manufacturing firm has discontinued production of a certain unprofitable product line. This created considerable excess production capacity. Management is considering to devote this excess capacity to one or more of three product 1,2 and 3. The available capacity on the machines which might limit output are given below: Machine type Available time (in machine hours per week) Milling machine 250 Lathe 150 Grinder 50 The number of machine hours required for each units of the respective product is given below; Machine type Productivity (in machine hours/unit) Product 1 Product 2 Product 3 Milling 8 2 3 Lathe 4 3 0 Grinder 2 0 1 The unit profit would be 20 birr, 6 birr and 8 birr for products 1,2 and 3. Find how much of each product the firm should produce in order to maximise profit ? Solve the problem by simplex method.

To calculate the number of containers that Heavey Compressors should be using, we need to determine the number of parts that need to be produced per day and divide it by the number of parts that can fit in each container.

100 parts per 8-hour day / 7 parts per container

= 14.2857 containers (round up to 15)

=15 containers

Therefore,  Heavey Compressors be using 15 containers.

Maximum inventory levels = reorder point + reorder quantity – [minimum consumption × minimum lead time].

= 100+15-[12.5x8]

= 115-100

= 15

Therefore, the maximum system inventory for this part is 15.

The maximum stock position is the largest number of goods a company can store to give its guests with service at the smallest possible cost. It's vital to keep force control in line with demand.

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Complete question:

Heavey Compressors uses a lean production assembly line to make its compressors. In one assembly area, the demand is 100 parts per eight-hour day. It uses a container that holds seven parts. It typically takes about five hours to round-trip a container from one work center to the next and back again. Heavey also desires to hold 20 percent safety stock of this part in the system. a. How many containers should Heavey Compressors be using? Do not round intermediate calculations. Round your answer up to the nearest whole number. containers b. Calculate the maximum system inventory for this part. Use the rounded value of the number of containers from part answer to the nearest whole number. parts c. If the safety stock percentage is reduced to zero, how would this impact the number of containers, all else being equal? calculations. Round your answer up to the nearest whole number. The number of containers will to containers.

Answer:

  $6400.42

Step-by-step explanation:

You want the value after 10 years of $5000.00 invested at 2.5%, compounded annually.

The formula for an amount earning compound interest is ...

  A = P(1 +r/n)^(nt)

where P is the amount invested at annual rate r compounded n times per year for t years.

Here, we have P=$5000, r=0.025, n=1, t=10, and the amount is ...

  A = $5000(1 +0.025)^(1·10) ≈ $6400.42

The final value after 10 years is $6400.42.

(Part 1)  The expression [tex]18\sqrt{8}[/tex] can be written in the form [tex]a\sqrt{b}[/tex] as [tex]36\sqrt{2}[/tex], where a and b are integers and b is as small as possible.

(Part 2)  The expression [tex]8\sqrt{18}[/tex] can be written in the form [tex]a\sqrt{b}[/tex] as [tex]16\sqrt{3}[/tex].

(Part 3)  The expression [tex]18\sqrt{8}[/tex] - [tex]8\sqrt{18}[/tex] + [tex]\sqrt{n}[/tex] is equal to ([tex]36\sqrt{2}[/tex] - [tex]16\sqrt{3}[/tex]) + [tex]\sqrt{n}[/tex].

(Part 1) Expressing [tex]18\sqrt{8}[/tex] in the form [tex]a\sqrt{b}[/tex], where a and b are integers and b is as small as possible:

We can factor [tex]\sqrt{8}[/tex] as [tex]\sqrt{2}[/tex]* [tex]\sqrt{4}[/tex]

So, [tex]18\sqrt{8}[/tex]= 18 * [tex]\sqrt{2}[/tex] *[tex]\sqrt{4}[/tex] = 18 *[tex]\sqrt{2}[/tex] * 2 = [tex]36\sqrt{2}[/tex]

So the expression [tex]18\sqrt{8}[/tex] can be written in the form [tex]a\sqrt{b}[/tex] as [tex]36\sqrt{2}[/tex], where a = 36 and b = 2.

(Part 2) Expressing [tex]8\sqrt{18}[/tex] in the form [tex]a\sqrt{b}[/tex], where a and b are integers and b is as small as possible:

We can factor [tex]\sqrt{18}[/tex] as [tex]\sqrt{3}[/tex] * [tex]\sqrt{6}[/tex]

So, [tex]8\sqrt{18}[/tex] = 8 * [tex]\sqrt{3}[/tex] * [tex]\sqrt{6}[/tex]

[tex]8\sqrt{18}[/tex]= 8 * [tex]\sqrt{3}[/tex] * [tex]\sqrt{2}[/tex] * [tex]\sqrt{3}[/tex]

[tex]8\sqrt{18}[/tex]= 8 *[tex]\sqrt{6}[/tex]* [tex]\sqrt{2}[/tex]

[tex]8\sqrt{18}[/tex]= 8 * [tex]\sqrt{2}[/tex] * [tex]\sqrt{6}[/tex]

[tex]8\sqrt{18}[/tex]= 8 * 2 * [tex]\sqrt{3}[/tex]

[tex]8\sqrt{18}[/tex]= [tex]16\sqrt{3}[/tex]

So the expression [tex]8\sqrt{18}[/tex] can be written in the form [tex]a\sqrt{b}[/tex] as [tex]16\sqrt{3}[/tex], where a = 16 and b = 3.

(Part 3) [tex]18\sqrt{8}[/tex] - [tex]8\sqrt{18}[/tex] + [tex]\sqrt{n}[/tex]:

We can substitute the simplified forms of [tex]18\sqrt{8}[/tex] and [tex]8\sqrt{18}[/tex] from parts 1 and 2 into this expression:

[tex]18\sqrt{8}[/tex] - [tex]8\sqrt{18}[/tex] + [tex]\sqrt{n}[/tex] = [tex]36\sqrt{2}[/tex] - [tex]16\sqrt{3}[/tex] + [tex]\sqrt{n}[/tex] = ([tex]36\sqrt{2}[/tex] - [tex]16\sqrt{3}[/tex]) + [tex]\sqrt{n}[/tex].

So, the expression [tex]18\sqrt{8}[/tex] - [tex]8\sqrt{18}[/tex] + [tex]\sqrt{n}[/tex] is equal to ([tex]36\sqrt{2}[/tex] - [tex]16\sqrt{3}[/tex]) + [tex]\sqrt{n}[/tex] where n is an unknown constant.

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The completed table is,

Input    (x)  -1       0    1   2

Output (y)    -6   -2   2    6

Function rule is the rule of writing the relationship between the two variables, one is dependent and another is independent.

We are given that, the output is 4 less than the input.

The table given in the problem is;

Input    (x)  -1   0  1  2

Output (y)

Thus we need to write such a function, which gives the value of (y) is 4 less than the value of (x), when we put this into the function.

y = 4x - 2

Complete the table using the above function rule;

At (x) equal to -1,

y = 4(-1)-2

y = -6

At (x) equal to 0,

y = 4(0)-2

y = -2

At (x) equal to 1,

y = 4(1)-2

y = 2

At (x) equal to 2,

y = 4(2)-2

y = 6

Hence, the function rule for the statement, "the output is 2 less than 4 times x the input is y = 4x - 2 and the completed table is,

Input    (x)  -1       0    1   2

Output (y)    -6   -2   2    6

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The probabilities are given as follows:

It is a distribution with two bounds, given by a and b, in which each possible outcome is equally as likely.

The probability of finding a value between c and d is:

[tex]P(c \leq X \leq d) = \frac{d - c}{b - a}[/tex]

Uniformly distributed between 11 minutes and 14 minutes, hence the bounds are given as follows:

a = 11, b = 14.

As the uniform distribution is a continuous distribution, the probability of an exact value, such as X = 11, is of zero.

The probability of a value between 11 and 13 is given as follows:

p = (13 - 11)/(14 - 11) = 2/3.

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Using Pythagorean theorem, the distance from where she started is 16.4km

The Pythagorean Theorem is a mathematical concept that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical terms, the theorem can be expressed as:

c^2 = a^2 + b^2,

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

c² = 10² + 13²

c² = 269

c = √269

c = 16.4 km

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Answer:

he months where Jacy's hometown gets at least 10.5 centimeters of rainfall are July (x = 7) and all the months after July, which are August (x = 8), September (x = 9), October (x = 10), November (x = 11), and December (x = 12).

Step-by-step explanation:

The given function is:

f(x) = 0.2x^2 - 1.8x + 6.5

where x represents the number of months passed and f(x) represents the average monthly rainfall in centimeters.

To find the months where the average rainfall is at least 10.5 centimeters, we need to set the function f(x) greater than or equal to 10.5 and solve for x:

0.2x^2 - 1.8x + 6.5 ≥ 10.5

0.2x^2 - 1.8x - 4 ≥ 0

Multiplying both sides by 5, we get:

x^2 - 9x - 20 ≥ 0

We can factor the left-hand side of the inequality as:

(x - 5)(x - 4) ≥ 0

The solution to this inequality is the set of values of x that make the inequality true. This includes the intervals where:

(x - 5) ≥ 0 and (x - 4) ≥ 0, which gives x ≥ 5

OR

(x - 5) ≤ 0 and (x - 4) ≤ 0, which gives x ≤ 4

Thus, the months where Jacy's hometown gets at least 10.5 centimeters of rainfall are July (x = 7) and all the months after July, which are August (x = 8), September (x = 9), October (x = 10), November (x = 11), and December (x = 12).

Jacy's hometown gets at least 10.5 centimeters of rainfall in the months of September, September of the following year, September of the year after that, and so on.

An inequality is a relation between two numbers or expressions that are not equal.

It can show which of them is greater or smaller by using symbols like < or >. It can also be a statement of fact about the order relationship of quantities.

According to the question given,

We need to solve the inequality:

A(t) ≥ 10.5

Substituting the given function, we get:

2.3sin(π/6)t ≥ 10.5

sin(π/6)t ≥ 4.57.

Since the sine function has a maximum value of 1, the inequality is only satisfied when:

sin(π/6)t = 1

Solving for t, we get:

π/6t = π/2 + 2πk or π/6t = 3π/2 + 2πk

where k is any integer.

Simplifying each equation, we get:

t = 3 + 12k or t = 9 + 12k, where k is any integer.

Therefore, the only solution is given by:

t = 9 + 12k

where k is any integer.

Hence, Jacy's hometown gets at least 10.5 centimeters of rainfall in the months of September, September of the following year, September of the year after that, and so on.

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Therefore, 4 people have been to Europe but not Asia.

A Venn diagram is a visual representation of sets and their relationships. It is used to show the similarities and differences between different sets of data. A Venn diagram consists of overlapping circles or other shapes, each representing a set. The size of each circle is proportional to the number of elements in the set it represents. The overlapping parts of the circles represent the elements that are shared by two or more sets. The non-overlapping parts represent the elements that are unique to each set. By analyzing the intersections and differences between sets in a Venn diagram, we can gain insights into the relationships between them. Venn diagrams are often used in mathematics, logic, statistics, and other fields to represent complex relationships between sets of data. They are also used in education to teach critical thinking and problem-solving skills.

Here,

Let E be the set of people who have been to Europe, A be the set of people who have been to Asia, and N be the set of people who have been to neither.

From the problem, we know that:

|E| = 5 (5 people have been to Europe)

|A| = 3 (3 people have been to Asia)

|E ∩ A| = 1 (1 person has been to both Europe and Asia)

To find the number of people who have been to Europe but not Asia, we need to find |E \ A|. We can use the formula:

|E \ A| = |E| - |E ∩ A|

Substituting the values we know:

|E \ A| = 5 - 1 = 4

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Answer:

hope this helps

The integral for the volume of the solid is ∫2π(eᵇ)(dy)

In calculus, finding the volume of a solid is an important concept. One method to do this is by using integration.

To find the volume of the solid obtained by rotating the region bounded by y=ln(x), y=0, and x=5 about the y-axis, we can use the method of cylindrical shells. This involves dividing the region into infinitely thin vertical strips and rotating each strip around the y-axis to create cylindrical shells.

To set up the integral, we can use the formula for the volume of a cylindrical shell, which is 2πrhΔx, where r is the distance from the y-axis to the edge of the strip, h is the height of the strip, and Δx is the thickness of the strip.

Since we are rotating around the y-axis, we need to express the curves in terms of y instead of x. Using the inverse of the natural logarithm function, we can rewrite

=> y=ln(x) as x=eᵇ.

Thus, the region is bounded by x=5, y=0, and

=> x=eᵇ.

Next, we need to find the distance from the y-axis to the edge of the strip, which is simply the x-value at a given y. This is given by r=e^y.

The height of the strip is simply Δy, the thickness of the strip. To find this, we can take the difference between two consecutive y-values. Thus, Δy=dy.

Putting it all together, the integral for the volume of the solid is:

=> ∫2π(eᵇ)(dy)

This integral represents the sum of the volumes of all the cylindrical shells that make up the solid. However, we still need to evaluate the integral to get the actual volume of the solid.

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Answer:

25

Step-by-step explanation: just like that

Answer: C

asdfasdasdasdfasdf

The appropriate method for the biologist to use for inference to the population is A) A one-sample t-interval for a population mean.

In this situation, the biologist wants to estimate the average weight of a population of dolphins living in a certain region of the ocean. The biologist will collect a random sample of dolphins and use the sample weights to create the estimate.

here, we have,

Therefore, the appropriate method for inference to the population is to use a one-sample t-interval for a population mean.

This method is used when we want to estimate the population mean using a sample mean and the standard deviation of the sample.

The t-interval takes into account the uncertainty of the estimate due to the random sampling process, which makes it an appropriate method for this situation.

Hence, The appropriate method for the biologist to use for inference to the population is A) A one-sample t-interval for a population mean.

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The correct question is:

A marine biologist wants to estimate the average weight of a population of dolphins living in a certain region of the ocean. The biologist will collect a random sample of dolphins and use the sample weights to create the estimate Which of the following is an appropriate method for the biologist to use for inference to the population?

A A one-sample t-interval for a population mean

B A one-sample t-interval for a sample mean

C A one-sample 2-interval for a population proportion

D A matched-pairs t-interval for a mean difference

E A two-sample t-interval for a difference between means

Answer:

259

Step-by-step explanation:

Answer:259

Step-by-step explanation:

202+57=259

Answer:

y= 2x+12

Step-by-step explanation:

y=mx+c

y= 2x+12

(0,y) (-4,4)

2=y-4/4

8=y-4

12=y

Answer: The measure of angle J is 40 degrees, 25 minutes, and 16 seconds, which can be expressed in decimal form as 40 + 25/60 + 16/3600 = 40.42111... degrees.

Since ∠J and ∠K are complementary, their measures add up to 90 degrees.

So, the measure of ∠K can be found by subtracting the measure of ∠J from 90 degrees:

∠K = 90 - 40.42111... = 49.57888... degrees

So, the measure of ∠K is approximately 49 degrees, 34 minutes, and 44 seconds.

Step-by-step explanation:

16%  percentage of the students have grade point averages that are greater than 2.11

Using the empirical rule, we know that for a bell-shaped distribution, approximately:

68% of the data falls within one standard deviation of the mean

95% of the data falls within two standard deviations of the mean

99.7% of the data falls within three standard deviations of the mean

To find the percentage of students with a grade point average greater than 2.11

We first need to calculate how many standard deviations away from the mean 2.11 is:

z = (2.11 - 2.56) / 0.45

= -1

This tells us that 2.11 is 1 standard deviation below the mean.

Since the distribution is symmetric.

The percentage of students with a GPA greater than 2.11 is the same as the percentage of students with a GPA less than 2.56 + 1*0.45, which is:

68% + 95% = 163%

So, approximately

100% - 163%

= 16% of the students have a GPA greater than 2.11.

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Answer:

Step-by-step explanation:

The zeros of a polynomial function are the values of x for which the function equals zero. For the given polynomial function p(x) = (2x - 1)(5x + 7)(8x + 9), the zeros can be found by setting each factor equal to zero and solving for x:

2x - 1 = 0, x = 1/2

5x + 7 = 0, x = -7/5

8x + 9 = 0, x = -9/8

Therefore, the zeros of the function are x = 1/2, x = -7/5, and x = -9/8.

In standard form, the polynomial function p(x) can be written as:

p(x) = 80x^3 + 122x^2 - 127x - 63

The relationship between the zeros of p(x) and its coefficients can be seen in Vieta's formulas. Vieta's formulas state that for a polynomial function of degree n with roots r1, r2, ..., rn, the coefficients of the polynomial can be expressed as:

a0 = (-1)^n * p0

a1 = (-1)^(n-1) * p1 / p0

a2 = (-1)^(n-2) * p2 / p0

...

an-1 = (-1) * pn-1 / p0

an = pn / p0

where p0 is the coefficient of the highest degree term (the leading coefficient), and the pi are the elementary symmetric polynomials, which are given by:

p1 = r1 + r2 + ... + rn

p2 = r1r2 + r1r3 + ... + rn-1rn

...

pn-1 = r1r2...rn-1 + r1r2...rn-2 + ... + rn-2rn-1

pn = r1r2...rn

Using Vieta's formulas, we can see that for the polynomial function p(x) given above, the coefficients are related to the zeros as follows:

a0 = -63

a1 = -127

a2 = 122

a3 = 80

And we can also see that:

a0 = (-1)^3 * p0 = -63

a1 = (-1)^(3-1) * p1 / p0 = -127

a2 = (-1)^(3-2) * p2 / p0 = 122

a3 = (-1)^(3-3) * p3 / p0 = 80

Therefore, the coefficients of the polynomial are related to the zeros through Vieta's formulas, which express the coefficients as functions of the zeros, and vice versa.

The set B' ∩ D' represents the event "the numbers do not add to 6." It contains 20 elements.

The complement of event B is B' = {2, 3, 4, 5, 7, 8, 9, 10, 11, 12}. The complement of event D is D' = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12}. Taking the intersection of their complements, we get B' ∩ D' = {2, 3, 4, 5, 7, 8, 10, 11}. This set represents the event "the numbers do not add to 6." It contains 8 elements out of the 36 possible outcomes of rolling two dice, so the probability of this event is 8/36 or 2/9.

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Answer:

-2

Step-by-step explanation:

Standard equation of a proportional relationship:

y = kx

We use the given point to find k.

4 = k × (-16)

k = 4/(-16)

k = -1/4

The equation of this proportional relationship is

y = -1/4 x

Now we use x = 8 in the equation just above to find its corresponding y value.

x = 8

y = -1/4 × 8

y = -2

Answer: -2

The air inside the beach ball with a radius of 20 inches is 33493.33 cubic inches.

The capacity of a sphere is its volume. It is the area that the sphere occupies. Cubic measurements of a sphere's volume include m3, cm3, in3, etc. The sphere has a round, three-dimensional shape. Its shape is determined by three axes: the x, y, and z axes. Sports like basketball and football are all examples of spheres with volume.

The volume of the cube is given as:

V = (4/3) πr³

Substituting the value of r = 20 inches we have:

V = (4/3)π(20)³

V = (4/3)(3.14)(20)³

V = 33493.33 cubic inches.

Hence, the air inside the beach ball with a radius of 20 inches is 33493.33 cubic inches.

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The correct answer is A, as the mean cost of all 12 trips can be calculated by adding up the total cost of all the trips and dividing by the total number of trips.

The proposed solution is wrong as the mean cost of all 12 trips cannot be calculated by taking the average of the means of the two employees. Taking the average of the means of the two employees assumes that they took the same number of trips and that their trip costs are equally weighted in the total cost calculation, which is not necessarily true.

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The number of parent chaperones required on the trip (p) variable is the dependent variable.

According to the question,

s = the number of band students on the trip

p = the number of parent chaperones required on the trip

As the question already said the number of students on the trip determines how many parent chaperones are required to go along, therefore s , the variable of number of student is a independent variable as it does not depends on another variable.

And 'p' ,  the number of parent chaperones required on the trip depend on number of students. Therefore 'p' is dependent variable.

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(a) Average for a student with 99 attendance is 81.1% and (b) The average for a student with 21 attendance is 78.8%.

Professor Cramer determines a final grade based on attendance, two papers, three major tests, and a final exam, the activities carry different weights.

Attendance is worth 3%,

each paper is worth 5%,

each test is worth 13%,

and the final is worth 48%.

(a) The average for a student with 99 on attendance, 75 on the first paper, 61 on the second paper, 94 on test 1, 86 on test 2, 77 on test 3, and 79 on the final exam

= [(99×3%) + (75×5%) + (61×5%) + (94×13%) + (86×13%) + (77×13%) + (79×48%)] ÷ [0.03 + 2(0.05) + 3(0.13) + 0.48]

= [2.97 + 3.75 + 3.05 + 12.22 + 11.18 + 10.01 + 37.92] ÷ 1

= 81.1 %

(b) The average for a student with the above scores on the papers, tests, and final exam, but with a score of only 21 on attendance

= [(21×3%) + (75×5%) + (61×5%) + (94×13%) + (86×13%) + (77×13%) + (79×48%)] ÷ [0.03 + 2(0.05) + 3(0.13) + 0.48]

= [0.63 + 3.75 + 3.05 + 12.22 + 11.18 + 10.01 + 37.92] ÷ 1

= 78.8 %

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14 boxes of erasers should Mr. Franklin need to buy.

A division is a process of splitting a specific amount into equal parts.

Given that Franklin wants to buy an eraser for every fourth-grade student

There are 12 erasers in each box.

There are 7 fourth-grade classes with 24 students in each class

The total number of students=7×24

=168

We need to find the number of boxes are required to buy an eraser for each student.

Since there are 12 erasers in each box, we can divide the total number of students  by 12 to find the number of boxes Mr. Franklin needs to buy:

number of boxes = total students / erasers per box

number of boxes = 168 / 12

number of boxes = 14

Hence, 14 boxes of erasers should Mr. Franklin need to buy.

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Answer: a

Step-by-step explanation:

It is determined that she has a sebaceous cyst. This encounter should be reported with code(s)  L72.9. So option B is correct.

The ICD-10-CM code for sebaceous cysts is unspecified. The other options are not specific to a sebaceous cyst or are not relevant to the given scenario.

L72.1 is for the epidermal cyst, L30.9 is for unspecified dermatitis, and L72.3 is for the inflamed sebaceous cyst.

ICD-10-CM codes are used for medical diagnosis coding and are important for medical record-keeping, insurance reimbursement, and data analysis.

Each code corresponds to a specific medical condition, and it is essential to choose the correct code that accurately represents the patient's diagnosis.

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Answer:

Step-by-step explanation:

it is 77.6

In a Quincy College class consisting of 15 students 12 or more have blue eyes.

If half the students at Quincy College have blue eyes, then the most probable event is that a randomly selected student from the college will have blue eyes.

This is because, if half the students have blue eyes, then there is a greater chance that a randomly selected student will have blue eyes than any other eye color.

For example, if we were to consider the event of a randomly selected student having green eyes, this would be less probable because fewer than half the students are likely to have green eyes.

Therefore, the most probable event is that a randomly selected student from Quincy College will have blue eyes.

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The system of equations to represent the situation are f + l = 20.5 and 6.25f + 9.5l = 149.25, solution is 14 pounds and 6.5 pounds

A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.

We are given that;

Cost of fish=$6.25

Cost of lobster=$9.50

Cost of 20.5 pounds =$149.25

Now, let's use the variables f and l to represent the pounds of fish and lobster sold, respectively. We can set up two equations based on the information given:

f + l = 20.5 (the total weight sold is 20.5 pounds)

6.25f + 9.5l = 149.25 (the total revenue made is $149.25)

Now we can solve the system of equations using any method we prefer. Here's one way to do it using substitution:

f + l = 20.5 (equation 1)

f = 20.5 - l (solve equation 1 for f)

6.25f + 9.5l = 149.25 (equation 2)

6.25(20.5 - l) + 9.5l = 149.25 (substitute f = 20.5 - l into equation 2)

128.13 - 6.25l + 9.5l = 149.25 (distribute the 6.25)

3.25l = 21.12 (combine like terms)

l = 6.5 (divide both sides by 3.25)

Now we can use this value of l to find f:

f + 6.5 = 20.5 (from equation 1)

f = 14 (subtract 6.5 from both sides)

Therefore, by the equations the fish market sold 14 pounds of fish and 6.5 pounds of lobster.

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1.20 If the system has three unknowns and R has three nonzero rows, then the system has at least one solution.

This statement is true.

1.21 It is only when the rank of the coefficient matrix is less than the number of unknowns that the system can have infinitely many solutions

1.22 The system below has an infinite number of solutions:

2x + 3y + 5z + 6 - 7 - 8v = 0

3x - 4y + 7z + + 8 + 5y = 0

-7x + 9y - 2z -- 4w - 5u + 2y = 0

--5x - 5y +92 +3w + 2u + 7y = 0

-9x + 3y - 9z+5w - 3u - 4y = 0

This statement is true.

When we perform row reduction on a system of linear equations, the resulting reduced row echelon form (R) will have the same number of nonzero rows as the rank of the coefficient matrix.

In other words, if R has three nonzero rows, then the rank of the coefficient matrix is also 3.

If the rank of the coefficient matrix is equal to the number of unknowns, then the system has a unique solution.

However, if the rank of the coefficient matrix is less than the number of unknowns, then the system has either no solution or infinitely many solutions.

But in this case, since the rank is equal to the number of unknowns, the system must have at least one solution.

1.21 If the system has three unknowns and R has three nonzero rows, then the system can have an infinite number of solutions.

This statement is false. If R has three nonzero rows, then the rank of the coefficient matrix is also 3.

If the rank of the coefficient matrix is equal to the number of unknowns, then the system has a unique solution.

It is only when the rank of the coefficient matrix is less than the number of unknowns that the system can have infinitely many solutions.

1.22 The system below has an infinite number of solutions:

2x + 3y + 5z + 6 - 7 - 8v = 0

3x - 4y + 7z + 8 + 5y = 0

-7x + 9y - 2z - 4w - 5u + 2y = 0

-5x - 5y + 92 + 3w + 2u + 7y = 0

-9x + 3y - 9z + 5w - 3u - 4y = 0

This statement is true.

To check if the system has infinitely many solutions, we need to check the rank of the coefficient matrix and the rank of the augmented matrix. In this case, the rank of the coefficient matrix is 3, which is less than the number of unknowns (5).

Also, when we perform row reduction on the augmented matrix, we get the following reduced row echelon form:

1 0 -1 0 1 0

0 1 2 0 -1 0

0 0 0 1 2 0

0 0 0 0 0 1

0 0 0 0 0 0

Since the rank of the augmented matrix is less than the number of unknowns, the system has infinitely many solutions.

The variables with free parameters are z, u, and y, which can take any value.

The other variables can be expressed in terms of these free parameters.

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