Compute each matrix sum or product if it is defined. If an expression is​ undefined, explain why. Let A= 3 0 −2 4 −5 2 ​, B= 8 −4 1 2 −3 −4 ​, C= 2 3 −3 2 ​, and D= 2 5 −2 5 .
−2​A, B−2​A, ​AC, CD

Answers

Answer 1

The solution is,

-2A = (-6, -8, 0, 8, 2, -8)

B - 2A = (2, -7, -4, 3, 4, -12)

AC is undefined.

CD = (3, 2, 12, 5)

What is matrix?

In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.

here, we have,

Given the matrices:

A = (3 4 0 -4 -1 4)

B = (8 1 -4 -5 2 -4)

C = (1 -1 3 1)

D = (3 -2 4 5)

We are required to compute the following

-2A, B - 2A, AC, CD

For -2A:

-2(3 4 0 -4 -1 4)

= (-6, -8, 0, 8, 2, -8)

For B - 2A:

Because B - 2A = B + (-2A),

we have:

(8 1 -4 -5 2 -4) + (-6, -8, 0, 8, 2, -8)

(2, -7, -4, 3, 4, -12)

For AC:

(3 4 0 -4 -1 4)(1 -1 3 1)

This is undefined.

For CD:

(1 -1 3 1)(3 -2 4 5)

= (3, 2, 12, 5)

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Related Questions

The table shows various values of a linear function f (x).


x –4 0 2 5 9 10
f (x) –11 1 7 16 28 31

What is f –1(10)?
31
28
3
9

Answers

The numeric value of the inverse function of f(x) at x = 10 is given as follows:

f –1(10) = 3.

How to define the linear function?

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.

From the table, the parameters are given as follows:

m = 3, as when x increases by 2, y increaes by 6.b = 1, as when x = 0, y = 1.

Hence the function is defined as follows:

y = 3x + 1.

To obtain the inverse function, we exchange the variables x and y and then isolate y, thus:

x = 3y + 1

3x = x - 1

y = (x - 1)/3

Hence the numeric value of the inverse function at x = 10 is given as follows:

y = (10 - 1)/3

y = 3.

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Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of squares of its sides.

Answers

We have proven that the sum of the squares of the diagonals of a parallelogram is equal to the sum of squares of its sides.

Parallelograms are quadrilaterals with opposite sides parallel to each other. They have many interesting properties, one of which is the relationship between the sum of the squares of the diagonals and the sum of squares of their sides. In this explanation, we will prove this relationship mathematically.

Let us consider a parallelogram with diagonals AC and BD, and sides AB, BC, CD, and DA.

To begin, let us draw line segments from the midpoints of each side to the midpoint of the opposite side. This divides the parallelogram into four congruent triangles, as shown below:

We can now use the Pythagorean theorem to find the length of each line segment. Let us denote the midpoint of AB as M, the midpoint of BC as N, and the midpoint of CD as P. Then we have:

AM² = AB²/4 + BM²

BN² = BC²/4 + CN²

CP² = CD²/4 + DP²

DM² = DA²/4 + AM²

Note that the line segments AM, BN, CP, and DM are half the length of the diagonals AC and BD. We can now add all these equations together:

=> AM² + BN² + CP² + DM² = (AB² + BC² + CD² + DA²)/4 + (AC² + BD²)/4

Rearranging this equation gives us:

=> AC² + BD² = 2(AM² + BN² + CP² + DM²) - (AB² + BC² + CD² + DA²)

Now, recall that the four triangles we divided the parallelogram into are congruent. Therefore, AM = BN = CP = DM, and AB = CD, BC = DA. Substituting these equalities into the above equation, we get:

=> AC² + BD² = 4(AM² + BN²) - 4(AB² + BC²)

We can further simplify this expression using the fact that the diagonals of a parallelogram bisect each other. Therefore, AM = CP and BN = DM. Substituting these equalities gives us:

AC² + BD² = 4(AM² + BN²) - 4(AB² + BC²)

= 4(AM² + BN² - AB² - BC²)

= 4(AN² + BM²)

Recall that AN and BM are half the length of the sides of the parallelogram. Therefore, we can write:

=> AC² + BD² = 4(AN² + BM²) = 4(AB² + BC² + CD² + DA²)

This completes the proof. We have shown that the sum of the squares of the diagonals of a parallelogram is equal to the sum of squares of its sides.

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Solve for x.
a) 7
b) -11
c) -6
d) -10

Answers

Answer:

b) - 11

Step-by-step explanation:

The two triangles ΔJKL and ΔGHK are similar since
LK = 2LH
JK = 2JK

(ratios of sides are same)

and the included angle ∠K is common to both triangles

So the sides are in the ratio 2:1 for larger to smaller triangle. Each side of the larger triangle is twice the length of the corresponding side of the smaller triangle

This means the ratio of side JL of the larger triangle ΔJKL must be twice the length of side GH of the smaller triangle ΔGHK

i.e.
JL = 2 x GH

We are given these sides as expressions in x
∴ x + 27 = 2(2x + 30)

x + 27 = 4x + 60

Subtract 4x from both sides:
x - 4x + 27 = 4x - 4x + 60

-3x + 27 = 60

Subtract 27 from both sides-3x = 60 - 27 = 33

==> 1

- 3x = 33

Divide by -3 both sides

-3x/-3 = 33/-3 = -11

x = -11

Answer b)


Do not be confused by the fact that x is negative. Many students get confused because they associate x as a length. x is just a variable used in the equations

The actual lengths will be positive
GH = 8 and JL = 16

You can work it out yourself

0. Alicia listens to a podcast for hour. She reads for hour. Then she cleans her room 2 1 5 for hour. Which activity did she spend the most time doing? 6​

Answers

The solution is, in cleaning she spend the most time doing.

What is  multiplication?

In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.

here, we have,

Alicia listens to a podcast for hour.

She reads for hour.

Then she cleans her room 2 1/5 for hour.

we know that,

1 hour = 60 mint.

so, we get,

1) 60 mint

2) 60 mint

3) 11/5 * 60 = 132 mint.

Hence, The solution is, in cleaning she spend the most time doing.

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Recruiters have completely stopped using newspaper ads as a tool for external recruitment because job seekers use the Internet almost exclusively even in smaller cities and towns.
a. True
b. False

Answers

Recruiters have completely stopped using newspaper ads as a tool for external recruitment because job seekers use the Internet almost exclusively even in smaller cities and towns. - False

Although the majority of job seekers today utilise the Internet as their main source of job searching, businesses and recruiters still use newspaper ads as a technique for external hiring In order to reach a larger pool of candidates, many businesses still utilise a combination of traditional print marketing, social media, and online job boards.

Newspaper ads still have certain benefits, though, such as reaching a different group of job seekers who might not be as active online. Additionally, newspapers are a more efficient approach to contacting potential candidates in smaller towns and cities where there may not be as much internet job search activity. Therefore, it is still a tool that some businesses and recruiters utilise in their hiring processes, particularly when attempting to connect with a larger and more varied pool of prospects.

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is finding the limit of a function as n tends to infinity the best way to find small o notation of that function

Answers

Yes, This statement "f(n)/g(n) as n goes to infinity = 0" is true, in this little o notation means that f(n) grows much slower than g(n) as n becomes large.

"f(n)/g(n) as n goes to infinity = 0" means that as n approaches infinity, the ratio of f(n) to g(n) gets arbitrarily small. This is a formal way of saying that g(n) grows faster than f(n), or equivalently, that f(n) is "smaller" than g(n) in the long run. Specifically, the little-o notation represents a formal definition of this concept: f(n) is said to be "little-o" of g(n) (written as f(n) = o(g(n))) if and only if f(n)/g(n) approaches 0 as n approaches infinity. In other words, if for any positive constant ε, there exists a positive constant N such that for all n > N, |f(n)/g(n)| < ε.

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____The given question is incomplete, the complete question is given below:

I'm just trying to understand how in little o notation this is true:

f(n)/g(n) as n goes to infinity = 0?

3. Dimitri's car has a fuel efficiency of 21 miles per gallon. His tank is full with 12 gallons of gas. Does he have
enough gas to drive from Cincinnati to Toledo, a distance of 202.4 miles?
Show your calculations, including at least one use of dimensional analysis. You choose how to round.

Answers

Yes, Dimitri has enough gas to drive from Cincinnati to Toledo. To calculate this, we can multiply the fuel efficiency by the amount of gas in the tank.

What is multiplication?

The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the components that are multiplied are referred to as the factors.

Fuel efficiency: 21 miles per gallon

Gas in the tank: 12 gallons

So, the total distance that Dimitri can drive is:

= 21 miles per gallon x 12 gallons

= 252 miles

Since the distance from Cincinnati to Toledo is 202.4 miles, and the total distance that Dimitri can drive is 252 miles, we can conclude that he has enough gas to make the trip.

Dimensional analysis:

= 21 miles/gallon x 12 gallons

= 252 miles (correct dimensions)

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PLEASE HELP!!!

Which angles are supplementary to each other?

A) ∠5 and ∠3
B) ∠5 and ∠11
C) ∠4 and ∠2
D) ∠11 and ∠10

PLEASE LOOK AT PICTURE!!!

Answers

Answer:

Angles 11 and 10 both make a 180-degree angle, but so do 5 and 3. I would go with 11 and 10

Answer:

D, <11 and <10

Step-by-step explanation:

Supplementary angles are angles that sum to 180°

Write an inequality that represents the statement “x is at most –5 or at least 7.”

Answers

Answer:

x ≤ - 5 or x ≥ 7

or, in interval notation

[ - ∞, -5] ∪ [7, ∞]

Step-by-step explanation:

From the question we get the following two inequalities
x is at most -5 ==>     x ≤ -5

x is at least 7 ==>       x ≥ 7 which can be rewritten as 7 ≤ x

This can be written as
x ≤ - 5 or x ≥ 7

In interval notation this would be
[ - ∞, -5] ∪ [7, ∞]

On the number line this would be represented as shown in the figure

A coin is flipped at the start of every game to determine if Team A (heads) or Team B (tails) will get the ball first.

Part A: Find the theoretical probability of a fair coin landing on heads. (2 points)

Part B: Flip a coin 25 times and record the frequency of each outcome. (4 points)

Part C: Determine the experimental probability of landing on heads. (4 points)

Part D: Compare the theoretical and experimental probabilities. Explain your answer. (2 points)

Answers

Part A: The theoretical probability of a fair coin landing on heads is 1/2.

Part B: The frequency of getting Heads is 12 and the frequency of getting tails is 13.

Part C: The experimental probability of landing on heads is 0.48

Part D: The theoretical probability is higher than the experimental probability.

What is probability?

Simply put, the probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or likelihood of various outcomes. Statistics is the study of events that follow a probability distribution.

A coin has two faces. One is head and other is tails.

If flip a coin, the outcome is {H,T}

The number of total outcomes is 2.

The number of frequency-getting heads is 1.

The number of frequency-getting tails is 1.

The theoretical probability of a fair coin landing on heads is 1/2.

Now flip a coin 25 times

The outcomes are

H,T,T,T, H,H,H, T,T,T, T,H,T, H,T,H,T,T, T, H, T, H,H,H,H

The frequency of getting Heads is 12 and the frequency of getting tails is 13.

The experimental probability of landing on heads is 12/25 = 0.48.

The theoretical probability is not the same as the experimental probability.

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Answer:

Part A: The theoretical probability of a fair coin landing on heads is 1/2.

Part B: The frequency of getting Heads is 12 and the frequency of getting tails is 13.

Part C: The experimental probability of landing on heads is 0.48

Part D: The theoretical probability is higher than the experimental probability.

Step-by-step explanation:

Find the height represented by x.

Answers

Answer:

find the given numbers on the number line

Determine whether the following arguments are best interpreted as being inductive or deductive. Also state the criteria you use in reaching your decision (i.e., the presence of indicator words, the nature of the inferential link between premises and conclusion, or the character or form of argumentation).
Contrary to the common notion that women tend to be chatty compared to men, little difference exists between the sexes in terms of talkativeness. Over a five-year period researchers placed unobtrusive microphones on 396 college students in various fields, at campuses in Mexico as well as in the United States. They found that both men and women spoke about 16,000 words per day.

Answers

This argument is best interpreted as inductive. The presence of empirical evidence in the form of research data suggests that the conclusion is based on empirical observations and generalization.

The argument uses evidence to make a generalization about the talkativeness of men and women, which is the characteristic of inductive reasoning. Additionally, the lack of logical or deductive inferential links between premises and conclusion supports the inductive interpretation of the argument.on:

Inductive reasoning is a type of reasoning in which a conclusion is drawn based on observations or evidence, rather than from logical deduction. Inductive reasoning involves making generalizations from specific observations or examples

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‼️please help‼️
Find Freedas weekly take-home pay (net pay)

Answers

Freeda's weekly take home pay based on the 48 hours a week Freeda works with an hourly rate of $13.75 and an over time rate of a time and a half, obtained by finding the difference between the sum of earnings and the sum of deduction is $813.75

What is a salary rate?

A salary rate is the amount received regular for a specified period.

The number of hours worked a week = 48 hours

Number of hours considered overtime = Time over 40 hours

Amount earned for normal working hours = $13.75 per hour

Amount she earns as bonus per week = $125

The deductions per week are;

FICA

Federal income tax = $5.50

State income tax = $2.75

401K = $10.00

Insurance = $8.00

Weekly take home (net) pay = Total weekly earnings - Total weekly deductions

Number of overtime hours Freeda works each week = 48 - 40 = 8

Freeda works 8 hours overtime each week

Her weekly earnings = 40 × 13.75 + 8 × 1.5 × 13.75 + 125 = 840

Freeda's weekly earning = $840

Weekly deductions = 5.5 + 2.75 + 10.00 + 8 = 26.25

Weekly take home (net) pay = $840 - $26.25 = $813.75

Freeda's weekly take home pay is $813.75

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A families dishwasher uses the same number of gallons of water for each load the table shows how many gallons of water G the dishwasher uses for in loads of dishes what equation can you use to represent the relationship shown in the table show York

Answers

An equation which you can use to represent the relationship shown in the table is y = 4.5x.

What is the point-slope form?

Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:

y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)

Where:

m represents the slope.x and y are the points.

Based on the information provided in the table (see attachment), we can logically deduce the following data points on the line:

Points on x-axis = (4, 5).

Points on y-axis = (18.0, 22.5).

At point (4, 18.0), a linear equation of this line in slope-intercept form can be calculated by using the point-slope form as follows:

y - 18.0 = (22.5 - 18.0)/(5 - 4)(x - 4)

y - 18.0 = 4.5(x - 4)

y = 4.5x - 18 + 18.0

y = 4.5x.

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What is the height h for the base that is 5/4 units long?

Answers

The height h for the base that is 5/4 units long is 3/5 cm.

What is Triangle?

A triangle is a two dimensional figure which consist of three vertices, three edges and three angles.

Sum of the interior angles of a triangle is 180 degrees.

Given is a triangle and the three side lengths.

We have the area of a triangle is,

Area of the triangle = [tex]\frac{1}{2}[/tex] × base × height

We have base = 5/4 cm

Area of the triangle can be found using the Heron's formula.

Heron's formula states that,

Area of a triangle = [tex]\sqrt{s(s-a)(s-b)(s-c)}[/tex]

where s is the semi perimeter = (a + b + c) / 2 and a, b and c are the side lengths.

Given 5/4 cm, 1 cm and 3/4 cm are the side lengths.

s = (5/4 + 1 + 3/4) / 2 = 3/2

Area = [tex]\sqrt{\frac{3}{2}(\frac{3}{2} -\frac{5}{4} )(\frac{3}{2}-1)(\frac{3}{2}-\frac{3}{4} ) }[/tex]

        = [tex]\sqrt{\frac{9}{64} }[/tex]

        = [tex]\frac{3}{8}[/tex] cm²

Substituting in the formula A = [tex]\frac{1}{2}[/tex] × base × height,

[tex]\frac{3}{8}[/tex] = [tex]\frac{1}{2}[/tex] × [tex]\frac{5}{4}[/tex] h

h = [tex]\frac{3}{5}[/tex] cm

Hence the height is 3/5 cm.

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For this item, all answers should be entered as whole numbers.

Gustavo made a fruit smoothie. He put 11 ounces of bananas, 5 ounces of blueberries, 3 ounces of strawberries, and 4 ounces of oranges in the smoothie.

Use whole numbers to complete the following statements.

The ratio of ounces of bananas to ounces of strawberries is 11:
3
.

So, there are nearly
ounces of bananas for each ounce of strawberries.

Answers

The ratio of bananas to strawberries is 11:3.

What is ratio?

A ratio is a comparison between two amounts that is calculated by dividing one amount by the other. The quotient a/b is referred to as the ratio between a and b if a and b are two values of the same kind and with the same units, such that b is not equal to 0. Ratios are represented by the colon symbol (:). As a result, the ratio a/b has no units and is represented by the notation a: b.

Given:

Number of bananas = 11 ounces,

Number of strawberries = 3 ounces

The ratio of bananas to strawberries

= 11/3

= 11:3.

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The table compares the average daily temperature and ice cream sales each day.


Temperature (°F) Ice Cream Sales
56.9 $201
62.3 $212
66.2 $218
68.4 $219
73.3 $228
74.6 $230
75.6 $233
75.9 $236
80.4 $245
86.8 $256


What is the slope of the line of best fit, where x represents the average daily temperature and y represents the total ice cream sales? (Round your answer to one decimal place.)
1.8
2.3
3.1
4.3

Answers

Answer:

1.8

Step-by-step explanation:

Enter the data provided into any of the regression calculators available free online and it will plot and provide the regression equation

The equation provided by one such calculator is
y = 1.8204x +  96.6608

where y = sales in $ and x = temperature in °F

From the above it can be seen that the slope of the line is 1.8204 which, rounded to 1.8 to one decimal place

Osvoldo has a goal of getting at least 30% of his grams of carbohydrates each day from whole grains. Today, he ate 220 grams of carbohydrates, and 55 grams were from whole grains.

Answers

Answer: 25%

Step-by-step explanation:

The answer is 25%. This is calculated by dividing the number of grams of carbohydrates from whole grains (55 grams) by the total number of carbohydrates eaten (220 grams). This gives 0.25, which can be expressed as 25%. Therefore, Osvoldo consumed 25% of his carbohydrates from whole grains today.

Answer:

Osvoldo only got 25% of his grams of carbohydrates from whole grains today, which is below his goal of 30%

Step-by-step explanation:

To find the percentage of Osvoldo's carbohydrates that came from whole grains, we can use the formula:

percentage = (part/whole) x 100%

where "part" is the number of grams from whole grains, and "whole" is the total number of grams of carbohydrates. So we have:

percentage = (55/220) x 100%

percentage = 0.25 x 100%

percentage = 25%

Therefore, Osvoldo only got 25% of his grams of carbohydrates from whole grains today, which is below his goal of 30%.

Prove that if x is a non-empty set of real numbers which is bounded above, then there is a sequence of real numbers in x converging to sup(x).

Answers

For all sufficiently large n, which shows that (xn) converges to sup(x) as desired. Therefore, if x is a non-empty set of real numbers that is bounded above, then there is a sequence of real numbers in x converging to sup(x).

Let x be a non-empty set of real numbers that is bounded above. Then, by the least upper bound property of the real numbers, sup(x) exists and is a real number.

For each positive integer n,

let xn be an element of x such that sup(x) - 1/n < xn ≤ sup(x).

Such an element xn exists because sup(x) is the least upper bound of x, so there must be elements of x arbitrarily close to sup(x).

We claim that the sequence (xn) converges to sup(x).

To see this, let ε > 0 be arbitrary.

Since xn ≤ sup(x) for all n,

we have sup(x) - xn ≥ 0, and so sup(x) - xn < ε for all n such that 1/ε is an integer. Thus, we have

|sup(x) - xn| < ε

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HELP ASAP………………………..

Answers

The required Levi has 28 ounces of pretzels in total.

What is simplification?

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
To solve this problem, you can use the expression:

total ounces of pretzels = number of bags × ounces of pretzels per bag

Substituting the given values, we get:

total ounces of pretzels = 6 bags × 14/3 ounces per bag

total ounces of pretzels = 28 ounces

Therefore, Levi has 28 ounces of pretzels in total.

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Andy's two nephews want to travel to Philadelphia with him to see a show. Andy has two free plane tickets and three tickets to the show. Which of the following best describes the two plane tickets?
A scarce resource

Answers

The amount of two plane tickets are a scarce resource, meaning there are not enough to provide for all three people.

A scarce resource is something that is in limited supply. In this case, the two plane tickets are the scarce resource because there are not enough to provide for all three people. This means that if Andy wants to travel to Philadelphia with his two nephews, only two of them can go. This is because while they may have three tickets to the show, they still need to find a way to get there. The two plane tickets are a scarce resource, as they are not enough for all three of them. This means that Andy must make a difficult decision about who can go and who has to stay home. The two plane tickets are the only way for two of them to get to Philadelphia, so they are a scarce resource in this situation.

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Candidate A makes 48 speeches. Candidate B makes 16 speeches. a. Write the ratio of speeches by candidate A to candidate B b. Simplify the ratio so the second quantity is 1. Show your work.

Answers

Answer:

a. 48:16

b. 3:1

Step-by-step explanation:

Part a asks for a ratio of the two speeches. You simply separate the two numbers with a colon to show that it's a ratio.

Part b asks that the second quantity (which is the speeches made by B) be 1. To do this, divide 16 on both sides. 48 divided by 16 is 3, and 16 divided by 16 is 1. Thus, 3:1

PLEASE HELP!!!!!!

which equation can be used to solve for x?

A) 135x = 180
B) 9x + 126 = 90
C) 9y = 126
D) 9x + 126 = 180

PLEASE LOOK AT PICTURE!!!!

Answers

Answer:

D) 9x + 126 = 180

Step-by-step explanation:

We know

The (9x) angle combined with the 126 degrees angle must make 180 degrees. Looking at all the options, we see that D is the only reasonable answer.

Compute the magnitude and phase spectra of the following signals (i.e. compute the Fourier coefficients and determine the magnitude and phase of each one of them). a. a[n] = 4 sin(in/3) b. x[n] = cos(2n7/3) + sin(2n7/5) c. x[n] = cos(2n7/3) sin(2n/5) (a trig. identity that might be useful: cos(x) sin(y) = 1/2 (sin(x + y) – sin(x – y))

Answers

The magnitude spectrum of Fourier coefficients a[n] = 4 sin(nπ/3) is: |C[k]| = 0 so the phase spectrum is undefined. Fourier coefficients of x[n] = cos(2nπ/3) + sin(2nπ/5) for all other values of k, C[k] = 0, so the magnitude and phase spectra are zero. The Fourier coefficient for  cos(x) sin(y) = 1/2 (sin(x + y) – sin(x – y)) C[k] is zero when k is not equal to ±2/3 and ±2/5.

To compute the Fourier coefficients of a[n] = 4 sin(nπ/3), we use the formula:

C[k] = (1/N) * Σ[n=0 to N-1] a[n] e^(-j2πkn/N)

where N is the period of the signal (in this case, N = 6 since sin(nπ/3) has a period of 6), and k is the frequency index.

For k = 0, we have:

C[0] = (1/6) * Σ[n=0 to 5] 4 sin(nπ/3) = (4/6) * (sin(0) + sin(π/3) + sin(2π/3) + sin(π) + sin(4π/3) + sin(5π/3))

C[0] = (4/6) * (0 + √3/2 + √3/2 + 0 - √3/2 - √3/2) = 0

For k = ±1, we have:

C[1] = (1/6) * Σ[n=0 to 5] 4 sin(nπ/3) e^(-j2πn/6) = (4/6) * (sin(0) - sin(π/3) - sin(2π/3) + sin(π) + sin(4π/3) - sin(5π/3))

C[1] = (4/6) * (0 - √3/2 + √3/2 + 0 + √3/2 - (-√3/2)) = 0

C[-1] = (1/6) * Σ[n=0 to 5] 4 sin(nπ/3) e^(j2πn/6) = (4/6) * (sin(0) - sin(π/3) - sin(2π/3) + sin(π) + sin(4π/3) - sin(5π/3))

C[-1] = (4/6) * (0 - √3/2 + √3/2 + 0 + √3/2 - (-√3/2)) = 0

For all other values of k, we have C[k] = 0. Therefore, the Fourier series of a[n] is:

a[n] = 0

The magnitude spectrum is:

|C[k]| = 0

The phase spectrum is undefined.

To compute the Fourier coefficients of x[n] = cos(2nπ/3) + sin(2nπ/5), we use the formula:

C[k] = (1/N) * Σ[n=0 to N-1] x[n] e^(-j2πkn/N)

where N is the period of the signal (in this case, N = lcm(3, 5) = 15 since both cos(2nπ/3) and sin(2nπ/5) have periods of 3 and 5, respectively), and k is the frequency index.

For k = 0, we have:

C[0] = (1/15) * Σ[n=0 to 14] (cos(2nπ/3) + sin(2nπ/5)) = (1/15) * (5 + 0 - 5 + 0 + 5 + 0 - 5 + 0 + 5 + 0 - 5 + 0 + 5 + 0 - 5 + 0 + 5) = 5/3

To compute C[k] for k ≠ 0 and k ≠ 5, we can use the trigonometric identity:

cos(x) sin(y) = 1/2 (sin(x + y) – sin(x – y))

Let x = 2kπ/3 and y = 2nπ/5, then:

cos(2kπ/3) sin(2nπ/5) = 1/2 (sin(2kπ/3 + 2nπ/5) – sin(2kπ/3 – 2nπ/5))

= 1/2 (sin(10knπ/15 + 6kπ/15) – sin(10knπ/15 - 2kπ/15))

= 1/2 (sin((2k + 3n)π/3) – sin((2k - n)π/3))

The first term is zero when (2k + 3n) is an odd multiple of 3, and the second term is zero when (2k - n) is an odd multiple of 3. Therefore, C[k] = 0 when k + 3n is odd or k - n is odd.

For k = 3n, we have:

C[3n] = (1/15) * Σ[m=0 to 14] (cos(2mπ/3) sin(2nπ/5))

= (1/30) * Σ[m=0 to 14] (sin((2m + 3n)π/3) – sin((2m - n)π/3))

= (1/30) * (sin(5nπ/3) – sin(nπ/3) + sin(7nπ/3) – sin(5nπ/3) + sin(9nπ/3) – sin(7nπ/3) + sin(11nπ/3) – sin(9nπ/3) + sin(13nπ/3) – sin(11nπ/3) + sin(15nπ/3) – sin(13nπ/3) + sin(17nπ/3) – sin(15nπ/3) + sin(19nπ/3) – sin(17nπ/3))

= (1/30) * (sin(nπ/3) – sin(19nπ/3)) = 0

Therefore, the only non-zero coefficients are C[0] = 5/3 and C[5] = -5/3. The magnitude and phase spectra are:

|C[0]| = 5/3, arg(C[0]) = 0

|C[5]| = 5/3, arg(C[5]) = π

For all other values of k, C[k] = 0, so the magnitude and phase spectra are zero.

To compute the Fourier coefficients of cos(x) sin(y) = 1/2 (sin(x + y) – sin(x – y))

x[n] = 1/2 (sin(2nπ/3 + π/2) - sin(2nπ/3 - π/2)) * 1/2 (sin(2nπ/5) - sin(-2nπ/5))

Using the formula for the Fourier coefficients of a sinusoidal signal:

C[k] = (1/N) Σ[n=0 to N-1] x[n] e^(-j2πnk/N)

we can compute the Fourier coefficients for x[n]:

C[k] = (1/N) Σ[n=0 to N-1] x[n] e^(-j2πnk/N)

= (1/N) [Σ[n=0 to N-1] 1/2 sin(2nπ/3 + π/2) e^(-j2πnk/N) - Σ[n=0 to N-1] 1/2 sin(2nπ/3 - π/2) e^(-j2πnk/N)] [Σ[n=0 to N-1] 1/2 sin(2nπ/5) e^(-j2πnk/N) - Σ[n=0 to N-1] 1/2 sin(-2nπ/5) e^(-j2πnk/N)]

= 1/4 [C1(k-2/3) - C1(k+2/3)] [C1(k-2/5) - C1(k+2/5)]

where C1(k) is the Fourier coefficient of the signal cos(2nπ/3), which is given by:

C1(k) = (1/N) Σ[n=0 to N-1] cos(2nπ/3) e^(-j2πnk/N)

= (1/N) Σ[n=0 to N-1] 1/2 [e^(-j2πnk/3) + e^(j2πnk/3)]

= 1/2 [δ(k-1/3) + δ(k+1/3)]

Therefore, the Fourier coefficient C[k] is zero when k is not equal to ±2/3 and ±2/5.

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a 5-card hand is (randomly) drawn from a standard 52 card deck. (a) what is the probability that the hand contains all clubs? (b) what is the probability that the hand contains three aces and two kings? (c) what is the probability that the hand contains 4 cards of the same suit, and 1 of another suit?

Answers

(a) The probability of drawing the first club is 13/52 (since there are 13 clubs in the deck out of 52 total cards).

After that, there are 12 clubs left out of 51 cards, so the probability of drawing a second club is 12/51.

Similarly, the probability of drawing a third club is 11/50, the probability of drawing a fourth club is 10/49, and the probability of drawing a fifth club is 9/48. Therefore, the probability of drawing a 5-card hand with all clubs is: (13/52) x (12/51) x (11/50) x (10/49) x (9/48) = 0.000495% (rounded to six decimal places)

(b) The number of ways to choose 3 aces out of 4 is (4 choose 3) = 4, and the number of ways to choose 2 kings out of 4 is also (4 choose 2) = 6. The remaining card can be any of the 44 non-aces and non-kings in the deck. Therefore, the number of 5-card hands that contain 3 aces and 2 kings is: 4 x 6 x 44 = 1056

The total number of 5-card hands is (52 choose 5) = 2598960. Therefore, the probability of drawing a 5-card hand with 3 aces and 2 kings is: 1056/2598960 = 0.04074% (rounded to five decimal places)

(c) There are four suits in a standard deck, so there are four ways to choose the suit that will have 4 cards in the hand. The number of ways to choose 4 cards from a single suit is (13 choose 4) = 715. The remaining card can be any of the 39 cards in the other three suits. Therefore, the number of 5-card hands that contain 4 cards of the same suit and 1 card of another suit is: 4 x 715 x 39 = 111540

The total number of 5-card hands is (52 choose 5) = 2598960. Therefore, the probability of drawing a 5-card hand with 4 cards of the same suit and 1 card of another suit is:

111540/2598960 = 4.29% (rounded to two decimal places)

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Frederick is making hot chocolate from a mix. The graph models, the relationship between tablespoons of mixing cups of water

Answers

The constant of proportionality is 2. (True)

The equation n = 2m represents this relationship. (True)

Frederick needs 5 cups of water for 10 tablespoons of the mic. (False)

Point (2, 4) means 2 tablespoons of ms is needed for 4 cups of water. (False)

What is a slope?

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction

If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,

then the equation of a line is

y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)

To find the slope;

m =  (y₂- y₁) / (x₂ - x₁)

Given:

Frederick is making hot chocolate from a mix.

The graph models, the relationship between tablespoons of mixing cups of water.

Let m be the variable that represents the number of cups of water and n is the number of tablespoons in the mix.

From the graph;

The line passes through (0, 0) and (1, 2),

So, the equation of the line is,

n - 0 = (0 - 2)/(0 - 1) m

n = 2m

Therefore, the constant of proportionality is 2.

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If a tree grows from 4 inches to 5 feet 6 inches in ten years. On average, how much did the tree grew every year?

Answers

1 centimeter I think

The density of this a box of tissues is 1 g/cm3. The mass of the object is 728 grams.
The length is 20.8cm, the width is 7cm, what is the height?

Answers

The formula for the volume of a rectangular box is V = l × w × h, where l is the length, w is the width, and h is the height. Since the density is given as 1 g/cm³, the mass of the object is equal to its volume in cubic centimeters (cc):

mass = volume × density

728 g = V × 1 g/cm³

V = 728 cm³

We are given the length and width of the box, so we can substitute these values into the formula for volume:

V = l × w × h

728 cm³ = 20.8 cm × 7 cm × h

Simplifying the right-hand side:

728 cm³ = 145.6 cm² × h

Dividing both sides by 145.6 cm²:

h = 728 cm³ ÷ 145.6 cm²

h = 5 cm

Therefore, the height of the box is 5 cm.

explain why the reading on a scale would be less after leaving the top floor and heading downward.

Answers

The reading on a scale could be less after leaving the top floor and heading downward due to gravity.

All objects that have mass or energy are subject to the fundamental natural force of gravity. It is the force that draws two objects together and maintains the orbits of things like planets, galaxies etc. The force of gravity is proportionate to an object's mass and inversely linked to the square of the distance among both. As a result, the gravity around two objects will be more considerable and closer if they are together.

Several events, like maintaining people and other objects on the surface of the Earth, creating ocean tides, and causing objects to fall to the ground when dropped, are caused by gravity. Due to its importance in helping to comprehend the mobility and behavior of celestial objects, it is also a fundamental idea in the sciences of astrophysics. Because gravity has an influence on the body, the reading on a scale could be lower as you descend from the top floor.

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determine the measure of the largest angle in a triangle with sides of 12, 9, and 4​

Answers

Answer:

130.8

Step-by-step explanation:

Use law of cosine

12^2=9^2+4^2-2*9*4cos

144=97-72cos

47=-72cos

(Use cosine inverse now)

Cos-1(47/-72)

Largest angle is 130.8 when rounded to the nearest then the

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