Use the Simplex algorithm to solve the following problem.Clearly state the solution. min 5x1 + 3x2 + 4x3 S/T 3x1 − 2x2 + x3 ≤ 16 2x1 + x2 + 5x3 ≥ 12 −4x1 + 2x2 + 2x3 = 11 xj ≥ 0

Answer:

$456

Step-by-step explanation:

12% = 0.12

3800 x 0.12 = $456

So, the down payment is  $456

The solution to the system of linear equations is (47/16, -161/40, 49/8).

To solve the system of linear equations using Gaussian elimination, we first need to write the equations in matrix form.

[2 -3 -5 | -6]
[-6 2 8 | 11]
[3 -1 3 | 5]

Next, we need to use elementary row operations to transform the matrix into reduced row echelon form (RREF). This means that the matrix will have a leading 1 in each row, and all other entries in that column will be 0.

Step 1: Divide the first row by 2 to get a leading 1.

[1 -3/2 -5/2 | -3]
[-6 2 8 | 11]
[3 -1 3 | 5]

Step 2: Add 6 times the first row to the second row to eliminate the -6 in the second row.

[1 -3/2 -5/2 | -3]
[0 5 7 | 7]
[3 -1 3 | 5]

Step 3: Subtract 3 times the first row from the third row to eliminate the 3 in the third row.

[1 -3/2 -5/2 | -3]
[0 5 7 | 7]
[0 7/2 13/2 | 14]

Step 4: Divide the second row by 5 to get a leading 1.

[1 -3/2 -5/2 | -3]
[0 1 7/5 | 7/5]
[0 7/2 13/2 | 14]

Step 5: Add 3/2 times the second row to the first row to eliminate the -3/2 in the first row.

[1 0 -1/2 | -1/2]
[0 1 7/5 | 7/5]
[0 7/2 13/2 | 14]

Step 6: Subtract 7/2 times the second row from the third row to eliminate the 7/2 in the third row.

[1 0 -1/2 | -1/2]
[0 1 7/5 | 7/5]
[0 0 4/5 | 49/10]

Step 7: Divide the third row by 4/5 to get a leading 1.

[1 0 -1/2 | -1/2]
[0 1 7/5 | 7/5]
[0 0 1 | 49/8]

Step 8: Add 1/2 times the third row to the first row to eliminate the -1/2 in the first row.

[1 0 0 | 47/16]
[0 1 7/5 | 7/5]
[0 0 1 | 49/8]

Step 9: Subtract 7/5 times the third row from the second row to eliminate the 7/5 in the second row.

[1 0 0 | 47/16]
[0 1 0 | -161/40]
[0 0 1 | 49/8]

Now the matrix is in RREF, and we can read off the solutions:

x = 47/16
y = -161/40
z = 49/8

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The [tex]4\frac{3}{8}-2\frac{1}{8}[/tex] [tex]=2\frac{1}{8}[/tex] This is the correct way to solve the equation it.

The meaning of an equations in algebra is a logical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

When we say we're "solving" an equation, we truly mean we're "solving" the issue or "answering" the question since, in most circumstances, a formula reflects a problem (or query) of some type. A mathematical phrase with two equal sides and an equal sign is called an equation.

Given,

[tex]4\frac{3}{8}-2\frac{1}{8}[/tex]

[tex]=2\frac{1}{8}[/tex]

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By answering the above question, we may state that Hence, the equation nonagon's last seven angles are all around 140.57 degrees.

When two assertions are connected by a mathematical equation, the equals symbol (=) is used to denote equality. In algebra, a mathematical statement that establishes the equivalence of two mathematical expressions is referred to as an equation. In the equation 3x + 5 = 14, for instance, the equal sign places a space between the integers 3x + 5 and 14. To comprehend the connection between the two phrases written on the opposing sides of a letter, utilise a mathematical formula. Frequently, the logo and the specific piece of software are the same. as in, for instance, 2x - 4 = 2.

S = (n - 2) × 180°

where n is the total number of sides, which in this instance is 9.

The internal angles of a nonagon's sum are as follows:

S = (9 - 2) × 180° = 1260°

Let x represent the dimensions of the nonagon's two equal angles. The nonagon's total angles may therefore be represented as follows:

1000° + 5x + 5x + 180° = 1260°

When we simplify the previous equation, we get:

10x = 80°

x = 8°

Each of the nonagon's two equal angles is 8°, and the total of the remaining seven angles is as follows:

1000° - 2x = 1000° - 2(8°) = 984°

984° ÷ 7 = 140.57° (rounded to two decimal places) (rounded to two decimal places)

Hence, the nonagon's last seven angles are all around 140.57 degrees.

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blank1 - Word Answer (-5,0)
blank2 - Word Answer (1,0)
blank3 - Word Answer (0,-5)

The x- and y-intercept(s) of an equation are the points where the graph of the equation crosses the x- and y-axis respectively. To find the x-intercept(s), we set y=0 and solve for x. To find the y-intercept, we set x=0 and solve for y.

a. x-intercepts:
0=(x+2)^2-9
9=(x+2)^2
±3=x+2
x=-5 or x=1
So the x-intercepts are (-5,0) and (1,0).

b. y-intercept:
y=(0+2)^2-9
y=4-9
y=-5
So the y-intercept is (0,-5).

Therefore, the correct answers are:
blank1 - Word Answer (-5,0)
blank2 - Word Answer (1,0)
blank3 - Word Answer (0,-5)

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21. tan^2 t/sec t = sec t-cos t
First, we can simplify the left-hand side of the equation by dividing tan^2 t by sec t. Remember that tan t = sin t/cos t and sec t = 1/cos t:
tan^2 t/sec t = (sin^2 t/cos^2 t)/(1/cos t) = sin^2 t/cos t
Now we can subtract cos t from both sides of the equation:
sin^2 t/cos t - cos t = sec t - cos t
Next, we can multiply both sides of the equation by cos t to get rid of the fraction:
sin^2 t - cos^2 t = sec t*cos t - cos^2 t
Finally, we can use the Pythagorean identity sin^2 t + cos^2 t = 1 to simplify the left-hand side of the equation:
1 - cos^2 t - cos^2 t = sec t*cos t - cos^2 t
0 = sec t*cos t - 2*cos^2 t
22. cot^2 t/csc t = csc t-sin t
First, we can simplify the left-hand side of the equation by dividing cot^2 t by csc t. Remember that cot t = cos t/sin t and csc t = 1/sin t:
cot^2 t/csc t = (cos^2 t/sin^2 t)/(1/sin t) = cos^2 t/sin t
Now we can subtract sin t from both sides of the equation:
cos^2 t/sin t - sin t = csc t - sin t
Next, we can multiply both sides of the equation by sin t to get rid of the fraction:
cos^2 t - sin^2 t = csc t*sin t - sin^2 t
Finally, we can use the Pythagorean identity sin^2 t + cos^2 t = 1 to simplify the left-hand side of the equation:
1 - sin^2 t - sin^2 t = csc t*sin t - sin^2 t
0 = csc t*sin t - 2*sin^2 t
24. (1−sin θ)/cos θ = sec θ−tan θ
First, we can simplify the left-hand side of the equation by dividing 1-sin θ by cos θ:
(1−sin θ)/cos θ = (1/cos θ) - (sin θ/cos θ) = sec θ - tan θ
So the equation is already simplified and both sides are equal.
25. (sin t/csc t) + (cos t/sec t) = 1
First, we can simplify the left-hand side of the equation by dividing sin t by csc t and cos t by sec t. Remember that csc t = 1/sin t and sec t = 1/cos t:
(sin t/csc t) + (cos t/sec t) = (sin t/(1/sin t)) + (cos t/(1/cos t)) = sin^2 t + cos^2 t
Now we can use the Pythagorean identity sin^2 t + cos^2 t = 1 to simplify the left-hand side of the equation:
1 = 1
So the equation is already simplified and both sides are equal.

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Answer:

Step-by-step explanation:

based on the median of the samples, what is a reasonable estimate of the number of students that bike to school? Round to the nearest whole number based on the median of the samples, what is a reasonable estimate of the number of students that bike to school? Round to the nearest whole number based on the median of the samples, what is a reasonable estimate of the number of students that bike to school? Round to the nearest whole number8957647h3ny4y4y7

A dilation with a scale factor of 8:7 creates an image that is eight times larger than the original.

A scale factor is a numerical value which is used to multiply or divide a given measurement. It is used to compare the sizes of two different objects, or to enlarge or reduce the size of an object. The scale factor can be used to calculate the area, perimeter, and volume of an object. For example, if an object is enlarged by a scale factor of 2, its area will increase by a factor of 4. Similarly, if an object is reduced by a scale factor of 0.5, its area will be reduced by a factor of 0.25.

This means that all of the points in the original image are moved away from the center of the dilation by a factor of eight. This creates an image that is eight times wider and seven times taller than the original. This type of dilation is also known as an enlargement. By enlarging the image, it is possible to see more detail and make the image easier to see. Additionally, it may also be possible to see features that were previously too small to be seen.

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The range of the data is 328. The 1st quartile, median and 3rd quartile is 147, 182, and 197, respectively. The IQR is 50. There is only one outlier in the data, which is 439.

i) The range of the data is the difference between the highest and lowest values. In this case, the range is 439 - 111 = 328.

ii) The 1st quartile is the 25th percentile of the data, the median is the 50th percentile, and the 3rd quartile is the 75th percentile. To find these values, we need to first sort the data in ascending order:
111, 131, 147, 151, 151, 182, 190, 197, 201, 209, 439
Then we can find the quartiles:
1st quartile = 147
Median = 182
3rd quartile = 197

iii) The interquartile range (IQR) is the difference between the 3rd and 1st quartiles. In this case, the IQR is 197 - 147 = 50.

v) Outliers are values that are more than 1.5 times the IQR above the 3rd quartile or below the 1st quartile. In this case, the upper bound for outliers is 197 + (1.5 * 50) = 272.5 and the lower bound is 147 - (1.5 * 50) = 71.5. The only value that falls outside of these bounds is 439, so there is one outlier in the data.

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Answer:

12/36 ÷ 2/2 = 6/18

12/36 ÷ 3/3 = 4/12

12/36 ÷  4/4 = 3/9

All of these equations are equal to one another since they are all being divided by a whole, they are all just being expressed differently.

I hope this helped & Good Luck <3 !!!

Answer:

12/36 ÷ 1 = 0.0333333.......

Step-by-step explanation:

36/12 ÷ 1 =3

12 divided by 36 gives 0.333...... and the result divided by 1 gives the same answer. The same process applies to 36 divided by 12, which gives 3. The result which is three is divided by 1 which gives the same three.

NB: Any number divided by 1 is still the same number. Example: 1200/1 = 1200

Hope it helps.

Answer:

Step-by-step explanation:

So all that you have to do is just multiply 15 by 110kg and you'll get a answer of 1500kg total.

From the given equation the value of r is 8.

A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.

To solve this equation for r, we need to isolate r on one side of the equation. First, we add 14 to both sides of the equation:

4r-14+14 = (9/4)r+14

4r = (9/4)r+14

Then, we subtract (9/4)r from both sides:

4r - (9/4)r = (9/4)r+14 - (9/4)r

(5/4)r = 14

Finally, divide both sides of the equation by (5/4) to isolate r:

r = 14/(5/4) = 8

Therefore, the value of r is 8.

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The result obtained from the long division of the polynomial is n² + 2n - 1 + 3/(n - 9).

The quotient of the polynomial is obtained by applying long division method as shown below.

        n²   +   2n   -   1

   ________________________

n - 9 √ (n³ - 11n² + 21n - 24)

            (n³ -  9n²)  

       ____________

               -2n² + 21n

               -2n² + 18n

               __________

                       3n - 24

                       3n - 27

                       ______

                            3

Therefore, the quotient is n² + 2n - 1 and the remainder is 3.

n³ - 11n² + 21n - 24 =  (n - 9)(n² + 2n - 1) + 3.

So when we divide the polynomial using long division method we would obtain the following result.

(n³ - 11n² + 21n - 24) ÷ (n - 9) = n² + 2n - 1 + 3/(n - 9)

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Find the quotient of this long division; (n³ - 11n² +21n-24)÷(n-9)

The sum of the two vectors is a vector with length 26.56 and an angle of 12.72°.

The vector ü can be represented in Cartesian coordinates as (16cos5°, 16sin5°) and the vector ü can be represented as (11cos25°, 11sin25°).

To find the sum of these two vectors, we add their corresponding coordinates:
(16cos5° + 11cos25°, 16sin5° + 11sin25°)

Using a calculator, we can approximate these values:
(25.82, 5.81)

Therefore, the sum of the two vectors is approximately (25.82, 5.81) in Cartesian coordinates. To find the length and angle of this vector, we can use the Pythagorean theorem and the inverse tangent function:

Length = √(25.82² + 5.81²)

≈ 26.56

Angle = tan⁻¹(5.81/25.82)

≈ 12.72°

So the sum of the two vectors is a vector with length 26.56 and an angle of 12.72°.

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(3, 5): Point A is located at the coordinates (3, 5) on the Cartesian plane, meaning it is 3 units to the right of the origin and 5 units above the origin.

Coordinates are points on a graph used to determine the position of a particular location. Coordinates can be expressed in two-dimensional (2D) or three-dimensional (3D) formats. In 2D, a point is determined by two numbers—its x and y values—which are the distances from the point to the origin along each axis. In 3D, an additional z-value is needed to indicate the distance from the origin along the third axis. Coordinates are used in a variety of applications, including navigation, mapping, surveying, and astronomy.

B. (2, -3): Point B is located at the coordinates (2, -3) on the Cartesian plane, meaning it is 2 units to the right of the origin and 3 units below the origin.

C. (-4, -2): Point C is located at the coordinates (-4, -2) on the Cartesian plane, meaning it is 4 units to the left of the origin and 2 units below the origin.

D. (0, 0): Point D is located at the coordinates (0, 0) on the Cartesian plane, meaning it is at the origin.

E. (-2, 1): Point E is located at the coordinates (-2, 1) on the Cartesian plane, meaning it is 2 units to the left of the origin and 1 unit above the origin.

F. (5, -4): Point F is located at the coordinates (5, -4) on the Cartesian plane, meaning it is 5 units to the right of the origin and 4 units below the origin.

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The polynomial 2x² - 10x + 15 will have a quotient 2x and remainder of 15 when divided by x - 5 and the result can be written in the form 2x + 15/(x - 5).

Using the long division method will require us to; divide, multiply, subtract, bring down the next number and repeat the process to end at zero or arrive at a remainder

We shall divide the polynomial 2x² - 10x + 15 by x - 5 as follows;

2x² divided by x equals 2x

x - 5 multiplied by 2x

2x(x - 5) = 2x² - 10x

subtract 2x² - 10x from 2x² - 10x + 15

2x² - 10x + 15 - (2x² - 10x)

2x² - 10x + 15 - 2x² + 10x = 15.

Therefore, the polynomial 2x² - 10x + 15 will have a quotient 2x and remainder of 15 when divided by x - 5 and the result can be written in the form 2x + 15/(x - 5).

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The factored form of the polynomial is f(x) = (x + 2)(x + 10)(x - 6) .

The given polynomial is f(x) = x4 + x3 + 2x2 - 22x - 60.

To find the zeros of the polynomial, set f(x) = 0 and solve for x. This gives us the following equation:

x4 + x3 + 2x2 - 22x - 60 = 0

The given zero is -2. To factor the polynomial, use the zero(-2) as one factor. To find the other factors, set (x + 2) = 0 and solve for x. This gives us the following equation:

x3 + 2x2 - 22x - 60 = 0

To solve this equation, use the Quadratic Formula. This gives us two factors, (x + 10) = 0 and (x - 6) = 0. Therefore, the factored form of the polynomial is:

f(x) = (x + 2)(x + 10)(x - 6)

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Answer:

Step-by-step explanation:

1+2x)5

First you minus the 1 with the 5

Which you'll get a four then divide it by 2

Which you'll get x=2

But then times it by 5

Then you get y=10

Answer:

[tex]\dfrac{d}{dx}\left(\sqrt{1\:+\:2x}\right)5 = \boxed{\dfrac{5}{\sqrt{1+2x}}}[/tex]

Step-by-step explanation:

Given [tex]y = \left(\sqrt{1\:+\:2x}\right)5[/tex]

we are asked to find [tex]\dfrac{dy}{dx}[/tex]

[tex]\dfrac{dy}{dx} = \dfrac{d}{dx}\left(\sqrt{1\:+\:2x}\right)5\\\\= 5\dfrac{d}{dx}\left(\sqrt{1+2x}\right)\\\\[/tex]

Find [tex]\dfrac{d}{dx}\left(\sqrt{1+2x}\right)[/tex]:
[tex]Let \;u = 1 + 2x\\\\f(u) = \sqrt(u)\\\\[/tex]

[tex]\mathrm{Apply\:the\:chain\:rule}:\quad \dfrac{df\left(u\right)}{dx}=\dfrac{df}{du}\cdot \dfrac{du}{dx}[/tex]

[tex]= \dfrac{d}{du}\left(\sqrt{u}\right)\dfrac{d}{dx}\left(1+2x\right)[/tex]

[tex]\dfrac{d}{du}\left(\sqrt{u}\right) = \dfrac{d}{du}\left(u^{\dfrac{1}{2}}\right)\\\\= \dfrac{1}{2}u^{\dfrac{1}{2}-1}\\\\= \dfrac{1}{2\sqrt{u}}\\\\\\[/tex]

Substitute back u = 1 + 2x
[tex]= \dfrac{1}{2\sqrt{1+2x}}[/tex]

[tex]\dfrac{d}{dx}(1 + 2x) =\dfrac{d}{dx}(1)} + \dfrac{d}{dx}{2x}\\\\= 0 + 2 \\\\= 2\\[/tex]

Therefore
[tex]\dfrac{dy}{dx} = \dfrac{d}{dx}\left(\sqrt{1\:+\:2x}\right)5\\\\= 5\dfrac{d}{dx}\left(\sqrt{1+2x}\right)\\\\[/tex]

[tex]= 5\cdot \dfrac{1}{2\sqrt{1+2x}}\cdot \:2\\\\= 5\cdot \dfrac{1}{\sqrt{1 + 2x}}\\\\=\dfrac{5}{\sqrt{1+2x}}[/tex]

the area of the shaded region:

[tex]= 2(the \ area \ of \ a \ quarter \ circle) - (the \ area \ of \ the \ square)\\[/tex]

[tex]= 2\Big(\dfrac{\pi 12^{2} }{4}\Big) - 12^{2} = \dfrac{144 \pi}{2} - 144 = 72 \pi - 144\\[/tex]

[tex]= 72 (\pi - 2) \ m^{2}[/tex]

Answer:

  72π-144 m²

Step-by-step explanation:

You want the area of a shaded region consisting of the overlap of two quarter circles in a 12 m square.

If we draw a diagonal from upper left to lower right through the figure, the shaded area is divided into two 90° segments of a circle of radius 12 m.

The formula for the area of a segment is ...

  A = 1/2r²(θ -sin(θ))

where θ is the measure of the central angle.

For θ = π/2 radians, this is ...

  A = 1/2r²(π/2 -1) . . . . . half the shaded area

Then the whole shaded area is ...

  2 × 1/2r²(π/2 -1) = (12 m)²(π/2 -1) = 72π -144 m²

The area of the shaded region is 72π -144 m².

__

Additional comment

If we expand the shaded area formula, we get ...

  A =1/2πr² -r²

This is recognizable as twice the area of a quarter circle, less the area of the square.

<95141404393>

it should be +60 not +78

when you multiply 18x by X - 3 you obtain 18x² - 54x . Subtract what you have gotten from 18x²-6x you obtain 60x

Answer:

write relation between work and power

We have:

Find the Range.

Substitute to F(x):

x = -1, x = 0, x = 1 and x = 2:

F(-1) = 3(-1) + 7 = -3 + 7 = 4

F(0) = 3(0) + 7 = 0 + 7 = 7

F(1) = 3(1) + 7 = 3 + 7 = 10

F(2) = 3(2) + 7 = 6 + 7 = 13

The Range:

The options which satisfy the given diagram are

m∠5 + m∠6 = 180°

m∠2 + m∠3 = m∠6

m∠2 + m∠3 + m∠5 = 180°

The sum of the interior angles of a triangle is 180°.

The exterior angle of a triangle is equal to the sum of its interior opposite angles.

Let the Triangle be ABC and the exterior angle be ∠D , then

∠A + ∠B + ∠C =  180°

∠A + ∠B = ∠D , ∠A and ∠B are interior opposite angles of the triangle

So, the Options that satisfied

m∠5 + m∠6 = 180° { linear pair}

m∠2 + m∠3 = m∠6 {exterior angle theorem}

m∠2 + m∠3 + m∠5 = 180° {Angle Sum Property}

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a. The explicit formula for the sequence is: an = a + (n - 1)d

b. There are 56 tiles in the 13th row.

What is arithmetic sequence?

An arithmetic sequence is one in which each phrase grows by adding or removing a certain constant, k.

a. Let's denote the number of tiles in the first row as a, and the common difference between consecutive rows as d. Then, the number of tiles in the second row would be a + d, the third row would be a + 2d, and so on. Since the second row has 12 tiles and the first row has 8 tiles, we have:

a + d = 12

a = 8

We can solve this system of equations to find the value of d:

a + d = 12

8 + d = 12

d = 4

Therefore, the explicit formula for the sequence is:

an = a + (n - 1)d

where a = 8 and d = 4.

b. To find the number of tiles in the 13th row, we can plug in n = 13 into the formula:

a13 = 8 + (13 - 1)4

a13 = 8 + 48

a13 = 56

Therefore, there are 56 tiles in the 13th row.

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Answer: a. an= 8+4(n-1)

b. a13=60

Step-by-step explanation:

There are 60 tiles in the 13th row.

a13=4(13)+8

52+8

The fraction of cake left with Bobby is 5/8.

Subtracting fractions include the subtraction of two or more fractions with the same or different denominators. Like fractions can be subtracted directly but for unlike fractions we need to make the denominators same first and then subtract them.

Given that, Bobby has 3/4 of a cake. He shared 1/8 slice with Morgan.

Slice of cake left = 3/4 -1/8

= 6/8 -1/8

= 5/8

Therefore, the fraction of cake left with Bobby is 5/8.

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Tracy's expression makes it easier to see how the areas of the rectangles are related i.e. 1/6 bh.

What is a Rectangle?

The opposite sides of a quadrilateral are equal and parallel, and all the angles are equal. Around us, there are several rectangle-shaped items. Two dimensions, the length and breadth, define each rectangle shape. The width and length of a rectangle are its longer and shorter sides, respectively.

Dimensions of top rectangle :

base = b

height = h

So, area = b × h

Dimensions of bottom rectangle :

base = b/3

height = h/2

So, area = b/3 × h/2 = (b × h) 6

Hence, area of bottom rectangle is  1/6th of that top rectangle.

Devin writes the area of the smaller rectangle as the product 1/3 b • 1/2 h. Tracy rewrites this as 1/6 bh.

Hence, Tracy's expression represents more easier way to see how the rectangle's area is related.

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There are 72 pupils in bertha if the ratio of 1 Bertha to form 1 Florence is 12:20 if there are 120 pupils in Florence.

The problem can be solved by using the concept of ratios and inverse ratios. It involves setting up a proportion based on the given ratio and using algebra to solve for the unknown quantity.

If the ratio of Bertha to Florence is 12:20, then the ratio of Florence to Bertha is 20:12 (inverse ratio).

Let the number of pupils in Bertha be x. Then, the number of pupils in Florence would be:

20/12 * x = 120

Simplifying:

x = 120 * 12/20

x = 72

Therefore, there are 72 pupils in Bertha.

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Claim: The current education system should be reformed to ensure that students of all backgrounds are able to pursue education and reach their full potential.

A claim is a statement made by an individual or organization that seeks to establish a legal right or entitlement to something. It is similar to a demand or request but is used in a legal context.

Points in support of the claim:

1. The current education system is not equitable and fails to provide equal opportunities to all students regardless of their background.

2. Students from low-income families are more likely to attend schools with fewer resources and less rigorous curriculums.

3. These students are often at a disadvantage when it comes to competing for college admissions and other opportunities.

4. Education reform can help to bridge the gap between different socioeconomic backgrounds and ensure that all students have access to quality education.

5. Reforms should focus on providing equal resources and opportunities to all students, regardless of their financial or social backgrounds.

6. This can help to create a more equitable education system, which will benefit all students, regardless of their background.

Conclusion:

Education reform is essential for ensuring that all students, regardless of their background, have access to quality education and the opportunities to reach their full potential. The current education system is not equitable and fails to provide equal opportunities to all students, leaving some at a disadvantage. Therefore, steps must be taken to reform the education system and bridge the gap between different socioeconomic backgrounds. This will help to create a more equitable education system that benefits all students, regardless of their backgrounds.

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So, the inverse of \(A\) is:
\[A^{-1} = \left[\begin{array}{ccc}
a^{-1}_{11} & a^{-1}_{12} & a^{-1}_{13} \\
a^{-1}_{21} & a^{-1}_{22} & a^{-1}_{23} \\
a^{-1}_{31} & a^{-1}_{32} & a^{-1}_{33} \\
\end{array}\right]\]

This is how we can compute the inverse of a matrix using the row reduction method.

To compute \(A^{-1}\) using the row reduction method, we need to first set up an augmented matrix with \(A\) on the left and the identity matrix on the right. Then, we will use elementary row operations to reduce the left side of the matrix to the identity matrix, which will result in the inverse of \(A\) on the right side. Here are the steps:

1. Set up the augmented matrix:
\[\left[\begin{array}{ccc|ccc}
a_{11} & a_{12} & a_{13} & 1 & 0 & 0 \\
a_{21} & a_{22} & a_{23} & 0 & 1 & 0 \\
a_{31} & a_{32} & a_{33} & 0 & 0 & 1 \\
\end{array}\right]\]

2. Use elementary row operations to reduce the left side of the matrix to the identity matrix. For example, we can use the following operations:
- Multiply the first row by a constant to make the first element 1
- Subtract a multiple of the first row from the second and third rows to make the first element of those rows 0
- Repeat these steps for the second and third columns until the left side of the matrix is the identity matrix

3. Once the left side of the matrix is the identity matrix, the right side of the matrix will be the inverse of \(A\):
\[\left[\begin{array}{ccc|ccc}
1 & 0 & 0 & a^{-1}_{11} & a^{-1}_{12} & a^{-1}_{13} \\
0 & 1 & 0 & a^{-1}_{21} & a^{-1}_{22} & a^{-1}_{23} \\
0 & 0 & 1 & a^{-1}_{31} & a^{-1}_{32} & a^{-1}_{33} \\
\end{array}\right]\]

So, the inverse of \(A\) is:
\[A^{-1} = \left[\begin{array}{ccc}
a^{-1}_{11} & a^{-1}_{12} & a^{-1}_{13} \\
a^{-1}_{21} & a^{-1}_{22} & a^{-1}_{23} \\
a^{-1}_{31} & a^{-1}_{32} & a^{-1}_{33} \\
\end{array}\right]\]

This is how we can compute the inverse of a matrix using the row reduction method.

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These are the Critical numbers of the function within the given interval.
Therefore, the answer is \(x=\frac{\pi }{6}, \frac{5 \pi }{6}\).

The Critical numbers of a function are the values of x that make the derivative of the function equal to zero. To find the eritical numbers of the given function, we need to find the derivative of the function and set it equal to zero.

The derivative of the function \(f(x)=-x+\sin (2 x)\) is \(f'(x)=-1+2\cos (2 x)\).

Setting the derivative equal to zero, we get:

\(-1+2\cos (2 x)=0\)

\(\cos (2 x)=\frac{1}{2}\)

Using the inverse cosine function, we get:

\(2 x=\cos ^{-1}\left(\frac{1}{2}\right)\)

\(2 x=\frac{\pi }{3}\) or \(2 x=\frac{5 \pi }{3}\)

Dividing both sides by 2, we get:

\(x=\frac{\pi }{6}\) or \(x=\frac{5 \pi }{6}\)

These are the Critical numbers of the function within the given interval.

Therefore, the answer is \(x=\frac{\pi }{6}, \frac{5 \pi }{6}\).

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