express the column matrix b as a linear combination of the columns of a. (use a1, a2, and a3 respectively for the columns of a.) a

The area of the shaded region is obtained as (π/√2) units².

What is area?

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.

The diagram is labelled.

The values for area of z and y needs to be obtained.

The result will be in the form |y| + z as when the integration will be done, y will come out to be negative.

Now, x + z is the area of rectangle ABCD.

Verify whether x and y are equal in magnitude -

[tex]\begin{aligned}& x=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos \theta}{\sin ^2 \theta} d \theta=\left[\frac{-1}{\sin \theta}\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} \\& =-1-(\sqrt{2}) \\& =\sqrt{2}-1\end{aligned}[/tex]

[tex]\begin{aligned}& y=\int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \frac{\cos \theta}{\sin ^2 \theta} d \theta=\left[\frac{-1}{\sin \theta}\right]_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \\& =-(\sqrt{2})-(-1) \\& =-\sqrt{2}+1\end{aligned}[/tex]

This is equal to |y| = x.

So, |y| + z is equivalent to writing x + z.

Now the formula for area of rectangle ABCD is -

Area = length × breadth

Area = √2 - [(3π/4) - (π/4)]

Area = √2 - (π/2)

Area = (π/√2)

Therefore, the area is found to be (π/√2) units².

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Answer: Go up 8 and to the right 1 starting at the point (0,4)

Step-by-step explanation: y=mx+b where m=8 (your slope) and b= 4 (your y-intercept)

Answer:To show that Lorenz curves are always concave up on the interval [0, 1], we can use the definition of a Lorenz curve, which is L(x) = xp. Taking the second derivative of L(x) with respect to x gives us L''(x) = p. Since p is always greater than 0, L''(x) is always greater than 0. This means that L(x) is always concave up on the interval [0, 1].

Table of Lorenz values for p = 1.2, 1.5, 2.1, 2.5, 3, and 5:

a. The value of p that corresponds to the most equitable distribution of wealth is 1, as this would mean that the proportion of wealth held by each portion of the population is equal.

b. The value of p that corresponds to the least equitable distribution of wealth is 5, as this would mean that a small portion of the population holds a large proportion of the wealth.

The Gini Index is a measure of income inequality, where a value of 0 represents perfect equality (everyone has the same income) and a value of 1 represents perfect inequality (one person has all the income). The Gini Index is calculated as the ratio of the area between the Lorenz curve and the line of equality to the total area beneath the line of equality.

To find A and B for the Gini index we can use the following integral:

A = ∫(L(x) - x) dx from 0 to 1

B = ∫(x - L(x)) dx from 0 to 1

We can then solve the integral for each specific function of L(x) = xp to find the specific value of A and B.

Step-by-step explanation:

Answer:....

Step-by-step explanation:

The probability that the next fruit is an apple is P = 0.8

We want to fnind the probability of randomly selecting an apple from the bag.

Remember that the probability is equal as the quotient between the number of apples and the total number of fruit on the bag.

Originally, there are:

4 apples.

3 peaches

2 plums.

The checkout person takes 2 plums and 1 peach, so now there are:

4 apples.

1 peach.

So there are 4 apples and 5 fruits in total

Then the probability of grabing an apple is:

P = 4/5 = 0.8

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Answer:

(4+9)+(4·9)=13+36=49

Step-by-step explanation:

True

The range of values for each expression is given as follows:

1. 8 ≤ a+b ≤ 16

2. -6 ≤ a - b ≤ 2

3. 15 ≤ ab ≤ 63

4. 1/3 ≤ a/b ≤ 7/5.

For the maximum values, we have that:

For the minimum values, we have that:

The problem asks for the range of values for each expression, considering 3 < a < 7 and 5 < b < 9.

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The probability of randomly selecting a marble that is not purple is 70%, so the correct option is A.

We know that there are:

For a total number of 2 + 4 + 15 + 9 = 30

There are a total of 30 marbles on the bag, the probability of randomly selecting a marble that is not purple, is equal to the quotient between the number of marbles that are not purple and the total number of marbles.

21 marbles are not purple, then the probability is:

P = (21/30)*100%

P = 0.7*100%

P = 70%

The correct option is a.

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If the equation for every x and x in the domain of f holds, then a function f is odd.  −f(x)=f(−x) − f ( x ) = f ( − x )

If the equation for every x and x in the domain of f holds, then a function f is odd. −f(x)=f(−x) − f ( x ) = f ( − x ) An odd function has a graph that, geometrically speaking, is rotationally symmetric with respect to the origin, meaning that the graph is unaffected by a 180° rotation of the origin.

An even function exists if an expression is obtained that is equivalent to f(x); an odd function exists if an expression is obtained that is equivalent to -f(x); and neither occurs, in which case it is neither.

An odd function is a function having a symmetrical graph about the origin. If a function doesn't display either symmetry, it cannot be either even or odd.

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The simplified expression of 1 minus the quantity 1 over cosecant squared x is 1 - 1/cosec^2(x).

The expression 1 minus the quantity 1 over cosecant squared x can be simplified by dividing the 1 over cosecant squared x into its components. The cosecant squared x can be written as 1/cosec x, and then the 1/cosec x can be written as 1/sin x, with the cosecant being the reciprocal of the sine. Therefore, the expression 1 minus the quantity 1 over cosecant squared x can be written as 1 - 1/sin x. To simplify this expression further, the denominator of 1/sin x can be squared, resulting in 1 - 1/sin^2(x). Since the cosecant is the reciprocal of the sine, cosec^2(x) can be substituted for sin^2(x), giving us the simplified expression 1 - 1/cosec^2(x).

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The complete question: which of the following is the simplified expression of 1 minus the quantity 1 over cosecant squared x end quantity question mark 1 - csc^2(x)

Answer:

cos^2x

Step-by-step explanation:

Very simple, 1-1/csc^2 x

= 1-sin^2 x Using the reciprocal Identity 1/csc x = sin x

= cos^2 x Using the Pythagorean Identity 1 - sin^2 x = cos^2 x

therefore, the answer is cos^2 x.

Hope this helped

The blanks has bee filled by using the correct identities of the algebra.

Part 4:

a. 5 + 5 - 6 = 4

b. 25 - 10 + 10 = 25

c. 24 + 16 - 16 = 24

d. 56 - 20 + 20 = 56

Part 5:  

a) a + b - b = a

b) c - d + d = c

c) e + f - f = e

d) g - h + h = g

Thus, the blanks has bee filled by using the correct identities of the algebra.

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The correct question is-

4. Using your knowledge of identities, fill in each of the blanks.

a. 5 + 5 - ___ = 4

b. 25 - ___ + 10 = 25

c. ___ + 16 - 16 = 24

d. 56 - 20 + 20 = ___

5. Using your knowledge of identities, fill in each of the blanks.

a) a + b - ___ = a

b) c - d + d = ____

c) e + ___ - f = e

d) ___ - h + h = g

Answer:

1.023 x [tex]10^{14}[/tex]    

Step-by-step explanation:

(3.1 x [tex]10^{8}[/tex])(3.3 x [tex]10^{3}[/tex])

(3.1 x 3.3)([tex]10^{8}[/tex] x [tex]10^{3}[/tex])

10.23 x  [tex]10^{13}[/tex]   When the bases are the same and we are multiplying we add the exponents

10.23 x [tex]10^{13}[/tex]  Is not in scientific notation because 10.23 is larger than 10

1.023 x [tex]10^{1}[/tex] x [tex]10^{13}[/tex]

1.023 x [tex]10^{14}[/tex]                                

The next number in the sequence is 151.

What is the successive differences?

The method of successive differences involves finding the differences between consecutive terms in a sequence and using that information to predict the next term in the sequence.

To use this method for the sequence 3, 7, 17, 33, 55, 83, 117:

Find the differences between consecutive terms:

7 - 3 = 4

17 - 7 = 10

33 - 17 = 16

55 - 33 = 22

83 - 55 = 28

117 - 83 = 34

Check if the differences are constant. In this case, they are increasing by 6 each time.

Use this information to predict the next term in the sequence:

117 + 34 = 151

So, the next number in the sequence is 151.

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See the diagram below.

=================================================

Explanation:

Pick two points on this diagonal line. I'll go for (0,-3) and (1,-1)

Apply the slope formula to those coordinates.

[tex](x_1,y_1) = (0,-3) \text{ and } (x_2,y_2) = (1,-1)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_{2} - \text{y}_{1}}{\text{x}_{2} - \text{x}_{1}}\\\\m = \frac{-1 - (-3)}{1 - 0}\\\\m = \frac{-1 + 3}{1 - 0}\\\\m = \frac{2}{1}\\\\m = 2\\\\[/tex]

A slope of 2, aka 2/1, means "move up 2, then right 1".

This "up 2, right 1" motion allows us to move from (0,-3) to (1,-1) as shown in the diagram below.

The factor of the trinomial (z² - 14z + 45) are,

⇒ (z - 5) (z - 9)

An algebraic equation with the second degree of the variable is called an Quadratic equation.

We have to given that;

The function is,

⇒ z² - 14z + 45

Now, We can factorize the trinomial as;

⇒ z² - 14z + 45

⇒ z² - (9 + 5)z + 45

⇒ z² - 9z - 5z + 45

⇒ z (z - 9) - 5 (z - 9)

⇒ (z - 5) (z - 9)

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The amount Murphy spent is solved to be $22.25

To calculate the amount Murphy spent the cost of the items and the units bought should be known. The calculation is done using the formula

= units bought * cost of each unit

From the question, we can deduce that

units bought = 5

cost of each = $2.50

amount spent = 5 * $2.5 = $12.5

units bought = 3

cost of each = $3.25

amount spent = 3 * $3.25 = $9.75

Total amount spent = spending for bottle water + spending for sandwiches

Total amount spent = $12.5 + $9.75

Total amount spent = $22.25

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The simple interest is I = $ 303.33  and the amount after 2 months with rate of 13 % is A = $ 14,303.33

Simple interest is a method of calculating interest that ignores the impact of compounding. While interest frequently compounds throughout the course of a loan's set periods, simple interest does not. Simple interest is calculated by multiplying the principal amount by the interest rate, times the number of periods.

Simple Interest = ( Principal Amount x Rate x Time Period ) / 100

Given data ,

Let the simple interest be represented as I

Let the amount with simple interest be A = P + I

Now , the principal P = $ 14,000

The rate of interest R = 13 %

The number of months = 2 months

The number of years = 2/12 years

Simple Interest I = ( Principal Amount x Rate x Time Period ) / 100

Substituting the values in the equation , we get

Simple Interest I = ( 14,000 x 13 x ( 2/12 ) ) / 100

On simplifying the equation , we get

Simple Interest I = $ 303.33

Now , the amount after 2 months = P + I

Amount = $ 14,000 + $ 303.33 = $ 14,303.33

Hence , the interest is $ 303.33

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In circle the length of PQ = 24 inches.

What is Circle?

A circle is a closed, two-dimensional object in which all points in the plane are equally spaced apart from the centre. The line of reflection symmetry is formed by each line that traverses the circle. Additionally, it possesses rotational symmetry around the centre for each angle.

Since AB= 25 inches and AC = 16 inches then

=> BC = AB-AC = 25-16 = 9 inches.

∠APB is a right angle because it is inscribed in a semicircle.

The three right triangles ᐃAPB, ᐃACP and ᐃPCB are all similar

because their corresponding angles are equal.  Therefore

=> [tex]\frac{AC}{PC}=\frac{PC}{BC}[/tex]

=> [tex]\frac{16}{PC}=\frac{PC}{9}[/tex]

=> [tex]PC^2 = 25*9 = 144 = 12^2[/tex]

=> PC = 12 inches

By symmetry , PC = QC then,

=> PQ = 12*2 = 24 inches.

Hence the length of PQ is 24 inches.

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The rectangles approximate ln(2) as 0.693. The approximation is an under-estimate, as the rectangles only cover a portion of the graph. The closest estimate is the fourth one, as it is closest to the value given by the calculator.

1. Slice the graph into 4 equal parts by drawing vertical lines at x=0.5, x=1, and x=1.5.

2. Find the midpoints of each slice by calculating the average of the x-values of each slice. The midpoints are x=0.25, x=0.75, x=1.25, and x=1.75.

3. Determine the height of each rectangle at the midpoints. The heights are y=0.253, y=0.572, y=0.854, and y=1.092.

4. Sketch in the rectangles using the midpoints and heights calculated.

5. Approximate ln(2) as the sum of the areas of the four rectangles, which is 0.693. This is an under-estimate.

6. The closest estimate is the fourth one, as it is closest to the value given by the calculator (0.693).

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The radius of the right circular cylinder is 4 inches and the height is 3.33 inches.

Radius of cone=6inches

height of cone = 10 inches

Let the radius and height of the right circular cylinder be x and y respectively.

According to triangle similarities,

10-y/x=10/6

⇒10-y/x=5/3

⇒30-3y=5x

⇒y=10-5x/3 ----1

The volume of the right circular cylinder is π[tex]r^{2}[/tex]h

v=  πx∧2y

substituting the value of y from 1

v=  π[tex]x^{2}[/tex](10-5x/3)

v=10πx^{2}- 5x∧3/3

We differentiate volume w.r.t radius

DV/dx= 20πx-5[tex]x^{2}[/tex]

setting derivative = 0

DV/dx= 20πx-5[tex]x^{2}[/tex]=0

5πx(4-x)=0

x=0 , x=4

[tex]D^{2}[/tex]V/Dx =20π-10πx

putting the maximum value of x in the second derivative we get,

X=Radius = 4inches

Y=height= 10-5x/3=3.33 inches

Therefore, The radius of the right circular cylinder is 4 inches and the height is 3.33 inches.

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Only the integer 12 is a divisor of every other integer which is a part of the set S.

This is a classic problem where we are asked to find such an integer that is going to be a divisor of the above set S of integers. The set S of integers consists of those integers that when squared are both divisible by 24 and 108. Now we know 24 can be written as 2*2*2*3 and similarly 108 can be written as 2*2*3*3*3. The smallest perfect square (n^2) which is a multiple of both 24 and 108 is 2^4*3^4, thus the smallest n is 2^2*3^2 = 36. So, only 12 is a divisor of all integers in S.

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Answer:

y-int: none, x-int: -1

Step-by-step explanation:

It crossed the x-axis at -1 so the x-int is, you guessed it, -1

As for the y-int assuming that its just a straight line, there is none. It never crosses the y-axis meaning that y-int DNE (or just none in you case)

Answer:

9 is correct

Step-by-step explanation:

2+3=5+4=9 , believe

We can start solving the problem by using a system of equations. Let's call the number of ounces of solution a x and the number of ounces of solution b y. From the problem statement, we know that:

x + y = 150 (because the total amount of the mixture is 150 ounces)

0.6x + 0.9y = 0.65(x + y) (because the mixture is 65% salt, and we can find the amount of salt in the mixture by multiplying the percentage by the total amount)

We can use the first equation to solve for one of the variables in terms of the other:

y = 150 - x

Now we can substitute this expression for y into the second equation:

0.6x + 0.9(150 - x) = 0.65(150)

0.6x + 135 - 0.9x = 97.5

-0.3x = -37.5

x = 125

Now we can use the first equation to find the value of y:

y = 150 - x

y = 150 - 125

y = 25

So the scientist should use 125 ounces of solution a and 25 ounces of solution b to obtain 150 ounces of a mixture that is 65% salt.

The probability that the total weight of the three randomly selected grapefruits is more than 3.4 pound will be equal to 0.842.  

From the data given in the question,

x > 3.4 lb

The mean is 1 lb.

The difference from the mean is 0.12 lb.

The z-score will be,

z = (3.4 - 1) / 0.12

z = 20

This is equivalent to an 84.20% or 0.842 probability.

The complete question is,

The weights of grapefruits of a certain variety are approximately normally distributed with a mean of 1 pound and a standard deviation of 0.12 pounds. Use scenario 6-9. what is the probability that the total weight of three randomly selected grapefruits is more than 3.4 pounds?

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The solution set of the original system is

[tex]$\left[\begin{array}{l}x_1 \\ x_2 \\ x_3 \\ x_4\end{array}\right]=\left[\begin{array}{c}-3 \\ -5 \\ 6 \\ -3\end{array}\right]$[/tex]

Step 1

We are given with a row reduced matrix form.:

[tex]$$M=\left[\begin{array}{rrrrr}1 & -2 & 0 & 3 & -2 \\0 & 1 & 0 & -4 & 7 \\0 & 0 & 1 & 0 & 6 \\0 & 0 & 0 & 1 & -3\end{array}\right]$$[/tex]

We have to find the solution set using the appropriate steps.

Step 2: The Augmented matrix Form

[tex]$$\left[\begin{array}{cccc}1 & -2 & 0 & 3 \\0 & 1 & 0 & -4 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{array}\right]\left[\begin{array}{l}x_1 \\x_2 \\x_3 \\x_4\end{array}\right]=\left[\begin{array}{c}-2 \\7 \\6 \\-3\end{array}\right]$$[/tex]

Step 3: System of Equations

[tex]$$\begin{aligned}& x_1-2 x_2+3 x_4=-2 \\& x_2-4 x_4=7 \\& x_3=6 \\& x_4=-3\end{aligned}$$[/tex]

Step 4: The Back-Substitution

From last two rows, we have

[tex]$$\begin{aligned}& x_3=6 \\& x_4=-3\end{aligned}$$[/tex]

Now substitute in row-2, to get value of $x_2$

[tex]$$\begin{aligned}& x_2-4(-3)=7 \\& x_2+12=7 \\& x_2=7-12 \\& x_2=-5\end{aligned}$$[/tex]

Now substitute in row-1, to get value of $x_1$

[tex]$$\begin{aligned}& x_1-2(-5)+3(-3)=-2 \\& x_1+10-9=-2 \\& x_1+1=-2 \\& x_1=-3\end{aligned}$$[/tex]

So,

[tex]$\left[\begin{array}{l}x_1 \\ x_2 \\ x_3 \\ x_4\end{array}\right]=\left[\begin{array}{c}-3 \\ -5 \\ 6 \\ -3\end{array}\right]$[/tex]

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using simple  product and quotient, The annual pay of the fourth worker is 49650.

PRODUCT - The outcome of multiplying two or more numbers is the product of those numbers. QUOTIENT - The outcome of dividing two numbers is the quotient of the two numbers.

Formulate the following supplied conditions: 46800 × 4 - 45900 - 44200 - 47450

Make a calculation for the following: 187200 - 45900 - 44200 - 47450.

Add up or subtract 141300, 44200, and 47450.

Calculate the difference or total of 97100 minus 47450.

Calculate the difference or sum: 49650

get the outcome: 49650

The annual pay of the fourth worker is 49650

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The smaller number is 15 and the larger number is 36.

Let x and y be the smaller and larger numbers, respectively.

From the given information, we know:

y = x + 21 (because the larger number is 21 more than the smaller number)

x + y = 51 (because the sum of the two numbers is 51)

We can substitute the first equation into the second equation to find the value of x:

x + (x + 21) = 51

x + x + 21 = 51

2x = 30

x = 15

Now that we know the value of x, we can use the first equation to find the value of y:

y = x + 21

y = 15 + 21

y = 36

So the smaller number is 15 and the larger number is 36.

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Answer: the answer is 90

Step-by-step explanation:There is only one number between 61 and 107 that is a multiple of 3, 5 and 9. So that number will be  90.

The property used to solve 7r + 2r is distributive property.

The distributive property states that an expression which is given in form of x (y + z) can be solved as x × (y + z) = xy + xz.

In other words,  the distributive property states that when a number is multiplied by the sum of two numbers, the first number can be distributed to both of those numbers .

Therefore,

7r + 2r

Using distributive law,

7r + 2r = r(7 + 2) = 9r

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