Write the dual of following problems:

(a) Maximize

Z = 7X1 + 5X2
Subject to:

X1 + 2X2 ≤ 6

4X1 + 3X2 ≤ 12

X1, X2 ≥ 0

(b) Maximize

Z= 3X1 + 4X2

Subject to:

5X1 + 4X2 ≤ 200

3X1 + 5X2 ≤ 150

8X1 + 4X2 ≥ 80

X1, X2 ≥ 0

[tex] \boxed{\sf Option \: A\text{:} \: \dfrac{1 + \sqrt{-32}}{4} = \dfrac{1}{4} + \sqrt{2}i} [/tex]

[tex] \\ \\ [/tex]

We are asked to express the given number in the form a + ib, also known as the algebraic form.

[tex] \\ \\ [/tex]

[tex] \\ \\ [/tex]

[tex] \sf z = a + ib \\[/tex]

Where:

✰ a and b are real numbers and are the complex number's real part and imaginary part, respectively.

[tex] \\ \\ [/tex]

✰ i is the imaginary unit number and can be expressed as follows:

[tex] \\ \sf i= \sqrt{-1}[/tex]

[tex] \\ \\ [/tex]

[tex] \\ \\ [/tex]

[tex] \sf z = \dfrac{1 + \sqrt{-32}}{4} = \dfrac{1 + \sqrt{32 \times (-1)}}{4} \\ \\ \\ \sf \diamond \: We \: know \: that \: \sqrt{ab} = \sqrt{a} \times \sqrt{b}. \\ \\ \implies \dfrac{1 + \sqrt{32 \times (-1)}}{4} = \dfrac{1 + \sqrt{32}\times \sqrt{-1}}{4}[/tex]

[tex] \\ \\ \sf \implies \dfrac{1 + \sqrt{32}\times \sqrt{-1}}{4} = \dfrac{1 + \sqrt{16 \times 2} \times \sqrt{-1}}{4} = \dfrac{1 + 4\sqrt{2} \times \sqrt{-1}}{4}[/tex]

[tex] \\ \\[/tex]

[tex] \diamond \: \sf Substitute \: \sqrt{-1} = i. \\ \\ \implies \sf \dfrac{1 + 4\sqrt{2}\times \sqrt{-1}}{4} = \dfrac{1 + 4\sqrt{2} \times i}{4} = \dfrac{1 + 4i\sqrt{2}}{4} [/tex]

[tex] \\ \\ \: \sf \diamond \: Simplify \: knowing \: that \: \dfrac{a + b}{c} = \dfrac{a}{c}+\dfrac{b}{c}. [/tex]

[tex] \\ \\ \\ \implies \: \sf \dfrac{1 + 4i\sqrt{2}}{4} = \dfrac{1}{4} + \dfrac{4i\sqrt{2}}{4} = \boxed{\sf \dfrac{1}{4} + \sqrt{2}i} [/tex]

[tex] \\ [/tex]

Therefore, the first option is correct.

[tex] \\ \\ \\ [/tex]

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The missing statement from his process;

RM = RL and MA = LA

Radii of the constructed arc are some length.

Angle bisectors are lines that cut across the angle under consideration. The term "bisect" means to divide into two equal portions. The considered angle is therefore divided in half by the bisected halves.

Given:

Oleg is writing proof of the construction of an angle bisector.

The missing statement from his process;

RM = RL and MA = LA

Radii of the constructed arc are some length.

From the reflexive property;

RA ≅ RA.

And ΔRMA ≅ ΔRLA, by SSS criterion.

And corresponding angles of congruent angles

∠MRA ≅ ∠LRA.

Therefore, ∠MRA ≅ ∠LRA.

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The solution is, the ratio of the number of students who went to the aquarium to the total number of students is 1:9.

The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.

here, we have,

24 students went to the zoo and 3 students went to the aquarium.

i.e. we get,

24 students went to the zoo

3 went to the aquarium

so, total no. of students = 24+ 3

                                       = 27

then, we have,

the ratio of the number of students who went to the aquarium to the total number of students is = 3 : 27

                                    = 1: 9

Hence, The solution is, the ratio of the number of students who went to the aquarium to the total number of students is 1:9.

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The formula for the area of a circle is A = πr^2, where r is the radius.

The area of the circle with a radius of 3.5 centimeters is approximately 38.5 square centimeters.

Substituting r = 3.5 centimeters and using π ≈ 3.14, we get:

A = 3.14 × (3.5)^2

A = 38.465 square centimeters (rounded to the nearest tenth)

Consequently, the radius of the circle is 3.5 centimetres, and its area is roughly 38.5 square centimeters.

The area of a circle is the measure of the region enclosed by the circle in a two-dimensional space. It is the amount of space inside the circle and is typically measured in square units, such as square centimeters (cm²) or square meters (m²). The formula to calculate the area of a circle is:

Area = π x radius²

Where π (pi) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter, which is approximately 3.14159, and radius is the distance from the center of the circle to any point on the circumference. The area formula can also be expressed in terms of the diameter:

Area = π x (diameter/2)²

This formula tells us that the area of a circle depends on the length of its radius or diameter, and it allows us to calculate the area of any circle given either of these two measurements.

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1) The Area of the first compound shape is given as  = 52cm

2) The area of the second shape is given as = 47cm

Note that the method for deriving this area of the shapes is by dividing them into "regular" shapes as shown in the attached images.

A compound form is created by combining simple shapes. It's also known as a "composite" or "complex" form.

For Shape A:

We have a triangle and a rectangle when the compound shape is de-regularized.

The dimensions of the triangle are:

Base 8cm

Height 3cm

Since the area of a triangle is given as:

1/2 B * H,

The area of the triangle is = 1/2 * 8 * 3cm

= 12cm

For the rectangle, the area of a rectangle is given as L * B. Since

L = 8cm and

B = 5cm

The area = 8* 5 = 40cm

The total area of Shape A = 40cm + 12cm

= 52cm

For Shape B:

We have two rectangles and one right-angled triangle when the shape is "converted".

The dimensions of rectangle 1 are:

L = 3

B = 2

Area = 6

The dimensions of rectangle 2  =
L = 8cm
B = 4cm

Area = 32cm

The dimension of the Right angled triangle is:

Base = 6cm
Height = 3cm (that is 7-4)

Thus:

The area of the triangle is:

1/2 * 6* 3
= 9cm

Thus the total area of the second shape is:

= 32 + 6 + 9

= 47cm

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Assuming the distribution is bell-shaped, the estimated standard deviation is 10.

Since the range is 60 and the distribution is bell shaped the N or Number will be 6

Thus, to calculate the standard deviation:

60/6 = 10

The standard deviation is a statistic that expresses the degree of variation or dispersion among a set of values. A low standard deviation suggests that values are typically close to the set's mean, whereas a high standard deviation suggests that values are dispersed over a wider range.

None of the standard deviations are "good" or "bad.". They serve as gauges of your data's dispersion. Sometimes, when using ratings scales, we want a wide spread because it shows that the range of the group we are rating is covered by our questions and ratings.

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The question which could not be answered based solely on the information in this data set is How many registered republicans voted for Gary Johnson in the 2016 presidential election? So, correct answer is option (d).

We have a large data consists of information on all registered republicans in the United States. The information of voters which present in data are Name, Gender, Age, Home address, Whether they voted or not in 2016 presidential election. We have to determine the question which answer is not possible on basis of this data. Let's consider the options one by one.

a) Here, gender and voters who vote or not in 2016 election, so it is easy to determine the percent of registered republican voters are male.

b) Address details and the information of voters who vote or not, helps to determine the state which had the most registered republican voters vote in the 2016 election.

c) The Age and registered voter information that he vote or not is used to determine the average age of registered republican voters.

d) Now, there is no information regarding the voters choice for voting their votes. So, it not determined that how many voters voted for Gary Johnson in the 2016 presidential election. This is write choice here.

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Answer:

Below

Step-by-step explanation:

For this , I will assume  i = [tex]\sqrt{-1}[/tex]

i^{11} + i^{16} + i^{21} + i^{26} + i^{31}  =

-  i         +1        +i            -1        - i       =     - i  

The directional derivative of f(x, y, z) = [tex]xy+z^{3}[/tex] at the point (2, 3, 1) in the direction of a vector making an angle of 3π/4 with the gradient of f at that point is √2 (3cos(3π/4) + 2sin(3π/4)).

To find the directional derivative of f(x, y, z) = xy + z^3 at the point (2, 3, 1) in the direction of a vector making an angle of 3π/4 with the gradient of f at that point, we need to calculate the gradient of f at that point and then use it to find the dot product with the given direction vector.

The gradient of f at the point (2, 3, 1) is given by ∇f(2, 3, 1) = (y, x, 3z^2) = (3, 2, 3).

The direction vector in the direction of 3π/4 with the gradient can be found as follows:

Let [tex]d = (cos(3\pi /4), sin(3\pi /4), 0)[/tex]

The directional derivative of f in the direction d at the point (2, 3, 1) is given by

∇f(2, 3, 1) . d = [tex](3, 2,3).(cos(3\pi /4), sin(3\pi /4),0)[/tex]

= [tex]3cos(3\pi /4)+2sin(3\pi /4)[/tex]

= √2 (3cos(3π/4) + 2sin(3π/4))

So the directional derivative of f in the direction of a vector making an angle of 3π/4 with the gradient of f at (2, 3, 1) is √2 (3cos(3π/4) + 2sin(3π/4)).

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The amount freedome ride charge more than Envo car in d miles is $20.05

word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math. If you've ever taken a math class, you've probably solved a word problem.

The total charge for Enzo car =$50.50 +8¢ = $50.50 + $0.08 = $50.58

The total charge for Freedom = $70.50 + 12¢

= $70.50 + $0.12

= $70.62

therefore in d miles ;

Enzo will charge $50.58d

freedome will charge

$70.62d

therefore the amount freedome will charge more than Enzo car is $70.62 - $50.58

= $20.05

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Answer:12.90

Step-by-step explanation:

The average number of likes will be 16.5. the correct option is D.

Mean is defined as the ratio of the sum of the number of data sets to the total number of data. The mean is calculated by adding all the data points and dividing it by the total number of the data.

Given that Bethany has a photo website where she places photos that she has taken so others can see them. Below is how many likes each of her current photos was given: 10, 13, 15, 15, 17, 29.

The average will be calculated as:-

Mean = ( 10 + 13 + 15 + 15 + 17 + 29 ) / 6

Mean = 99 / 6

Mean = 16.5

Therefore, the average number of likes will be 16.5. the correct option is D.

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In both cases, pulley C is the fastest one since its radius is the smallest one.

In this problem we find two cases of power transmition, the first example consists in a rigid transmition system formed only by pulleys and the second example consisting in a flexible transmition system formed by belts and pulleys.

Ideally, power is conserved throughout every system and rotational velocity of the pulley is inversely proportional to its radius. We need to determine does rotate faster in each case.

n ∝ 1 / R

Thus, we find the following results:

Case 1: Pulley C is the fastest pulley since it has the smallest radius.

Case 2: Pulley C is the fastest pulley since its has the smallest radius.

There is no additional information (i.e. any angular speed of pulley, geometric dimensions in the second case) that may allows us to determine whether if rigid transmition system or a flexible transmition system is faster.

The representation only allows us to determine what pulley is the fastest within a system.

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The number of bags of potatoes and carrots did the farmer sell in all is 75 bags.

A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

Given that, a farmer is selling  potatoes and carrots into bags, 3/5 were bags of potatoes and the rest were bags of carrots. He sold 45 bags of potatoes and 3/4 of the bags of carrots, and had 1/4 of the bags left.

The bags of potato = 3x/5

The bags of carrot = 2x/5

Therefore, according to question,

x-45-3x/5×2x/5 = x/4

7x/10-x/4 = 45

18x/40 = 45

x = 45×40/18

x = 100

Total bags sold = 100×2/5+3/5+45 = 75

Hence, the number of bags of potatoes and carrots did the farmer sell in all is 75 bags.

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The solution is,  the answers to f(g(5)), g(f(78)), and h(g(f(2))) are 8, 60, and 83 respectively.

Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable.

here, we have,

Given the expressions, f(x)=√x+5, g(x)=3x-7, and h(x)=5x, the answer to f(g(5)) is 8.

This is because when g(5) is substituted into f(x) we get f(g(5))=f(3x-7)=√(3x-7)+5. When 5 is substituted for x, this simplifies to f(g(5))=√(-2)+5 which equals 8.

The answer to g(f(78)) is 60. This is because when 78 is substituted into f(x) we get f(78)=√78+5 which simplifies to f(78)=9. When 9 is substituted into g(x) we get g(f(78))=3x-7 which simplifies to g(f(78))=3(9)-7 which equals 60.

Finally, the answer to h(g(f(2))) is 83. This is because when 2 is substituted into f(x) we get f(2)=√2+5 which simplifies to f(2)=3. When 3 is substituted into g(x) we get g(f(2))=3x-7 which simplifies to g(f(2))=3(3)-7 which equals 8. When 8 is substituted into h(x) we get h(g(f(2)))=5x which simplifies to h(g(f(2)))=5(8) which equals 83.

In summary, the answers to f(g(5)), g(f(78)), and h(g(f(2))) are 8, 60, and 83 respectively.

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Answer option 5) 0.77 is correct. The power of the test against this alternative is 0.77, given the calculations of the Scientist.

The power of a statistical test is the probability of correctly rejecting a false null hypothesis. In this case, the null hypothesis is rejected in favor of a specific alternative, and the scientist has calculated that the probability of making a type II error (failing to reject the null hypothesis when the alternative is actually true) is 0.23.

Therefore, the power of the test is the complement of the type II error probability, which is 1 - 0.23 = 0.77. This means that the test has a 77% chance of correctly rejecting the null hypothesis when the alternative is actually true. Answer option 5) 0.77 is correct.

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The water station would be located at 2.92 miles. Round to the nearest tenth of a mile, the answer is 2.9 miles.

Distance is a numerical value that quantifies "how much ground an object has traveled" while in motion. Distance is calculated as the sum of a person's movement and time.

Call the distance between the starting point and the water station x. 5 - x meters separate from the water station from the finish line.

We may write the following equation because we are aware that the ratio between these two lengths is 7:5.

x / (5 - x) = 7/5

Expanding and solving for x, we get:

5x = 7(5) - 7x

12x = 35

x = 35/12 = 2.92 miles

Therefore, the distance of the water station would be 2.9 miles.

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The correct statement about the quadratic function is given as follows:

C. The second difference would be constant.

A quadratic function is modeled as follows:

y = ax² + bx + c.

Then the first difference, which is the first derivative of the quadratic function, is obtained as follows:

y = 2ax + b.

The second difference, which is the second derivative of the quadratic function, is obtained as follows:

y = 2a.

The coefficient a is a constant, hence the second difference would be constant and the correct option is given by option C.

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Answer:

4(1) - 5 = -1

4(2) - 5 = 3

4(3) - 5 = 7

Step-by-step explanation:

Replace variable "n" with values of "n".

4n-5 when n = 1 is 4(1) - 5

4(1) - 5 = -1

4(2) - 5 = 3

4(3) - 5 = 7

Answer:

x=2⁰ AED=78⁰

Step-by-step explanation:

180⁰-108⁰= 78⁰

angle AED= 78⁰

108⁰+50⁰= 158⁰

180⁰-158⁰=22⁰

22⁰+78⁰=100⁰

100⁰+ 78⁰=178⁰

180⁰-178⁰= 2⁰

x=2⁰

The number of bacteria after 8 hours is approximately 2,713.

Let's apply the exponential growth formula:

N = N0 * e(rt),

N is the number of bacteria at a particular period. N0 is the initial bacterial population, e is the mathematical constant (about equal to 2.71828), r is the growth rate, and t is the amount of time that has passed.

The information provided in the problem can be used to determine the growth rate:

N1 = N0 * e(r*2)

N2 = N0 * e(r*4)

To get eliminate N0, divide the second equation by the first equation:

N2/N1 = e(r4) / e(r2)

385/135 = e(r4) / e(r2)

If we condense this expression, we get:

e(r*2) = 385/135

e(r*2) = 2.8519

When we take the natural logarithm of both sides, we get:

r*2 = ln(2.8519)

r = ln(2.8519)/2

r = 0.5457 (rounded to 4 decimal places)

The number of bacteria after 8 hours can now be determined using the exponential growth algorithm once more:

N = N0 * e(rt)

N = 135 * e(0.5457*8)

N = 2,713 (rounded to the nearest whole number)

Therefore, the number of bacteria after 8 hours is approximately 2,713.

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The value of f(0) is -2.

Option A is the correct answer.

A function has an input and an output.

A function can be one-to-one or onto one.

It simply indicated the relationships between the input and the output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

From the graph, we see that,

f(x) = -1 for -1 ≤ x < 0

Similarly,

f(x) = -2 for 0 ≤ x < 1 _____(1)

Now,

The value of f(x) at x = 0.

From (1),

f(0) = -2

Thus,

The value of f(0) is -2.

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The slopes of the lines are listed below:

Case 1: m = 1

Case 2: m = - 4 / 5

Case 3: m = - 4

Case 4: m = 5

Case 5: m = 2 / 7

Case 6: m = - 1

Case 7: m = 0

Case 8: m = NaN (Not a Number)

Case 9: m = 3 / 5

Case 10: m = - 8 / 3

In this question we find ten cases of lines, whose slopes must be determined according to the following rules:

Now we proceed to determine the slope for each case:

Case 1

m = (5 - 0) / [0 - (- 5)]

m = 1

Case 2

m = (- 3 - 1) / [0 - (- 5)]

m = - 4 / 5

Case 3

m = (- 3 - 1) / (1 - 0)

m = - 4

Case 4

m = [1 - (- 4)] / (1 - 0)

m = 5

Case 5

m = (3 - 1) / [0 - (- 7)]

m = 2 / 7

Case 6

m = (0 - 3) / (3 - 0)

m = - 1

Case 7

m = 0

Case 8

m = NaN (Not a Number)

Case 9

m = [- 1 - (- 4)] / (5 - 0)

m = 3 / 5

Case 10

m = (- 4 - 0) / (1.5 - 0)

m = - 4 / 1.5

m = - 8 / 3

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The solution is, a = 0 & b = -15.

A complex number is a number of the form a + bi, where a and b are real numbers ; This way, a complex number is defined as a polynomial.

here, we have,

z₁ = 3 cis(pi) and z₂ =5 cis(pi/2)

we know, cisx = cosx + i sinx

so, we get,

z₁ = 3 cis(pi) = 3(-1)

and z₂ =5 cis(pi/2) = 5( i.1)

If z₁ x z₂ = a+bi.

or, -3 * 5i = a+ ib

or, -15i = a +ib

Hence, The solution is, a = 0 & b = -15.

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Answer:

m<B = 58 degrees, m<C = 63 degrees

Step-by-step explanation:

let's find y!

angles B and A are complementary, so they should equal 90

therefore, B + A = 90

substitute

4y + 8y - 6 =90

12y = 96

y=8, and m<B = 8y-6; so m<B = 58

angles C and D are equal because of vertical angles

therefore, C = D

substitute

7x=5x+18

2x=18

x=9

m<C = 7x, so m<C = 63

½ of a 12 inches pizza represents the bigger portion.

It is the close curve of an equidistant point drawn from the center. The radius of a circle is the distance between the center and the circumference.

Let r be the radius of the circle. Then the area of the circle will be

A = πr² square units

½ of a 12-inch pizza, then the area is given as,

A = (π / 2) x (12 / 2)²

A = 56.52 square inches

30% of a 9-inch pizza. Then the area is given as,

A = 0.30 x π (9 / 2)²

A = 19.0755 square inches

½ of a 12 inches pizza represents the bigger portion.

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Answer:

2:1

Step-by-step explanation:

If we examine the two figures we see that the length of each side of Figure B is twice the length of the corresponding side of Figure A

For example side of length 44 in Figure B corresponds to side of length 22 in Figure A

Scale Factor: Final Dimension/Initial Dimension =

Length of Figure B side/Length of Figure A corresponding side

44/22 = 2/1 or 2:1

So the scale factor from A to B is 2 or 2:1

Answer:

15:30 or 3:30

Step-by-step explanation:

19:12=18:72

18:72-3:42= 15:30

15:30 or 3:30

The time when Grace started reading her chemistry book is calculated as  15:30 in 24-hour format.

To find out what time Grace started reading her chemistry book in 24-hour format,

Simply subtract 3 hours and 42 minutes from 19:12.

Convert 19:12 to minutes: 19 hours × 60 minutes/hour + 12 minutes

= 1152 minutes.

Subtract 3 hours and 42 minutes from 1152 minutes:

1152 minutes - (3 hours × 60 minutes/hour + 42 minutes)

= 1152 - (180 + 42)

= 930 minutes.

Convert the remaining minutes back to hours and minutes:

930 minutes ÷ 60 minutes/hour

= 15 hours with a remainder of 30 minutes.

Combine the results:

Grace started reading her chemistry book at 15:30 in 24-hour format.

Therefore, time when Grace start her reading of chemistry book is 15:30 in 24-hour format.

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According to the information, the mean of these values would be: 169

To calculate the mena of this series of numbers we must add all the values and multiply them by the number of values available. In this case, we would have to divide it by 10. The procedure is shown below:

According to the above, the mean of this operation would be 169.

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The distance that the light travels in the scientific notation will be 2.79 × 10⁹ miles.

Speed is defined as the length traveled by a particle or entity in an hour. It is a scale parameter. It is the ratio of length to duration.

We know that the speed formula

Speed = Distance/Time

The light travels at a speed of 1.86 × 10⁵ miles per second. And it takes about 15,000 seconds for the light to reach Neptune.

Then the number of miles is given as,

1.86 × 10⁵ = d / 15,000

d = 1.86 × 10⁵ × 15,000

d = 2.79 × 10⁹ miles

The distance that the light travels in the scientific notation will be 2.79 × 10⁹ miles.

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