribution of the sample mean is equal to the population mean, and the standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size.
For the first random sample of n=16 piston rings, the center of the sampling distribution of the sample mean diameter is still 10 cm, as this is the population mean.
However, the standard deviation of the sampling distribution is equal to the population standard deviation of 0.07 cm divided by the square root of 16, which is 0.0175 cm. Therefore, the standard deviation of the sampling distribution of the sample mean diameter for the first random sample is 0.0175 cm.
For the second random sample of n=64 piston rings, the center of the sampling distribution of the sample mean diameter is still 10 cm, but the standard deviation of the sampling distribution is equal to the population standard deviation of 0.07 cm divided by the square root of 64, which is 0.00875 cm.
Therefore, the standard deviation of the sampling distribution of the sample mean diameter for the second random sample is 0.00875 cm.
Finally, we are asked which of the two random samples is more likely to have a sample mean diameter within 0.01 cm of 10 cm. Since the standard deviation of the sampling distribution of the sample mean decreases as the sample size increases, the second random sample of n=64 piston rings is more likely to have a sample mean diameter within 0.01 cm of 10 cm.
This is because the decreased variability of the sampling distribution means that the sample means are more tightly clustered around the population mean of 10 cm. Therefore, we can conclude that the second random sample is more precise than the first random sample.
In statistics, a random sample is a subset of a larger population that is selected in such a way that each member of the population has an equal chance of being included in the sample.
Random sampling is a crucial tool for drawing conclusions about a population based on a smaller subset of data. In this problem, we will explore the sampling distribution of the sample mean for two different random samples of piston rings.
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Juan worked 2.5 hours and earned $37.00. Lisa was paid $42.00 for 3 hours of work. Which of the following statements is true?
Juan made a lower rate of pay because he made less money.
Lisa made a better rate of pay because she made more money.
Lisa said she made a better rate of pay because her rate was $14 an hour.
Juan said he made a better rate of pay because he was paid $14.80 an hour.
Answers
Answer:
Juan said he made a better rate of pay because he was paid $14.80 an hour.
Step-by-step explanation:
Juan made more because he makes $14.80 an hour, whereas Lisa makes only $14 an hour.
Enjoy your day! :D
Bilquis owns a small business selling used books. She knows that in the last week 49 customers paid cash, 42 customers used a debit card, and 32 customers used a credit card.
Based on these results, express the probability that the next customer will pay with a credit card as a decimal to the nearest hundredth.
Answers
Answer:
The probability that the next customer will pay with a credit card will be 0.26 or 26%.
What is probability?
Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.
Bilquis owns a small business selling used books. She knows that in the last week 49 customers paid cash, 42 customers used a debit card, and 32 customers used a credit card.
Then the probability that the next customer will pay with a credit card will be
P
=
32
123
�
=
0.26
P=
123
32
P=0.26
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Step-by-step explanation:
I hope this helps :-)
Hand me over the answers of B and C or perish
Answers
C.the y intercept represents how much money it costs per 1 ticket
Hope this helps
Have a great day/night
what is (-1/2) - (-2/5) ?
help please .
Answers
Answer:
-1/10
Step-by-step explanation:
Answer:
-1/10
Step-by-step explanation:
First you need a common demonator of 10
so
-1/2 --> -5/10
and -2/5 --> -4/10
using this the equation is:
-5/10 - (-4/10) which equals
-5/10 + 4/10= -1/10
rewrite 4 x 4 x 4 x 4 As a exponent
rewrite A x A x A as an exponent
Answers
Answer:
Step-by-step explanation:
\(4^4\\A^3\)
Answer:
\(4^{4}\) and \(A^{3}\)
Step-by-step explanation:
The equation Y-3 = -2(x+ 5) is writen in point-slope form. What is the y-intercept of the ine?
O 13
O-7
O2
8
Answers
Answer:
-7
Step-by-step explanation:
the equation can be simplified to y = -2x - 7
the last number of the equation represents the y-intersept
hope this helped!
What is the answer????????????
Answers
Answer:
43.8
Step-by-step explanation:
Remark
Are there ever a lot of assumptions!!
The rectangles are all the same sizethe triangles are the same sizeThe rectangles are indeed rectangles.The triangles are isosceles.The two equal sides are both equal to the width of the rectangles.Area of the rectangles
A = L * W
L = 4 m
W = 3 m
Area = 4*3 = 12
There are 3 rectangles so the total area = 3*12 = 36
Area triangles
Area = 1/2 b * h
b = 3
h = 2.6
Area = 1/2 * 2.6 * 3
Area = 1/2 7.8
But there are 2 of them so the area for the 2 is
Area = 2* 1/2 7.8
Area = 7.8
Total area = 36 + 7.8
Total area = 43.8
Consider the economy whose data appear in the table below. Working-age population 100,000 Labor force 60,000 Unemployed 12,000 Instructions: Round your answers to one decimal place.
a. The unemployment rate is ___%.
b. The labor-force participation rate is ___ %.
Answers
a. The unemployment rate is 20%.
b. The labor-force participation rate is 60%.
According to the question,
Working age population- 100,000
Labour force - 60,000
Unemployed - 12000 instructions
a) The unemployment rate is the percentage of the labor force that is unemployed and it is calculated as follows:
Unemployment Rate = \(\frac{Number of unemployed People}{Labor force}\) × 100
When an unemployed population is 12,000 and the labor force is 60,000,
the unemployment rate is = \(\frac{12000}{60000}\) × 100 = 20%
b) labor force participation rate is the percentage of working adult population that participates in labor either by actively looking for a job, or working. It is calculated as follows:
Labor Force Participation Rate = \(\frac{Labor force}{working age population}\) × 100
When the labor force is 60,000 and the working-age population is 100,000
Labor Force Participation Rate = \(\frac{60000}{100000}\) × 100 = 60%
So, a. The unemployment rate is 20%.
b. The labor-force participation rate is 60 %.
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Dear Garden Designer,
Shed
I've ordered this shed. It is 2 meters wide, 3.25 meters
long and 2.8 meterstal
I have moved into a house with a smal garden that needs a total redesign
Please design my garden for me. I have attached an accurate scale drawing of my
garden to this emai. I ve listed below some features I want in the garden. I will email
you later about some other things I also want.
To start, please could you draw these features accurately on the plan, showing where
you think they should go in the garden. Send me your plan with an explanation of your
thinking
Best wishes,
Decking for barbeques
I want some decking near the patio doors
It should be big enough to seat at least six people
Mandy
Circular pond
I would like a circular pond.
I'd like its area to be about 7 m.
Path and Borders
I would like some flower borders. These should not be
more than one motor wide as I find wider ones difficult
to look after.
I'd like a gravel path 1 meter wide to go from the shed
to the house and from the garden gate to the house
I will cover the rest with grass
Answers
Answer:
Step-by-step explanation:MATHEMATICAL GOALS
This lesson unit is intended to help assess how well students are able to interpret and use scale
drawings to plan a garden layout. This involves using proportional reasoning and metric units.
COMMON CORE STATE STANDARDS
This lesson relates to the following Standards for Mathematical Practices in the Common Core State
Standards for Mathematics, with a particular emphasis on Practices 1, 3, 4, 5, and 6:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
8. Look for and express regularity in repeated reasoning.
This lesson gives students the opportunity to apply their knowledge of the following Standards for
Mathematical Content in the Common Core State Standards for Mathematics:
7.G: Draw, construct, and describe geometrical figures and describe the relationships between
them.
Solve real-life and mathematical problems involving angle measure, area, surface area,
and volume.
7.EE: Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
7.RP: Analyze proportional relationships and use them to solve real-world and mathematical
problems.
INTRODUCTION
This lesson unit is structured in the following way:
• Before the lesson, students work individually on a task designed to reveal their current levels of
understanding. You review their responses and create questions to help them improve their work.
• At the start of the lesson, students reflect on their individual responses, before producing a
collaborative improved solution to the task. Then, in the same small groups students analyze
sample responses. They then discuss as a whole-class the methods they have seen and used.
• In a follow-up lesson, students reflect on their work. If time allows, an extension task is available.
MATERIALS REQUIRED
• Each student will need a copy of Design a Garden and Garden Plan, some blank paper, a miniwhiteboard, pen, and eraser, and the How Did You Work? questionnaire. The Garden Plan should
be copied at exactly 100% scale so the measurements are accurate. If this is not possible,
photocopy the rules on S-3, one rule per student, which should then match the Garden Plan
measurements. It will be useful to have spare copies of the Garden Plan.
• Each small group of students will need a new copy of the Garden Plan, the Assistants’ Methods, a
glue stick, felt-tipped pen, and a sheet of poster paper. For the optional extension Mandy’s Second
Email will be needed. Provide short rules, meter rules, string, protractors, scissors, glue, card,
plain paper, graph paper, and colored pencils for students who choose to use them.
TIME NEEDED
20 minutes before the lesson, a 110-minute lesson (or two 60-minute lessons), and 10 minutes in a
follow-up lesson. Actual timings will depend on the needs of your students.
Given the two equations f(x) = 4 * (3) ^ x and g(x) = 3x + 5 5. What is the sum between the equations y-intercepts .
Answers
Find the value of the power.
93 =
Answers
Hey there!
Assuming you meant:
9^3
IF SO, LETS SOLVE FOR IT!
9^3
= 9 * 9 * 9
= 81 * 9
= 729
Therefore, your answer is most likely:
[729] So, the base number is probably 5 and the exponent is probably 3
Random note:
Your base number is always that number that will be repeated by what the exponent is telling it!
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
T/F. he triple exponential smoothing method uses seasonality variations in the analysis of the data.
Answers
False. The triple exponential smoothing method does consider seasonality variations in the analysis of the data, along with trend and level components, to provide accurate forecasts.
The statement is false. Triple exponential smoothing, also known as Holt-Winters method, is a time series forecasting method that incorporates trend and seasonality variations in the analysis of the data, but it does not specifically use seasonality variations.
Triple exponential smoothing extends simple exponential smoothing and double exponential smoothing by introducing an additional component for seasonality. It is commonly used to forecast data that exhibits trend and seasonality patterns. The method takes into account the level, trend, and seasonality of the time series to make predictions.
The triple exponential smoothing method utilizes three smoothing equations to update the level, trend, and seasonality components of the time series. The level component represents the overall average value of the series, the trend component captures the systematic increase or decrease over time, and the seasonality component accounts for the repetitive patterns observed within each season.
By incorporating these three components, triple exponential smoothing can capture both the trend and seasonality variations in the data, making it suitable for forecasting time series that exhibit both long-term trends and repetitive seasonal patterns.
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What error did Tracy make? She used AG = 2EG, but the correct statement is 2AG = GE. She found that x = 4, but the correct value for x is 2. She used GF = 2BG, but the correct statement is BG = 2GF. She found that GF = 64, but the correct value of GF is 8.
Answers
Answer:
C. She used GF = 2BG, but the correct statement is BG = 2GF.
Step-by-step explanation:
credits to the other person!
Answer:
C.
Step-by-step explanation:
Edge 2021
in the equation of exchange, if m = $1.5 trillion, v = 7, and p = 1.05, then:
Answers
Thus, if M = $1.5 trillion, V = 7, and P = 1.05, the real output of the economy (Q) is $10 trillion.
The equation of exchange is an economic equation that relates the money supply (M) with the price level (P), the velocity of money (V), and the real output of the economy (Q). The equation is M x V = P x Q.
Using the given values of M = $1.5 trillion, V = 7, and P = 1.05, we can solve for Q by rearranging the equation as Q = M x V / P.
Plugging in the numbers, we get:
Q = ($1.5 trillion x 7) / 1.05 = $10 trillion
Therefore, if M = $1.5 trillion, V = 7, and P = 1.05, the real output of the economy (Q) is $10 trillion.
It's worth noting that the equation of exchange is a theoretical model and may not reflect actual economic conditions. In practice, changes in any one of the variables (M, V, P, or Q) can affect the others, and there may be other factors at play that influence the economy.
Nonetheless, the equation of exchange provides a useful framework for thinking about the relationship between the money supply, prices, and economic activity.
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Complete question
in the equation of exchange, if m = $1.5 trillion, v = 7, and p = 1.05, then:
Find real output.
how is 2/3 equal 66.6? show your work!
Answers
Answer:
Step-by-step explanation:
Dividing 3 into 2 results in the infinite (repeating) decimal fraction 0.6666....
0 . 6
3 / 20
18
------------
20
and so on: this pattern repeats itself indefinitely
y = 2/3 x-1
y = -1/3 x + 2 System of equation
Answers
Answer:
See Below
Step-by-step explanation:
y= 2/3x - 1
y= -1/3x +2
Use substitution
2/3x -1 = -1/3 +2
+1/3 +1/3
x-1 = 2
+1 +1
x = 3
Plug that in
y = 2/3(3/1) -1
3 cancel out leaving it with
y = 2-1
y = 1
(3,1)
A doctor finds that patients inhales once every 3 seconds. The same patient's heart beats 82 times per minute while at rest. How many times does this patient inhale in a 24-hour day?
Answers
Using arithmetic operations it is obtained that the patient inhales approximately 29,520 times in a 24-hour day.
What is arithmetic operation?
A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.
We can start by converting the patient's breathing rate from seconds to minutes, since we have the heart rate in minutes.
Use the arithmetic operation of multiplication and division -
The patient inhales once every 3 seconds, which is equivalent to -
(1 inhale / 3 seconds) x (60 seconds / 1 minute) = 20 inhales per minute
Next, we can use the patient's heart rate to find the number of breaths per heartbeat, and then multiply by the heart rate to get the total number of breaths per minute -
Breaths per heartbeat = 1 inhale / 4 heartbeats (since each heartbeat corresponds to 4 phases of the cardiac cycle, during which the patient inhales and exhales once)
Breaths per minute = (Breaths per heartbeat) x (Heart rate)
= (1/4) x 82
= 20.5
So the patient takes approximately 20.5 breaths per minute.
Multiplying this by the number of minutes in a day (24 hours x 60 minutes per hour = 1440 minutes) gives -
Breaths per day = (Breaths per minute) x (Minutes per day)
= 20.5 x 1440
= 29,520
Therefore, the number of inhales is 29,520.
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59 is 60% of what number
Answers
Answer:
duh. 98.33 yeah yeah yeah yeah yeah yeah
Let f(x)=√x with f:R→R. Discuss the properties of f. Is it injective, surjective, bijective, is it a function? Why or why not? Under what conditions change this?Explain using examples.I'm having some trouble figuring out this equation.
Answers
The function f(x)=√x with f:R→R is injective and surjective but not bijective.
Injective (or one-to-one): f(x) is injective, which means that if f(x1) = f(x2), then x1 = x2. In other words, if the square root of x1 is equal to the square root of x2, then x1 and x2 are equal.
Surjective (or onto): f(x) is surjective, which means that for every y in the codomain (R), there exists an x in the domain (R) such that f(x) = y. In other words, the square root of any real number can always be expressed as a real number.
Bijective: f(x) is not bijective. The square root function is not bijective because it is not defined for negative values of x.
Function: A function f is a function if for every x in the domain of f, there is exactly one y in the codomain of f such that f(x) = y. In other words, each x in the domain is mapped to exactly one y in the codomain.
In the case of f(x) = √x, it is a function. For every x in the domain of f, which is [0,∞), there is exactly one y in the codomain of f, which is [0,∞), such that f(x) = y.
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7.
If 3 tickets to a certain show cost $13.20, what would 7 tickets cost?
What is this answer?
Answers
Answer:
$ 30.8Step-by-step explanation:
ratio and proportion:
3 tickets = 7 tickets
$13.20 x
then cross multiply: 3 (x) = 13.20 (7)
simplify: 3x = 92.4
solve for x = 92.4 / 3
x = $ 30.8
therefore,
7 tickets costs $ 30.8
imagine that we rolled a fair, six-sided die 1,000 times. out of 1,000 rolls, how many times do you think the die would come up even (2, 4, or 6)?
Answers
Probability of Number of outcome is an even in likely situation is given as 512
Given that;
Number of time die roll = 1,000
Find:
Probability of Number of outcome is an even
Computation:
Probability of an even = 3 / 6
Probability of an even = 1 / 2
Probability of Number of outcome is an even = [Probability of an even]1000
Probability of Number of outcome is an even = [1/2]1000
Probability of Number of outcome is an even = 500
So most likely outcome is 512
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Help ASAP I need answers now !!!!pls
Answers
Answer:
????? what
Step-by-step explanation:
Answer:
what is the question hlp me first plzzzzzzzzzzzzzzz
Step-by-step explanation:
Simplify the expression by combining like terms.
3.6x+5.9−2.2−1.7x
Enter your answer as an expression, like this: 42x+53
Answers
1.9x+3.7
Explanation:
3.6x + (-1.7x)= 1.9x
5.9 - 2.2= 3.7
Assume the random variable x is normally distributed with mean μ 82 and standard deviation σ= 5, Find the indicated probability P(x< 80) Plxe 80)= [ (Round to four decimal places as needed.)
Answers
The probability P(x < 80) for a normally distributed random variable x with a mean μ = 82 and a standard deviation σ = 5 is approximately 0.3446 when rounded to four decimal places.
Standard deviation is: It is a measure of how spread out numbers are. It is the square root of the Variance, and the Variance is the average of the squared differences from the Mean.
To find the probability P(x < 80) for a normally distributed random variable x with a mean μ = 82 and a standard deviation σ = 5,
To find this probability, follow these steps:
1. Calculate the z-score for x = 80. The z-score is given by the formula: z = (x - μ) / σ
2. Look up the z-score in a standard normal distribution table (also known as a z-table) to find the corresponding probability.
Step 1: Calculate the z-score
z = (80 - 82) / 5 = -2 / 5 = -0.4
Step 2: Look up the z-score in the z-table
Looking up a z-score of -0.4 in a z-table, we find a corresponding probability of approximately 0.3446.
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The probability P(x < 80) for a normally distributed random variable x with a mean μ = 82 and a standard deviation σ = 5 is approximately 0.3446 when rounded to four decimal places.
Standard deviation is: It is a measure of how spread out numbers are. It is the square root of the Variance, and the Variance is the average of the squared differences from the Mean.
To find the probability P(x < 80) for a normally distributed random variable x with a mean μ = 82 and a standard deviation σ = 5,
To find this probability, follow these steps:
1. Calculate the z-score for x = 80. The z-score is given by the formula: z = (x - μ) / σ
2. Look up the z-score in a standard normal distribution table (also known as a z-table) to find the corresponding probability.
Step 1: Calculate the z-score
z = (80 - 82) / 5 = -2 / 5 = -0.4
Step 2: Look up the z-score in the z-table
Looking up a z-score of -0.4 in a z-table, we find a corresponding probability of approximately 0.3446.
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*ECONOMICS* please help me :(((
Answers
Answer:
B.
Step-by-step explanation:
Monopolistic Competition is a type of imperfect competition such that many producers sell products that are differentiated from one another and hence are not perfect substitutes. Therefore B. would be correct since Monopolistic Competition is Many sellers with a featured product that's differentiated to the buyer.
Which theorem would be used to write 15x + 12 + 117 = 180 in the diagram below?
a. Alternate Interior Angles Theorem
b. Corresponding Angles Theorem
c. Consecutive Exterior Angles Theorem
d. Vertical Angles Theorem
Answers
Answer:
C. Consecutive Exterior Angles Theorem states that if the transversal passes through the two parallel lines then any two exterior angles are congruent
Answer:
B
Step-by-step explanation:
The numbers 8,x,9 and 36 are in proportion.Find x
Answers
If you can give the proportion then it would be better.
I am asuming
8:x::9:36
Then 8:x::1:4
So x = 32
Answered by Gauthmath must click thanks and mark brainliest
Answer:
x = 32
Step-by-step explanation:
\(\frac{8}{x}\) = \(\frac{9}{36}\) = \(\frac{1}{4}\) ( cross- multiply )
x = 8 × 4 = 32
for what real values of $c$ is $4x^2 14x c$ the square of a binomial?
Answers
The real values of $c$ for which $4x² + 14xc + c$ is the square of a binomial are $c = 0$ and $c = \frac{4}{49}$.
To determine the real values of $c$ for which the expression $4x² + 14xc + c$ is the square of a binomial, we need to find the conditions under which it can be factored as $(ax + b)²$, where $a$ and $b$ are constants.
Expanding $(ax + b)²$, we have $(ax + b)² = a²x² + 2abx + b²$. Comparing this with $4x² + 14xc + c$, we can set up the following equations:
$a² = 4$ (equating the coefficients of $x²$)
$2ab = 14c$ (equating the coefficients of $x$)
$b² = c$ (equating the constant terms)
From the first equation, we find that $a = \pm 2$. Substituting this value into the second equation, we have:
$2(\pm 2)b = 14c$
$\Rightarrow \pm 4b = 14c$
$\Rightarrow b = \pm \frac{14c}{4}$
$\Rightarrow b = \pm \frac{7c}{2}$
Finally, substituting the values of $a$ and $b$ into the third equation, we get:
$\left(\pm \frac{7c}{2}\right)² = c$
$\Rightarrow \frac{49c²}{4} = c$
$\Rightarrow 49c² = 4c$
$\Rightarrow 49c² - 4c = 0$
$\Rightarrow c(49c - 4) = 0$
Now, we can solve this equation to find the values of $c$:
$c = 0$: This is one possible solution.
$49c - 4 = 0 \Rightarrow 49c = 4 \Rightarrow c = \frac{4}{49}$: This is another possible solution.
Therefore, the real values of $c$ for which $4x² + 14xc + c$ is the square of a binomial are $c = 0$ and $c = \frac{4}{49}$.
Learn more about binomial here:
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Carland Tim start a yard cleaning business, During the first week they
purchased 2 shovels and 2 brooms for $45. They also purchased garbage bags
for $13. Carl cleaned the Kingston's Front and Backyard and Tim cleaned the
Murphys Driveway
Answers
Answer:
They spend = $58
They earned = $35
Step-by-step explanation: