show that if a and b are square, upper triangular matrices. then so is ab. (hint: the ith column of ab is a suitable linear combination of the columns of a)

A + B  =0  => {u + v, au} is linearly independent set of two vectors

=>  {u + v, au}is a basis for V.

Let u and v be distinct vectors of a vector space V and {u, v} is a basis for V and a and b are nonzero scalars.

We have to show: both {u + v, au} and {au, bv} are also bases for V.

[tex]u\neq v,a,b\neq 0[/tex]

{u, v}  is a basis for V => {u + v , au} and {au, bu} are bases for V

From basis {u , v} we get dimension of V

{u , v} is a basis for V

=>  V is a 2-dimensional vector space and (.) [tex]\alpha u+\beta v=0= > \alpha =\beta =0[/tex]

Show  {u + v, au}  is a basis for V:

{u + v, au} :

A(u + v) + B(au) = 0 <=> Au + Av + Bau = 0

<=> (A + Ba)u + (B)v = 0

By the step (in second step)

A + Ba = 0 , [tex]a\neq 0[/tex]

B = 0

=> A + B  =0  => {u + v, au} is linearly independent set of two vectors

=>  {u + v, au}is a basis for V.

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A compound event can be described for flipping a coin two times and getting at least 1 head.

Compound event is defined as the events in which there are more than one event happens together.

All the outcomes of the events sum up the probability to 1.

We have to describe a compound event where the probability is in between 50% and 80%.

That is, probability is in between 0.5 and 0.8.

That is, there is more than half of the chance to occur the event.

Suppose that we flip a coin two times.

Sample space = {HH, HT, TH, TT}

Find the probability of getting at least 1 head.

There are 3 outcomes out of 4 of getting at least 1 head.

Probability = 3/4 = 75%

Hence there is a 75% probability for getting at least 1 head when flipped a coin two times.

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The common volume of the two spheres is 5πr³/6. Since the two spheres are identical, the volume of the common portion is twice the volume of the spherical cap.

Since the distance between the centers of the two spheres is r/2, a cross-section of the two spheres and the plane perpendicular to their centers of symmetry will form an isosceles triangle with height h=r/2 and base 2√(r²-(r/2)²). The intersection of the two spheres will be a spherical cap with height h and radius r. The volume of this spherical cap is

V_cap = (1/3)π[tex]h^2[/tex](3r-h).

Since the two spheres are identical, the volume of the common portion is twice the volume of the spherical cap, so

Substituting h=r/2, we get

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Answer:

Quadrant II

Step-by-step explanation:

sin (∅) > 0 means sin (∅)is positive

sec (∅) < 0 means sec (∅) is negative

thus,

∅ lies in quadrant II

Answer:

11

Step-by-step explanation:

[tex]\sqrt{64}[/tex] -10 + 13

8 - 10 + 13

-2 + 13

11

Out of the following the statement  has a reference angle of 30° and is in quadrant one or two-  is true of the location of the terminal side of an angle 0 whose sine value is 1/2.

Since the reason behind this is that the sine value of 1/2  can be written as sin(30°), this means that the terminal side of the angle is located at 30°, which is either quadrant one or two. Quadrant one is the upper right quadrant of the coordinate plane, and quadrant two is the upper left quadrant.

The reference angle of an angle is the angle between the terminal side of the angle and the positive x-axis. In this case, the reference angle would be  30°. The reference angle of an angle is important because it is used to determine the quadrant in which the terminal side of the angle is used to determine the quadrant in which the terminal side of the angle is located. The reference angle must be less than 90° in order for the terminal side to be located in the first and second quadrants.

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We can be 95% confident that the true mean playing time of all Willie Nelson recordings is between 48.89 and 53.71 minutes.

Using a t-multiplier for a 95% confidence interval with 39 degrees of freedom (n-1), we get t* = 2.022.

The margin of error is t* times the standard error of the mean, which is s / sqrt(n), where n is the sample size.

Thus, the margin of error is 2.022 * (5.8 / sqrt(40)) = 2.41 minutes.

To construct the confidence interval, we add and subtract the margin of error to the sample mean:

Lower limit = 51.3 - 2.41 = 48.89

Upper limit = 51.3 + 2.41 = 53.71

Therefore, we can be 95% confident that the true mean playing time of all Willie Nelson recordings is between 48.89 and 53.71 minutes.

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The solution is, the value of x = 5+ 8√2, 5- 8√2

Simplify simply means to make it simple. In mathematics, simply or simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving.

here, we have,

given that,

(x - 5)² +15=143

we know, (a-b)^2 = a^2 -2ab + b^2

so, we get,

x^2 - 10x + 25+15 = 143

or, x^2 -10x -103 = 0

now, solving we get,

x = 5+ 8√2, 5- 8√2

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We cannot deduce anything about the p-value for age in the regression summary from just the 90% and 95%. cis for the parameter associated with age.

The 90% and 95% cis for the parameter associated with age tell us the range of values that the parameter could take, given the data. This range does not tell us anything about how likely the parameter is to take any of these values.

To determine how likely it is to take any of these values, we need to look at the p-value for age in the regression summary. The p-value tells us the probability that the parameter associated with age is different from zero.

It is the level of significance and is used to determine if the parameter is statistically significant. It is the only way to determine the likelihood of the parameter taking any of the values in its 90% and 95% cis.

Therefore, we cannot deduce anything about the p-value for age in the regression summary from just the 90% and 95% cis for the parameter associated with age.

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Answer:

Step-by-step explanation:

If James gave each cousin 3 cupcakes, there would be 25 left. James is 71 cupcakes short of being able to give each cousin 15 cupcakes. You want to know the number of cousins and the number of cupcakes.

The given relations between the number of cupcakes (n) and the number of cousins (c) are ...

  n -3c = 25 . . . . . 3 each ⇒ 25 left

  n -15c = -71 . . . . . 15 each ⇒ 71 short

Subtracting the second equation from the first, we have ...

  (n -3c) -(n -15c) = (25) -(-71)

  12c = 96

  c = 8 . . . . . . divide by 12

  n = 3c +25 = 24 +25 = 49

(a) James has 8 cousins.

(b) James baked 49 cupcakes.

Hence, The value of y = 12 when x = -6.

Given that , Y varies directly with x .

If y=−4 when x=2,

Find y when x=−6.

According to the question,

This is a direct variation , y = kx

If,

y = −4  when x=2,

y = kx

-4 = 2k

k = -2

To find ;

y when x = −6

y = kx

y = (-2) (-6)

y = 12

The mean and the standard deviation for the data-set are given as follows:

mean = 23.8; standard deviation = 5.2.

The mean of a data-set is given by the sum of all the observations in the data-set divided by the number of observations, hence:

Mean = (8.2 + 13.4 + 18.6 + 23.8 + 29 + 34.92 + 39.4)/7

Mean = 23.8.

The standard deviation is given by the square root of the sum of the differences squared between each observation and the mean, divided by the number of observations, hence:

s = sqrt([(8.2-23.8)² + (13.4-23.8)² + (18.6-23.8)² + (23.8-23.8)² + (29-23.8)² + (34.92-23.8)² + (39.4-23.8)²]/7)

s = 5.2.

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Answer:

second

Step-by-step explanation:

The expression that represents the number of inches the plant grew is 8 - x.

An expression is combination of variables, numbers and operators.

A plant measured x inches tall last week and 8 inches tall this week.

We have to find the expression which represent the plant grew this week.

The number of inches the plant grew can be represented by the difference between its height this week and its height last week.

Eight minus x is the required expression.

8 - x

Hence,  the expression that represents the number of inches the plant grew is 8 - x.

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Complete question

A plant measured x inches tall last week and 8 inches tall this week. Represent the number of inches the plant grew

A. X - 8

B. 8 - X

For the expression not chosen, describe a situation that the expression might represent.

Answer: A gallon of stain covers 70 square feet, so 8 gallons would cover 8 x 70 = 560 square feet. If Joaquin must apply 2 coats to the deck, he needs enough stain to cover the deck twice, so he needs 560 square feet x 2 = 1120 square feet.

Of the deck sizes listed, the largest one that Joaquin can fully cover with 2 coats of stain is 14 x 20 = 280 square feet, which is less than 1120 square feet. This means that Joaquin can fully cover all of the listed deck sizes if he must apply 2 coats of stain.

Step-by-step explanation:

Answer: Jason made $16

Step-by-step explanation:

Start by using the ratio of 2:3 to find the fraction of the total profit that Jason earned: Jason's share: 2/(2+3) = 2/5

Caleb's share: 3/(2+3) = 3/5

We can then use this fraction to calculate how much money Jason made:

Jason's share = (2/5) x $40 = $16

Therefore, Jason made $16 from the lemonade stand profits.

The option which is not a requirement for the permutation rule is d) Order is not taken into account (rearrangements of the same items are considered to be the same).

Permutations is the number of arrangements of r objects taken out of n objects.

The requirements for permutation include the folowing

Since these are the requirements for a permutation, the options a, b and c are requirements for a permutation while option d is not.

So, the option which is not a requirement is d) Order is not taken into account (rearrangements of the same items are considered to be the same).

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2.5 hours spent in each car wash would cost $40

From the question, we have the following parameters that can be used in our computation:

y = 10x + 20

y = 15x + 5

A situation for the system of equations is the total charges of different car wash per hour

So, the solution would be

The number of hours spent in each car wash for the total amount to be the same

From the graph, the solution is (2.5, 40)

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Answer:

a) Prime factorization of 396: 2² × 3¹ × 11¹

b) Factors of 396: 88 and 9.

Step-by-step explanation:

a) To find the prime factorization of 396, we can start by dividing it by the smallest prime number, which is 2:

396 ÷ 2 = 198

Now we can write:

396 = 2 × 198

Next, we can divide 198 by 2 again:

198 ÷ 2 = 99

So we have:

396 = 2 × 2 × 99

Now we need to find the prime factorization of 99. We can start by dividing it by 3:

99 ÷ 3 = 33

So we can write:

396 = 2 × 2 × 3 × 33

Now we need to find the prime factorization of 33. We can start by dividing it by 3 again:

33 ÷ 3 = 11

So we have:

396 = 2 × 2 × 3 × 11 × 1

We don't need to include the factor of 1, so we can write the prime factorization of 396 in index form as:

396 = 2² × 3¹ × 11¹

b) To find which two of the given numbers are factors of 396, we need to check if all the prime factors of each number are also prime factors of 396.

The prime factorization of 88 is 2³ × 11. Both 2 and 11 are factors of 396, so 88 is a factor of 396.

The prime factorization of 121 is 11². 11 is a factor of 396, but there is no factor of 121 that matches the remaining prime factors of 396 (2² × 3¹), so 121 is not a factor of 396.

The prime factorization of 9 is 3². Both 3 and 2 are factors of 396, so 9 is a factor of 396.

The prime factorization of 14 is 2 × 7. 7 is not a factor of 396, so 14 is not a factor of 396.

The prime factorization of 22 is 2 × 11. Both 2 and 11 are factors of 396, so 22 is a factor of 396.

Therefore, the two numbers that are factors of 396 are 88 and 9.

Answer:

7)  216 ft²

8)  106 yd²

9)  200 m²

Step-by-step explanation:

To calculate the area of each given figure, subtract the area of the cut-out rectangle (marked in blue on the attached diagram) from the area of the larger rectangle.

[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]

Note: The height of the larger rectangle is not complete. I have assumed it is 12 ft.

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&=21 \cdot 12- 6\cdot 6\\&=252-36\\&=216\; \sf ft^2\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&= 17\cdot 10- (17-9)\cdot 8\\&= 17\cdot 10- 8\cdot 8\\&=170-64\\&=106\; \sf yd^2\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&= 15\cdot 20- 10\cdot 10 \\&=300-100\\&=200\; \sf m^2\end{aligned}[/tex]

Part A: the system of inequalities has a definite solution to it.

Part B: Yes the point is a part of Solution Area

What is Linear Inequality?

In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality. It shows the data which is not equal in graph form.

Solution:

The inequalities 4x + 2y ≤ 16

And x + y ≥ 4 is attached below please refer to it.

Part A: The graph of both inequalities intersects each other therefore the system of inequalities has a definite solution to it.

Part B: Yes point (2, 3) is included in the solution area, the following is the mathematical proof.

Using substitution 4*2 + 2*3 ≤ 16

8+6 ≤ 16

14 ≤ 16

And 2+3 ≥ 4

5 ≥ 4

Hence, the point is included in the solution area.

Part C:

Let’s assume a point 1,1 in the solution set so it implies that only the inequality 4x + 2y ≤ 16 is satisfied. Thus, not giving a feasible solution.

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The percentage decrease in the temperature is 87.13 %.

The percentage formula is dividing the required value by the total value and multiplying by 100.

Example:

Required percentage value = a

total value = b

Percentage = a/b x 100

Example:

50% = 50/100 = 1/2

25% = 25/100 = 1/4

20% = 20/100 = 1/5

10% = 10/100 = 1/10

We have,

The temperature of a mixture changes from 121(1/4)°F to 15(3/5)°F.

This means,

Initial temperature = 121(1/4) = 485/4

Final temperature = 15(3/5) = 78/5

The change in temperature.

= 78/5 - 485/4

=  (312 - 2425)/20

= 2113/20

= - 105(13/20)

This means,

There is a decrease in temperature.

Now,

The percentage decreased.

= 105(13/20) / (485/4) x 100

= (2113/20) / (485/4) x 100

= 2113/20 x 4/485 x 100

= 2113/2425 x 100

= 87.13%

Thus,

The percentage decrease in the temperature is 87.13 %.

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The problem involves finding a constant c that makes each given function a valid probability mass function given are f(x) = x/c, x=1,2,3,4; f(x) = cx, x=1,2,3,...,10; f(x) = c(x+1)², x=0,1,2,3; f(x) = x/c, x=1,2,3,...,n; f(x) = (x+1)(x+2)/[x(x+3)], x=0,1,2,3,...

(a) We know that the sum of all the probabilities in a pmf must be equal to 1.

Therefore, we have:

1/c + 2/c + 3/c + 4/c = 1

Solving for c, we get:

c = 10

The pmf can be depicted as:

x f(x)

1 0.1

2 0.2

3 0.3

4 0.4

(b) Again, we know that the sum of all the probabilities in a pmf must be equal to 1. Therefore, we have:

c(1 + 2 + 3 + ... + 10) = 1

Solving for c, we get:

c = 1/55

The pmf can be depicted as:

x f(x)

1 1/55

2 2/55

3 3/55

... ...

10 10/55

(d) We can use the same approach as in part (a) to find c:

c(1 + 4 + 9) = 1

Solving for c, we get:

c = 1/14

The pmf can be depicted as:

x f(x)

0 1/14

1 1/6

2 1/3

3 4/14

(e) Similar to part (a), we have:

1/c + 2/c + 3/c + ... + n/c = 1

Solving for c, we get:

c = (n+1)n/2

The pmf can be depicted as:

x f(x)

1 1/(n+1)

2 2/(n+1)

3 3/(n+1)

n n/(n+1)

(f) We can use the hint given in the problem:

f(x) = 1/(x + 1) - 1/(x + 2)

We can see that this is a telescoping series, where all terms cancel out except for the first and the last term. Therefore, we have:

f(0) = 1/1 - 1/2 = 1/2

f(1) = 1/2 - 1/3 = 1/6

f(2) = 1/3 - 1/4 = 1/12

f(3) = 1/4 - 1/5 = 1/20

The pmf can be depicted as:

x f(x)

0 1/2

1 1/6

2 1/12

3 1/20

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Answer: Let's assume that each of the 12 collectors donates n paintings. Then the total number of paintings donated will be 12 * n. We know that the total number of paintings donated by all collectors is 48, so we have the equation:

12 * n = 48

Dividing both sides by 12, we find that each collector donates n = 4 paintings. So, if Sergio has p paintings in his collection, after his donation he will have p - 4 paintings in his collection. The final answer is:

p - 4

Step-by-step explanation:

a) The probability of selecting a subject with a positive test result given that the subject does not use drugs is equals to the 0.09473.

b) The probability of selecting a subject with a negative test result given that the subject uses drugs is equals to the 0.12.

The confusion matrix of the pre employment drug screening data is present above. It contains positive test result, Negative test result and Total.

Subject uses drugs 44 6 = 50

Subject does not use drugs 90 860 = 950

Similarly , 90 + 44 = 134 , 860 + 6 = 886 and 950 + 50 = 1000.

a) We have to determine the probability of selecting a subject with a positive test result that the subject does not use drug. Let A be the event that the subject with a positive test result and B be the event that subject does not use drug. We have to determine P(A/B). From conditional probability, P(A/B) = P(A and B)/P(B)

= 90/950 = 0.09473

Hence, the probability of selecting a subject with a positive test result given that the subject does not use drug is 0.09473.

b) We have to calculate the probability of selecting a subject with a negative test result that the subject use drugs. Let R be the event that the subject with a negative test result and S be the event that subject uses drug. We have to calculate the value of P(R/S). By using Bayes theorem, P(R/S) = P(R and S) /P(S)

= 6/50 = 0.12

Hence, the probability of selecting a subject with a negative test result given that the subject uses drug is 0.12.

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Complete question:

The above table complete the question.

2. this data came from a pre-employment drug screening. positive test result negative test result (drug use is indicated) (drug use is not indicated) subject uses drugs 44 6 (true positive) (false negative) subject is not a drug user 90 860 (false positive) (true negative) (a) false positive find the probability of selecting a subject with a positive test result given that the subject does not use drugs. (b) false negative find the probability of selecting a subject with a negative test result given that the subject uses drugs.

For the function for the temperature H = f(t) , for cup of coffee placed on the Kitchen counter , then the derivative f'(t) will be Negative .

The Temperature Function in degree Celsius for a cup of coffee placed on the kitchen counter will be negative because if cup of coffee is placed on the counter there is a loss of heat from cup of coffee to surroundings as temperature increases .

If the cup of coffee is losing heat to the surroundings, the temperature of the coffee will decrease over time.

The derivative of the temperature function, f'(t), will be negative. it means that as time increases, the rate of change of the temperature will be negative, indicating a decrease in temperature.

Therefore , the derivative function f'(t) is negative .

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The given question is incomplete , the complete question is

The temperature, H , in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by H = f(t) , where t is in minutes since the coffee was put on the counter .

Is f'(t) positive or negative ?

The flight takes a total of  3 hours 45 minutes

From the question, we have the following parameters that can be used in our computation:

Inital time = 3:30 pm EST

Final time = 4:15 pm PST

When converted, we have

Inital time = 12:30 pm PST

Final time = 4:15 pm PST

So, we have

Difference = 4:15 pm PST -  12:30 pm PST

Evaluate the difference

Difference = 3 hours 45 minutes

Hence, the diration is 3 hours 45 minutes

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From the given data the probability that a respondent 18-29 years and 30+ years are 0.536 and 0.654 espectively. The probability that they answer yes (i.e. global warming will have an impact on them during their lifetime) is 0.265. There also appears to be a difference between ages 18-29 and 30+ regarding concern over global warming.

The probability that a respondent 18-29 years of age thinks that global warming will not pose a serious threat during his/her lifetime is:

P(18-29 and No) = 293/(134 + 293 + 117) = 0.536

The probability that a respondent 30+ years of age thinks that global warming will not pose a serious threat during his/her lifetime is:

P(30+ and No) = 432/(131 + 432 + 178) = 0.654

For a randomly selected respondent, the probability that they answer yes (i.e. global warming will have an impact on them during their lifetime) is:

P(Yes) = (134 + 131)/(134 + 131 + 293 + 432 + 117 + 178) = 0.265

Yes, there appears to be a difference between ages 18-29 and 30+ regarding concern over global warming. The probability of a respondent 30+ years of age thinking that global warming will not pose a serious threat during their lifetime (0.654) is higher than the corresponding probability for respondents 18-29 years of age (0.536). This suggests that older respondents are less concerned about global warming than younger respondents. However, a formal statistical test would be needed to confirm this difference.

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In the linear equation, The value of the x and y are 11/6, -5 and  18/4.

The definition of a linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line.

here, we have,

We have to find that the value of the x and the y.

From the given information:

The equations are :

(4y - 18)

(6x - 11)

(x + 5)​

Now take all equation equal to zero.

Then

4y - 18 = 0

y = 18/4

And then

6x - 11 = 0

x = 11/6

And then

x + 5 = 0

x = -5.

So, In the linear equation, The value of the x and y are 11/6, -5 and  18/4.

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