when two or more objective function coefficients are changed simultaneously, further analysis is necessary to determine whether the optimal solution will change.

The n(B) represents the number of blue cards, n(S) represents the number of cards that that have solid or shaded shapes is n(T∩B∩S).

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Evaluating the function in the given values we will see that the correct option is the first one.

g(1) = 2.2966; g(2) = 14.8932

Here we want to evaluate the function below:

g(x) = 1.8x^3 − 0.0034x + 0.5

in x = 1 and x = 2.

To do so, wejust needto replace the variableby the correspondent number, then we will get:

g(1) = 1.8*1^3 − 0.0034*1 + 0.5 = 2.2966

Andthe other value is:

g(2) =1.8*2^3 − 0.0034*2 + 0.5 = 14.8932

Then the correct option is the first one:

2.2966; 14.8932

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One vertical angle is (8x+7) = 115, the other vertical angle is (10x-20) = 115, and the adjacent angle is (2y-15) = 2(64) - 15 = 113.

Vertical angles are a pair οf οppοsite angles fοrmed by the intersectiοn οf twο lines. Since the angles are οppοsite tο each οther, they have the same measure. Therefοre, we can set the expressiοns fοr the twο vertical angles equal tο each οther and sοlve fοr x:

8x + 7 = 10x - 20

Subtracting 8x from both sides:

7 = 2x - 20

Adding 20 to both sides:

27 = 2x

Dividing by 2:

x = 13.5

So one vertical angle is 8x + 7 = 8(13.5) + 7 = 115 and the other vertical angle is 10x - 20 = 10(13.5) - 20 = 115.

To find y, we can use the adjacent angle (2y - 15) and set it equal to one of the vertical angles. Let's use 8x + 7:

8x + 7 = 2y - 15

Adding 15 to both sides:

8x + 22 = 2y

Dividing by 2:

4x + 11 = y

So the value of y is 4x + 11, which we can now calculate by substituting x = 13.5:

y = 4(13.5) + 11 = 64

Therefore, one vertical angle is (8x+7) = 115, the other vertical angle is (10x-20) = 115, and the adjacent angle is (2y-15) = 2(64) - 15 = 113.

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To calculate a margin of error, you multiply the standard error by the number of standard deviations from the mean, which is also called the critical value.

A margin of error is a measure of the uncertainty or potential error associated with a statistical estimate or measurement. It is commonly used in survey research and other statistical studies to indicate the degree of confidence one can have in the findings based on the sample data.

The margin of error is typically expressed as a range of values that is expected to include the true population parameter with a certain level of confidence. The margin of error is influenced by several factors, including the sample size, the variability of the data, and the level of confidence desired.

The standard error is another important concept in statistics that is closely related to the margin of error. It measures the variability of the sample mean or proportion and represents the standard deviation of the sampling distribution.

The critical value is the number of standard deviations from the mean that corresponds to a certain level of confidence in the sampling distribution. For example, a 95% confidence level corresponds to a critical value of 1.96 standard deviations from the mean in a normal distribution.

In summary, the margin of error is a measure of the precision and reliability of a statistical estimate or measurement, and it is calculated by multiplying the standard error by the number of standard deviations from the mean (the critical value) based on the desired level of confidence.

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An empty set is a set that does not contain any element in itself.

A set is a well defined organized form of particular things. For example, the set of all natural number, the set of student with more than 90 percent marks in the college, the set of people with black eyes in a class. The particular things in the set are called elements of the set.

Now, an empty set is any set that does not contains any element inside it. For example, if there is no one with the black eyes in the room and we want to make a set of people with black eyes then there will no element in such set and that set will be called an empty set.

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We know that a parabola is defined as the set of all points that are equidistant from the focus and the directrix. Let (x, y) be an arbitrary point on the parabola. Then, the distance from the point to the focus is given by:

√[(x - 4)² + (y + 2)²]

The distance from the point to the directrix is given by the absolute value of the difference between the y-coordinates:

| y - (-3) | = | y + 3 |

Since the point is equidistant from the focus and directrix, we can set these distances equal to each other:

√[(x - 4)² + (y + 2)²] = | y + 3 |

We can square both sides to eliminate the absolute value:

(x - 4)² + (y + 2)² = (y + 3)²

Expanding the right side and simplifying:

x² - 8x + 16 + y² + 4y + 4 = y² + 6y + 9

Simplifying further and collecting like terms:

x² - 8x + y² + 4y = -11

Completing the square for the x and y terms, we add and subtract the square of half the coefficient of x and y respectively:

(x - 4)² - 16 + (y + 2)² - 4 = -11

(x - 4)² + (y + 2)² = 9

Therefore, the equation of the parabola with focus (4, -2) and directrix y = -3 is:

(x - 4)² + (y + 2)² = 9.

The radius of the smallest circle is 5 units

Three concentric circles are given in which the two largest circles have radii of 12 and 13. It is given that area of the ring between the two largest circles equals the area of the smallest circle and we need to find the radius of the smallest circle.

Let the radius of the smallest circle be r.

Area of smallest circle = πr²

Area of outermost circle = π(13)² = 169π

Area of middle circle = π(12)² = 144π

Area of the ring between the two largest circles = 169π - 144π = 25π

Area of smallest circle = Area of the ring between the two largest circles

                                          πr² = 25π

                                            r² = 25

                                            r = 5

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Answer:

B.

Step-by-step explanation:

Answer: B

Step-by-step explanation:

Answer: 3.9%

Step-by-step explanation:

We can use the formula for continuous compounding to solve this problem:

A = P * e^(r*t)

where A is the final amount, P is the initial principal, r is the interest rate, and t is the time in years.

In this case, we know that P = $1,800, A = $2,400, and t = 6 years. We want to solve for r.

First, we can rearrange the formula to isolate r:

r = (ln(A/P)) / t

where ln represents the natural logarithm.

Plugging in the values we know, we get:

r = (ln(2400/1800)) / 6

r = 0.0392

The correct answer for the required values is 1825.5, 1,812.5, 165.2, 1,690, 1,935, and 245 respectively.

To find the mean, median, standard deviation, first quartile, third quartile, and interquartile range (IQR) of the full-time equivalent number of students (FTES) at Lake Tahoe Community College for the given years, we can solve it using the following formulas for various aspects:

Mean: μ = (Σx)/n

Median: Middle value of the data set

Standard deviation: σ = sqrt[Σ(x-μ)²/n]

First quartile (Q1): Median of the lower half of the data set

Third quartile (Q3): Median of the upper half of the data set

IQR: Q3 - Q1

Year         FTES

2005-06  1,585

2006-07  1,690

2007-08  1,735

2008-09  1,935

2009-10   2,021

2010-11         1,890

1. μ = (Σx)/n = (1,585 + 1,690 + 1,735 + 1,935 + 2,021 + 1,890)/6 = 1,825.2

Therefore the mean is 1,825.2

2. To find the median, arrange the data in ascending order:

1,585, 1,690, 1,735, 1,890, 1,935, 2,021

Since there are an even number of values, the median is the average of the two middle values, which are 1,812.5 and 1,812.5. So, the median is:

Median = (1,812.5 + 1,812.5)/2 = 1,812.5

3. σ = sqrt[Σ(x-μ)²/n]

= sqrt[((1,585-1,825.2)² + (1,690-1,825.2)² + (1,735-1,825.2)² + (1,935-1,825.2)² + (2,021-1,825.2)² + (1,890-1,825.2)²)/6]

= 165.2

4. To find the first quartile (Q1), find the median of the lower half of the data set, which consists of the values 1,585, 1,690, and 1,735.

Since there are an odd number of values, the median is the middle value, which is 1,690. Therefore, Q1 = 1,690.

5. To find the third quartile (Q3), we need to find the median of the upper half of the data set, which consists of the values 1,890, 1,935, and 2,021.

Since there are an odd number of values, the median is the middle value, which is 1,935. Therefore, Q3 = 1,935

6. IQR = Q3 - Q1

Substituting the calculated values in the equation

IQR = 1935 - 1690

= 245

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No, In order for a set W is not a subspace of[tex]R^3[/tex]because it does not satisfy the closure property required for a subspace, which is closure under addition and scalar multiplication.

In order for a set W to be a subspace of[tex]R^3[/tex], it must satisfy two requirements. The first requirement is closure under addition, which means that when two elements of W are added together, the resulting sum must also be an element of W. The second requirement is closure under scalar multiplication, which means that when an element of W is multiplied by a scalar, the resulting product must also be an element of W. However, the set W defined in the question does not satisfy either of these requirements, which means it is not a subspace of[tex]R^3[/tex] with the standard operations. For example, when two elements of W are added together, the resulting sum does not necessarily satisfy the constraints of the set W. This means that the set W is not closed under addition, and therefore is not a subspace of [tex]R^3.[/tex] Furthermore, when an element of W is multiplied by a scalar, the resulting product does not necessarily satisfy the constraints of the set W. This means that the set W is not closed under scalar multiplication, and therefore is not a subspace of[tex]R^3.[/tex]

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According to the information, the number of the riddle is 583.

To find the number of the riddle we must follow the clues provided by the riddle. First we have a 5 in the ones space. Later, we have in the tens the result of the sum of 5 plus 3, so it would be 8 in the tens place. Finally, in the hundreds place we have two numbers less than five, that is, 5 - 2 = 3, so it would be 3 in the hundreds place.

According to the information above, the number of the puzzle would be 583.

Note: This question is incomplete. Here is the complete information:

I have a 5 in my ones place. The digit in my tens place is 3 plus the digit in my ones place. The digit in my hundreds place is 2 less than the digit in my ones place. What number am I?

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Cost of lemon to fill four 9-inch lemon pies is $7.84

The division is one of the basic arithmetic operations in math in which a larger number is broken down into smaller groups having the same number of items. Division is the reciprocal of multiplication.

The store is selling lemons at $0.49 each.

Each lemon yields about 2 tablespoons of juice.

Four 9-inch lemon pies, each requiring half a cup of lemon juice.

Amount of juice to be filled in four 9-inch pies = 4*0.5

                                                                             = 2 cups of juice.

For two cups of juice = 2*16=32 tablespoons.

For 32 tablespoon of juices we require, 32/2=16 lemons.

Cost of 16 lemons = 16*0.49

                              =$7.84.

Hence, cost of lemon to fill four 9-inch lemon pies is $7.84.

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The side length of each piece is given as follows:

[tex]20\sqrt{2}[/tex] inches.

The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.

For this problem, the parameters are given as follows:

Hence the side length of each piece is obtained as follows:

x² + x² = 40²

2x² = 40²

x² = 800

x = sqrt(2 x 400)

[tex]x = 20\sqrt{2}[/tex]

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Gulf Coast Electronics should order 13,000 capacitors from Boston Components, 50,000 capacitors from Able Controls, and 14,000 capacitors from Lyshenko Industries to obtain 77,000 acceptable-quality reservoir capacitors at a minimum cost of $204,425.

To minimize the total cost of obtaining 75,000 acceptable-quality reservoir capacitors, we can formulate the following linear program:

Objective function:

Minimize: 2.35B + 2.6A + 2.85L

Subject to:

B + A + L = 77000 (total number of capacitors required)

B ≥ 0.1A (at least 10% of the volume given to Able Controls should be awarded to Boston Components)

L ≥ 29500 (at least 29,500 capacitors should be ordered from Lyshenko)

B ≤ 32500 (maximum of 32,500 capacitors should be ordered from Boston Components)

A ≤ 50000 (maximum of 50,000 capacitors should be ordered from Able Controls)

L ≤ 48500 (maximum of 48,500 capacitors should be ordered from Lyshenko)

0 ≤ B, A, L (non-negativity constraints)

Solving this linear program using a linear programming software, we get:

B = 13000

A = 50000

L = 14000

Correct Question :

Gulf Coast Electronics is ready to award contracts to suppliers for providing reservoir capacitors for use in its electronic devices. For the past several years, Gulf Coast Electronics has relied on two suppliers for its reservoir capacitors: Able Controls and Lyshenko Industries. A new firm, Boston Components, has inquired into the possibility of providing a portion of the reservoir capacitors needed by Gulf Coast. The quality of products provided by Lyshenko Industries has been extremely high; in fact, only 0.4% of capacitors provided by Lyshenko had to be discarded because of quality problems. Able Controls has also had a high quality level historically, producing an average of only 1% unacceptable capacitors. Because Gulf Coast Electronics has had no experience with Boston Components, it estimated Boston Components’ defective rate to be 12%. Gulf Coast would like to determine how many reservoir capacitors should be ordered from each firm to obtain 77000 acceptable-quality capacitors to use in its electronic devices. To ensure that Boston Components will receive some of the contract, management specified that the volume of reservoir capacitors awarded to Boston Components must be at least 10% of the volume given to Able Controls. In addition, the total volume assigned to Boston Components, Able Controls, and Lyshenko Industries should not exceed 32500, 50000, and 48500 capacitors, respectively. Because of Gulf Coast’s long-term relationship with Lyshenko Industries, management also specified that at least 29500 reports should be ordered from Lyshenko. The cost per capacitor is $2.35 for Boston Components, $2.6 for Able Controls, and $2.85 for Lyshenko Industries.

Formulate and solve a linear program for determining how many reservoir capacitors should be ordered from each supplier to minimize the total cost of obtaining 75,000 acceptable-quality reservoir capacitors. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. Enter "0" if your answer is zero. If the constant is "1" it must be entered in the box.

Let B = number of capacitors ordered from Boston Components

Let A = number of capacitors ordered from Able Controls

Let L = number of capacitors ordered from Lyshenko Industries

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The equation of the linear function represented by the table is y=-x+9. Therefore, option A is the correct answer.

The general equation of a straight line is y=mx+c, where m is the gradient, and y = c is the value where the line cuts the y-axis. This number c is called the intercept on the y-axis.

The given table is

  x        y

 -5       14

 -2       11

  1        8

  4       5

From the given table, the coordinate points are (-5, 14) and (-2, 11)

Here, slope(m) =(11-14)/(-2+5)

= -1

Substitute m=-1 and (x, y)=(-5, 14) in y=mx+c, we get

14=-1(-5)+c

c=9

So, the equation is y=-x+9

Therefore, option A is the correct answer.

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Mancy's monthly earning of $3,000, rent of r, feeding cost of r/2, and utilities cost of (r - 100), indicates that her transportation cost is obtained by the function;

t = $3,100 - 2.5·r

A function defines the method by which an input variable is mapped unto an output variable.

The amount Nancy earns per month = $3,000

The amount she pays as rent = 2 × The cost of her food

Amount spent on utilities and personal items = Her rent - 100

the amount she spends on transportation = The amount remaining from her salary after the other expenditures

The cost of her rent = r

The cost of her transportation = t

Therefore;

Cost of her rent, r = 2 × The food cost

The food cost = r/2

Utilities and personal items cost = r - 100

Transportation cost, t = Monthly earning - Rent - Food - Utilities

t = 3000 - r - r/2 - (r - 100)

The function that models her transportation cost is therefore;

t = $3,100 - 2.5·r

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The graph of the above equation is given as attached. The two points on the graphs are: (0, -2) and (3, -4)

A graph is a visual representation of data or mathematical functions, typically shown on a coordinate system consisting of an x-axis (horizontal) and a y-axis (vertical).

Graphs are relevant in math because they allow us to easily see and understand relationships and patterns in data, equations, and functions.

In particular, graphs help us to:

Visualize and interpret data - Graphs make it easier to identify trends, patterns, and relationships in data sets, allowing us to draw meaningful conclusions and make informed decisions.

Communicate mathematical ideas - Graphs provide a clear and concise way to communicate mathematical ideas and concepts, making them useful for presentations and publications.

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Answer: 11/3

Step-by-step explanation: 2 1/5= 11/5

11/5 ÷ 3/5= 11/3

(keep change and flip) change the division sign to multiplication and the 3/5 would become 5/3

write it down visually

Answer: 3 2/3

In order to solve this we must turn our fractions into whole numbers.

2 1/5 as a whole number is 2.20

3/5 as a whole number is .6

Next we must divide our two whole numbers into one another.

2.20/.6 is 3.66666667

666666667 as a fraction is 2/3.

This means that 3/5 can fit into 2 1/5 3 whole times and 2/3 of a whole. This means you will need 3 whole boxes and 2/3 of a box.

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

Answer: #1 108 #2 526. #3 512

Step-by-step explanation:

6*6=36 +12*6 = 108

22*3=66 + 20*23= 460 460+66=526

16*16=256 + 32*8=256 256+256=512

Answer:

1)  108 cm²

2)  906 ft²

3)  512 m²

Step-by-step explanation:

To calculate the area of each given composite figure, divide the figure into two rectangles and sum the area of the two rectangles.

[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]

Separate the figure into two rectangles by drawing a horizontal line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 6\cdot 6+12 \cdot 6\\&=36+72\\&=108\; \sf cm^2\end{aligned}[/tex]

Separate the figure into two rectangles by drawing a vertical line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 22\cdot23 + 20\cdot 20\\&=506+400\\&=906\; \sf ft^2\end{aligned}[/tex]

Separate the figure into two rectangles by drawing a horizontal line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 16\cdot16 + 32\cdot 8\\&=256+256\\&=512\; \sf m^2\end{aligned}[/tex]

Each force in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force would be in 32.54 degrees.

The given data includes the magnitudes and Cartesian vector expressions of three forces: F1 = 350 N, F2 = 100 N, and F3 = 250 N.

To find the Cartesian vector expression of F1, the magnitude is multiplied by the sine and cosine components of the vector's direction angle, resulting in (224.97j + 268.11k) N.

To find the Cartesian vector expression of F2, the magnitude is multiplied by the cosine components of the vector's direction angles, resulting in (70.7i + 50j - 50k) N.

To find the Cartesian vector expression of F3, the magnitude is multiplied by the cosine components of the vector's direction angles, resulting in (125i - 176.77j + 125k) N.

The resultant force vector is found by adding the Cartesian vector expressions of F1, F2, and F3, resulting in (195.7i + 98.2j + 343.11k) N.

The magnitude of the resultant force vector is found by taking the square root of the sum of the squares of its x, y, and z components, resulting in 407.02 N.

The direction angle of the resultant force vector along the x-axis is found by taking the inverse cosine of the ratio of its x-component to its magnitude, resulting in 61.26 degrees.

The direction angle of the resultant force vector along the y-axis is found by taking the inverse cosine of the ratio of its y-component to its magnitude, resulting in 76.04 degrees.

The direction angle of the resultant force vector along the z-axis is found by taking the inverse cosine of the ratio of its z-component to its magnitude, resulting in 32.54 degrees.

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The question is -

The cables attached to the screw eye are subjected to the three forces shown. Express each form in a Cartesian vector and determine the magnitude and coordinate direction angles of the resultant force

In hyperbola, If a is negative, left does the parabola open.

What is an example of a hyperbola?

A hyperbola is a collection of points in a plane whose distances from two fixed points, referred to as its foci (plural of focus), varies by a constant amount. For instance, the picture displays a hyperbola with foci at (-3,0) and (3,0). The point at illustrates its constant distance as being 4. (-3,-2.5).

                                Open and closed hyperbolas are the two different varieties. A hyperbola widens on the x-axis at one end and the y-axis at the other.

The equation below describes a parabola.

                  x= av

If a is negative, left does the parabola open.

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To verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.

To prove that,

[tex]limn → ∞ \frac{2n+1}{5n+4} = \frac{2}{5} [/tex]

we use the limit definition then we need to prove the statement

[tex]∀ε>0,∃N∈Ns.t.n>N⟹ \frac{2n+1}{5n + 4} [/tex]

we already have,

[tex]∣ \frac{2n+1}{5n+4}− \frac{2}{5} ∣=∣− \frac{3}{5(5n+4)} ∣= \frac{3}{5(5n+4)} [/tex]

For every positive ε, we choose that N=⌈1/ε⌉, where we are using the ceiling function to locate the smallest integer number larger than the fraction 1/ε. This gives us a natural number, proving that the limit's quantified statement definition is accurate:.

[tex]∀ε>0,n>N⟹n> \frac{1}{ε} \\ ⟹1n<ε \\ ⟹ \frac{3}{5(5n+4)} < \frac{3}{25n} <ε \\ ⟹∣ \frac{2n+1}{5n+4} − \frac{2}{5} ∣<ε.[/tex]

Hence, proved.

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Answer:

Step-by-step explanation:

If the stamp's value doubles every decade, then the value of the stamp after x decades will be $40 multiplied by 2 raised to the power of x.

Therefore, the equation for the function that approximates the stamp's value x decades from now is:

V(x) = $40 x 2^x

Where V(x) represents the stamp's value x decades from now.

The required  the distance from point A to point B is approximately 855.31 feet.

The angle of elevation is the angle created between the line of sight and the horizontal. The angle created is an angle of elevation if the line of sight is upward from the horizontal line.

Let's call the distance from point A to the lighthouse "x" and the height of the lighthouse "h" (h = 111 feet). We can use the tangent function to relate the angle of elevation to the distance and height:

tan(5°) = h/x

Solving for x, we get:

x = h/tan(5°) ≈ 1268 feet

Now let's call the distance from point B to the lighthouse "y". We can use the same tangent function to relate the angle of elevation from point B:

tan(15°) = h/y

Solving for y, we get:

y = h/tan(15°) ≈ 412.69 feet

The distance between points A and B is just the difference between x and y:

1268 - 412.69 ≈ 855.31 feet

So the distance from point A to point B is approximately 855.31 feet.

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Complete question:

A boat is heading towards a lighthouse, whose beacon-light is 111 feet above the water. From point A, the boat's crew measures the angle of elevation to the beacon, 5º, before they draw closer. They measure the angle of elevation a second time from point B at some later time to be 15°. Find the distance from point A to point B. Round your answer to the nearest tenth of a foot if necessary.​

The x can take any value less than or equal to -20/21 and satisfy the given inequality. The solution to the inequality is x ≤ -20/21.

The inequality -3(9x+20) ≥ 15x-20 is solved using various properties of inequality such as the distributive property, addition and subtraction properties, and division property. The solution is x ≤ -20/21. Each step is justified with the appropriate property of inequality. The final answer means that x can take any value less than or equal to -20/21 and satisfy the given inequality.

-3(9x+20) ≥ 15x-20 (Given inequality)

-27x - 60 ≥ 15x - 20 (Distributed -3 on the left side)

-42x - 60 ≥ -20 (Subtracted 15x from both sides)

-42x ≥ 40 (Added 60 to both sides)

x ≤ -20/21 (Divided by -42, and flipped the inequality)

Properties of inequality used:

Given inequality

Distributive property of multiplication over addition

Subtraction property of inequality

Addition property of inequality

Division property of inequality (flipped the inequality since dividing by a negative number)

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Answer: Without the coupon, Mariana would have to pay $6.25 per pound of chicken, so for 5 pounds, the cost would be:

5 * $6.25 = $31.25

With the coupon, she would receive a discount of $3.25, so the final cost would be:

$31.25 - $3.25 = $28.00

To write an expression for the cost to buy p pounds of chicken, we can use the formula:

C = 6.25p - 3.25

where C is the cost in dollars and p is the number of pounds of chicken purchased. Note that the expression assumes that at least one pound is purchased, as indicated in the question.

Step-by-step explanation:

In order to find the pivot columns of matrix A, the Gaussian elimination can be usd to reduce it to row echelon form. The resultant matrix is

| 2 3 -1 19 2 |

| 0 3 1 11 7 |

| 0 0 0 1 -2 |

The pivot columns are the columns that correspond to the pivot positions in the row echelon form. In this case, the first and second columns are pivot columns.

To express each of the other columns as linear combinations of the pivot columns, we can use the following procedure:

For each non-pivot column, write its coefficients as a column vector b.

Create a matrix B by placing the column vectors b next to each other.

Solve the system Ax = b using back substitution.

For the first non-pivot column [ -1, 2, -1 ]^T, we have:

| 2 3 | | x1 | | -1 |

| 0 3 | * | x2 | = | 2 |

| 0 0 | | x3 | | -1 |

From the third row, we get x3 = -1. Substituting this into the second row, we get 3x2 = 2, or x2 = 2/3. Substituting x2 and x3 into the first row, we get 2x1 + 3(2/3) - 1(-1) = -1, or x1 = -3. Therefore, the first non-pivot column can be expressed as a linear combination of the pivot columns as:

[-1, 2, -1]^T = -3[2, 0, 2]^T + 2/3[3, 3, -1]^T

For the second non-pivot column [19, -8, 15]^T, we have:

| 2 3 | | x1 | | 19 |

| 0 3 | * | x2 | = | -8 |

| 0 0 | | x3 | | 15 |

From the third row, we get x3 = 15. Substituting this into the second row, we get 3x2 = -8 - 3(15), or x2 = -17. Substituting x2 and x3 into the first row, we get 2x1 + 3(-17) - 1(15) = 19, or x1 = 23. Therefore, the second non-pivot column can be expressed as a linear combination of the pivot columns as:

[19, -8, 15]^T = 23[2, 0, 2]^T - 17[3, 3, -1]^T

For the third non-pivot column [2, 5, -14]^T, we have:

| 2 3 | | x1 | | 2 |

| 0 3 | * | x2 | = | 5 |

| 0 0 | | x3 | | -14 |

From the third row, we get x3 = -14. Substituting this into the second row, we get 3x2 = 5 + 3(14), or x2 = 47. Substituting x2 and x3 into the first row, we get 2x1 + 3(47) - 1(-14) = 2, or x1 = -60.

Therefore, the values we get are  x1 = -60, x2 = 47, x3 = -14 and the resultant matrix will be

| 2 3 -1 19 2 |

| 0 3 1 11 7 |

| 0 0 0 1 -2 |

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a. The feasible region is a triangle in the x1-x2 plane with vertices at (0,0), (1,0), and (2/3,1/3).

b. The slack variables x3 and x4 are zero on the boundaries of the feasible region.

c. The maximum value of x is 2/3 and is attained at x1 = 2/3 and x2 = 1/3.

d. The requirement space is the region satisfying the given constraints, and feasibility means that a solution exists within this region.

For a. To sketch the feasible region, we first find the boundary curves of the region by setting each constraint equal to zero. The resulting curves are x2 = 0, -x1 + 3x2 = 0, and -3x1 + 2x2 = 0. The feasible region is the area in the x1-x2 plane that satisfies all the constraints, which is a triangle with vertices at (0,0), (1,0), and (2/3,1/3).

For b. The slack variables x3 and x4 are introduced to convert the inequality constraints to equations. They represent the amount by which the left-hand side of each constraint is less than the right-hand side. When a slack variable is zero, it means that the corresponding inequality is an equality and the point lies on the boundary of the feasible region. In this case, x3 = 0 corresponds to the line x2 = 0, and x4 = 0 corresponds to the line -x1 + 3x2 = 0.

For c. To solve the problem geometrically, we find the corner points of the feasible region and evaluate the objective function at each point. The maximum value of x occurs at the point (2/3,1/3), where x = 2/3.

For d. The requirement space is the region of the x1-x2 plane that satisfies all the constraints. Feasibility means that there exists a solution to the problem within this region. The feasible region is a subset of the requirement space and is defined by the intersection of the constraint curves.

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Answer:

What are some of the options?

Step-by-step explanation: