A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. It is known that the standard deviation in the cutting length that this machine produces is 0.20 feet. A sample of 75 cut sheets yields a mean length of 12.25 feet. This sample will be used to obtain a confidence interval for the mean length cut by machine. Referring to Scenario 1, the Z value to use in obtaining the 95% confidence interval is approximately .
A. 2.58.
B.1.96.
C. 1.75.
D. 1.645.

Explanation:

5(y+3)-11=-y-6 Given

5(y+3)=-y+5 . . . . addition property of equality (11 is added)

5y+15=y+5 . . . . . distributive property (5 is distributed)

5y=-y-10 . . . . . . . addition property of equality (-15 is added)

6y=-10 . . . . . . . . . addition property of equality (y is added)

y=-5/3 Division Property of Equality/Reduce

Answer:

C

Step-by-step explanation:

First, when fractions get raised to an exponent, both the numerator and the denominator get raised to the power.

When exponents get raised to a power, the powers multiply.

E.g. [tex](x^2)^3=(x^2)(x^2)(x^2)=(x*x)(x*x)(x*x)=x^6[/tex]

In this case,

[tex](\frac{4^3}{5^-^2})^5=\frac{4^(^3^*^5^)}{5^(^-^2^*^5^)}=\frac{4^1^5}{5^-^1^0}\\[/tex]

Note that [tex]x^-^n=\frac{1}{x^n}[/tex]

So, 1/5^(-10) = 5^10

So, our answer is C

let's firstly convert both fractions with the same denominator, by simply multiplying one by the denominator of the other, let's proceed,

[tex]\bf \cfrac{1}{5}\cdot \cfrac{3}{3}\implies \cfrac{3}{15}~\hspace{7em}\cfrac{1}{3}\cdot \cfrac{5}{5}\implies \cfrac{5}{15} \\\\[-0.35em] ~\dotfill\\\\ \boxed{\cfrac{3}{15}}\rule[0.35em]{10em}{0.25pt}~~\cfrac{4}{15}~~\rule[0.35em]{10em}{0.25pt}\boxed{\cfrac{5}{15}}[/tex]

well, low and behold, 4/15 doesn't simplify further and is right between those two, and its denominator is not 4.

Answer:

Answered below

Step-by-step explanation:

Sheet 1: Question 3

Vertically opposite angles are equal so you will equate the angles given,

∠LPN = ∠OPM

7 + 13x = -20 + 16x

27 = 3x

x = 9

Sheet 1: Question 4

Vertically opposite angles are equal so you will equate the angles given,

∠ABD = ∠EBC

2x + 20 = 3x + 15

-x = -5

x = 5

Sheet 1: Question 5

Step 1: Find the value of x

Vertically opposite angles are equal so you will equate the angles given,

∠SOP = ∠ROQ

5x = 4x + 10

x = 10

Step 2: Find angles

Angle SOP = 5x = 5(10) = 50°

Angle ROQ = 50° (because it is vertically opposite to angle SOP)

Angle SOR = 180 - 50 (because all angles on a straight line are equal to 180°)

Angle SOR = 130°

Angle POQ = 130° (because it is vertically opposite to angle SOR)

Sheet 1: Question 6

Angle 1 = 72° (because vertically opposite angles)

∠4 + ∠1 + 41 = 180° (because all angles on a straight line are equal to 180°)

∠4 + 72 + 41 = 180

∠4 = 67°

∠3 = 41° (because vertically opposite angles)

∠2 = 67° (because vertically opposite angles)

Sheet 2: Question 3

Step 1: Find the value of x

Sum of complementary angles is equal to 90°

Angle A + Angle B = 90°

7x + 4 + 4x + 9 = 90°

11x = 90 - 13

11x = 77

x = 7

Step 2: Find angle A and angle B using x

Angle A: 7x + 4

7(7) + 4

Angle A = 53°

Angle B: 4x + 9

4(7) + 9

Angle B = 37°

Sheet 3: Question 3

Step 1: Find the value of x

Sum of supplementary angles is equal to 180°.

Angle A + Angle B = 180°

3x - 7 + 2x + 2 = 180°

5x = 185

x = 37

Step 2: Find angle A and angle B using x

Angle A: 3x - 7

3(37)-7

Angle A = 104°

Angle B: 2x + 2

2(37) + 2

Angle B = 76°

Sheet 3: Question 4

Sum of supplementary angles is equal to 180°.

Step 1: Find x

1/4(36x-8) + 1/2(6x-20) = 180°

Take LCM

[36x - 8 + 2(6x - 20)]/4 = 180°

36x - 8 +12x - 40 = 180 x 4

48x - 48 = 720

48x = 768

x = 16

Step 2: Find both angles with the help of x

Angle 1: 1/4(36x-8)

1/4[36(16)-8] = 568/4

Angle 1 = 142°

Angle 2: 1/2(6x-20)

1/2[6(16)-20] = 76/2

Angle 2 = 38°

Sheet 4: Question 1

All angles on a straight line are equal to 180°

Angle z + 138° = 180°

Angle z = 180 - 138

Angle z = 42°

Sheet 4: Question 2

Linear pair 1: 5 and 7 (because both angles are on a straight line and are equal to 180°)

Linear pair 2: 6 and 8 (because both angles are on a straight line and are equal to 180°)

Sheet 4: Question 3

Step 1: Find the value of x

All angles on a straight line are equal to 180° or linear pairs are equal to 180°

Angle LMO + Angle OMN = 180°

7x + 20 + 10 + 5x = 180°

12x = 180 - 30

x = 150/12

x = 12.5

Step 2: Find angles using the value of x

Angle LMO: 7x + 20

7(12.5) + 20

Angle LMO = 107.5°

Angle OMN: 10 + 5x

10 + 5(12.5)

Angle OMN = 72.5°

Sheet 4: Question 4

Linear pairs are equal to 180°.

Angle 1 + Angle 2 = 180°

1/3(27x-6) + 1/2(6x-20) = 180°

Take LCM = 6

[2(27x-6) + 3(6x-20)]/6 = 180

54x - 12 + 18x - 60 = 1080

72x - 72 = 1080

72x = 1152

x = 16

!!

Answer:

Step-by-step explanation:

Given that an advertiser wishes to estimate the proportion of adults in Utah who already own a gym membership.

Confidence level wanted = 90%

Critical value Z for 90% = 1.645

p is not known hence for maximum std error we know p =0.5

Let us take p =0.5

Margin of error = [tex]1.645*\sqrt{\frac{pq}{n} } =\\1.645*0.5(\frac{1}{\sqrt{n} } ) 0.10\\n=67.65\\n~68[/tex]

Sample size can be 68

Answer:

b. 20%

Step-by-step explanation:

The distribution of  number of hours worked by students is:

0 - 10          =         20

10 - 20        =         80

20 - 30       =         200

30 - 40       =         100

Total number of students = 400

We need to find the percentage of students working between 10 and 20 hours. From the above table we can see that number of students working between 10 and 20 hours is 80. Now we need to convert this into percentage. What we need to evaluate is what percentage of 400 is 80.

The formula for percentage is:

[tex]\frac{\text{Concerned Value}}{\text{Total Value}} \times 100\%[/tex]

Concerned value is 80, total value is 400. So using these in the formula we get:

[tex]\frac{80}{400} \times 100%\\\\  = 20\%[/tex]

Thus, 20% of the students are working between 10 and 20 hours.

The percentage of statistics students working between 10 and 20 hours per week is 20%, which is calculated by dividing the frequency of the group (80 students) by the total number of students (400) and multiplying by 100.

The question is asking us to find the percentage of statistics students who work between 10 and 20 hours per week, given the frequency distribution provided in Exhibit 2-1. According to the exhibit, 80 students work between 10 and 20 hours per week. Since the total number of students is 400, we calculate this percentage by taking the number of students in the given category (80 students) and dividing it by the total number of students (400), then multiplying by 100 to get a percentage.

The calculation is:

Percentage = (80 ÷ 400) × 100

Percentage = 0.20 × 100

Percentage = 20%

Therefore, the percentage of students working between 10 and 20 hours is 20%, which corresponds to option b.

Answer:

The answer is 54.5

Step-by-step explanation:

The median of a set of observations is the value in the middle in case there is an uneven number of observations or the average between the two values in the middle.

First we arrange the times from less time to longer times.

50 - 52 - 57 - 230 .

The values in the middle are  52 - 57

And the average between this two values is 54.5. Thus that is the median legth of time.

Answer:

The median length of time required to review an application is 54.5 minutes.

Step-by-step explanation:

The times (in minutes) that several underwriters took to review applications for similar insurance coverage are 50, 230, 52, and 57.

Arranging in ascending order we get

50,52,57,230

Median length of time = [tex]\frac{52+57}{2}=54.5[/tex]

Hence, the median length of time required to review an application is 54.5 minutes.

Answer:

For solar: [tex]S(x)=6240+75x[/tex]

For traditional: [tex]T(y)=850+320y[/tex]

Step-by-step explanation:

Cost to install solar water heater = $6240

The average annual cost to run the solar water heater is about = $75

Total cost to run x years : [tex]S(x)=6240+75x[/tex]

A traditional gas water heater costs about = $850

The average annual cost to run = $320

Total cost to run y years : [tex]T(y)=850+320y[/tex]

1. The total cost of a solar water heater, including installation and annual running costs, can be calculated as $6240 + ($75 * years).

2. The total cost of a traditional gas water heater, including installation and annual running costs, can be calculated as $850 + ($320 * years).

Given that,

Solar water heater:

Installation cost: $6240 (after rebates)

Annual running cost: $75

Traditional gas water heater:

Installation cost: $850

Annual running cost: $320

Let's create the total cost functions for each water heater as a function of the number of years.

For the solar water heater, the total cost (TC) can be calculated as:

TC_solar(years) = installation cost + (annual running cost * years)

Given the installation cost of $6240 and annual running cost of $75,

The total cost function for the solar water heater becomes:

TC_solar(years) = $6240 + ($75 * years)

For the traditional gas water heater, the total cost (TC) can be calculated as:

TC_gas(years) = installation cost + (annual running cost * years)

Given the installation cost of $850 and annual running cost of $320,

So, The total cost function for the gas water heater becomes:

TC_gas(years) = $850 + ($320 * years)

These equations allow you to determine the total cost of each water heater based on the number of years they are used.

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Answer:

a testable proposition that explains an observed phenomenon or answers a question

Step-by-step explanation:

Basically an hypothesis is a theory given as a solution to a question proposed by a problem in from of you, but that it needs testing before it can be taken as truth or not

A hypothesis is an educated guess or a testable prediction drawn from observations. It is a tentative statement about reality that can be verified through experiments. If a hypothesis explains a large body of experimental data, it may evolve into a theory.

A hypothesis is a testable prediction about how the world will behave if our idea is correct. Observations and questioning may lead a scientist to form a hypothesis. In essence, a hypothesis is an educated guess or a tentative statement about reality that can be tested to determine its validity.

An example of a hypothesis is 'if I study all night, I will get a passing grade on the test'. The hypothesis, the educated guess in this scenario, then undergoes a process of testing through experimentation, calculation, and comparison with the experiments of others.

If a hypothesis is capable of explaining a large body of experimental data, it can reach the status of a theory. A theory in scientific terms is a well-substantiated, comprehensive, testable explanation of particular aspects of nature. However, theories can be modified if new data becomes available. This process of questioning, observing, hypothesizing, testing, and eventually forming a theory is known as the scientific method.

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Answer:

the 3 yard line

Step-by-step explanation:

Answer:

See explanation

Step-by-step explanation:

7.8 9.1 9.5 10.0 10.2 10.5 11.1 11.5 11.7 11.8

12.2 12.2 12.5 13.1 13.5 13.7 13.7 14.0 14.4 14.5

14.6 15.2 15.5 16.0 16.0 16.1 16.5 17.2 17.8 18.2

19.0 19.1 19.3 19.8 20.0 20.2 20.3 20.5 20.9 21.1

21.4 21.8 22.0 22.0 22.4 22.5 22.5 22.8 22.8 23.1

23.1 23.2 23.7 23.8 23.8 23.8 23.8 24.0 24.1 24.1

24.5 24.5 24.9 25.1 25.2 25.5 26.1 26.4 26.5 26.7

27.1 29.5

A. The five-number summary is

B. The interquartile range is

[tex]Q_3-Q_1=23.8-14.2=9.6[/tex]

C. See attached diagram

D. The distribution is not symmetric, the left half shows more data than the right part

Answer:

A triangle with three sides that are each equal in length to those of another triangle, for example, are congruent. This statement can be abbreviated as SSS. Two triangles that feature two equal sides and one equal angle between them, SAS, are also congruent.

Step-by-step explanation:

Answer:

21

Step-by-step explanation:

You were close to de answer but you rounded in a wrong way

If we calculate [tex]\sqrt{16^{2} + 14^{2}}[/tex]  we get 21.2602916255

We need to round to the nearest whole number, it means that we have we only have two options: 21 or 22

We need to remember the rounding rules:

We see that 1 is followed by 2, it means that we have to round the number down, so our answer is 21

Answer:

Step-by-step explanation:

Easy. its [tex]x^{2} \int\limits^a_b {x} \, dx \\ \\ \left \{ {{y=2} \atop {x=2}} \right. \\ \neq \frac{x}{y} \beta \al\left \{ {{y=2} \atop {x=2}} \right. pha \neq \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] \geq \\ \left \{ {{y=2} \atop {x=2}} \right. \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right][/tex]

The inequality that represents the graph is:

                        [tex]y\geq x+3[/tex]

By looking at the graph we observe that the graph is a solid line which passes through the point (-1,0) and (1,2).This means that the inequality is a inequality with a equality sign(i.e. not strict)

This means that the equation of line is:

[tex]y-2=\dfrac{2-0}{1-(-1)}\times (x-(-1))\\\\y-2=\dfrac{2}{2}\times (x+1)\\\\y-2=x+1\\\\y=x+1+2\\\\y=x+3[/tex]

The shaded region is away from the origin.

This means that the inequality does not pass the zero test.

Hence, the inequality is:

               [tex]y\geq x+3[/tex]

Answer:

Option C 41%

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The number of people who own at most three t-shirts costing more than $19 each is equal to the number of people who own one t-shirt costing more than $19 each, plus the number of people who own two t-shirt costing more than $19 each, plus the number of people who own three t-shirt costing more than $19 each

Observing the graph

The number of people who own one t-shirt costing more than $19 each is 5

The number of people who own two t-shirt costing more than $19 each is 17

The number of people who own tree t-shirt costing more than $19 each is 23

The number of people who own at most three t-shirts costing more than $19 each is

[tex](5+17+23)=45\ people[/tex]

To find out the percentage divided the number of people who own at most three t-shirts costing more than $19 by one hundred eleven people (total people that shopped in a store) and then multiply by 100

[tex](45/111)100= 40.54\%[/tex]

Round to the nearest whole number

[tex]40.54\%=41\%[/tex]

The question involves the application of statistical principles to determine the percentage of people who own at most three expensive t-shirts. Assuming 59% is the correct answer, this means 66 out of the 111 people surveyed own three or fewer such t-shirts. The examples in the question references demonstrate the principles of economics, but aren't directly related to the question.

The question is about determining the percentage of people who own at most three t-shirts costing more than $19 each from a sample of 111 people. Given that the possible percentages provided are 21% and 59%, the process involves the application of statistical principles, particularly dealing with percentages in relation to a specific sample size. This example fits into a branch of mathematics known as statistics, which deals with data analysis, collection, interpretation, presentation, and organization.

Let's assume from the reference that 59% is the correct answer, this conveys that of the 111 people surveyed, approximately 66 people own three or fewer such t-shirts. This is obtained by calculating 59% of 111, which equals about 65.49 and rounds to 66.

In the references provided, there are examples of how the cost of t-shirts affects purchasing decisions. This is related to the principles of economics which involve opportunity cost and budget constraints, but is not directly tied to answering the initial question.

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Answer:

5r + 2p + 6

Step-by-step explanation:

4r+r = 5r

9-3 = 6

Answer:

  3

Step-by-step explanation:

A 6-inch piece of string can be cut into 12 pieces that are 1/2 inch each. If Lee needs 15 pieces, she needs 3 more.

___

Each inch is divided into 2 pieces. That's what 1/2 inch means. 6×2 = 12.

Answer:

  Domain of set B: {2, 4, -4, 0}

Step-by-step explanation:

The domain of the function whose ordered pairs are listed in set B is the set of first numbers of those pairs: {2, 4, -4, 0}.

_____

Comment on the question

A "set" does not have a domain. A "function" has a domain. To make any sense of this question, we have to interpret the question to mean the function described by the ordered pairs in the set.

If the [tex]f(x)[/tex] and [tex]t(x)[/tex] are even function then [tex]fo\ t\ (x)[/tex] is an even function, if [tex]f(x)[/tex] and [tex]t(x)[/tex] are odd function then the function [tex]fo\ t\ (x)[/tex] is an odd function and if [tex]f(x)[/tex] is even and [tex]t(x)[/tex] is odd then the function [tex]fo\ t\ (x)[/tex] is an even function.

Further explanation:

An even functrion satisfies the property as shown below:

[tex]\boxed{f(-x)=f(x)}[/tex]

An odd functrion satisfies the property as shown below:

[tex]\boxed{f(-x)=-f(x)}[/tex]

Consider the given composite function as follows:

[tex]\boxed{fo\ t\ (x)=f\left(t(x))\right}[/tex]

If both the function [tex]f(x)[/tex] and [tex]t(x)[/tex] are even function.

[tex]\begin{aligned}fo\ t\ (-x)&=f\left(t(-x))\right\\&=f\left(t(x))\right\\&=fo\ t\ (x)\end{aligned}[/tex]

From the above calculation it is concluded that,

[tex]\boxed{fo\ t\ (-x)=fo\ t\ (x)}[/tex]

This implies that the composite function [tex]fo\ t\ (x)[/tex] is an even function.

If both the function [tex]f(x)[/tex] and [tex]t(x)[/tex] are odd function.

[tex]\begin{aligned}fo\ t\ (-x)&=f\left(t(-x))\right\\&=f\left(-t(x))\right\\&=-fo\ t\ (x)\end{aligned}[/tex]

From the above calculation it is concluded that,

[tex]\boxed{fo\ t\ (-x)=-fo\ t\ (x)}[/tex]

This implies that the composite function [tex]fo\ t\ (x)[/tex] is an odd function.

If the function [tex]f(x)[/tex] is even and [tex]t(x)[/tex] is odd.

[tex]\begin{aligned}fo\ t\ (-x)&=f\left(t(-x))\right\\&=f\left(-t(x))\right\\&=fo\ t\ (x)\end{aligned}[/tex]

From the above calculation it is concluded that,

[tex]\boxed{fo\ t\ (-x)=fo\ t\ (x)}[/tex]

This implies that the composite function [tex]fo\ t\ (x)[/tex] is an even function.

Your first two tests add up to:

85 + 89 = 174

To average 87 for 3 tests, the tests need to add up to : 87 x 3 = 261

To average 91, the 3 tests need to equal: 91 x 3 = 273

So to average 87, the third test needs to be: 261 - 174 = 87

To average 91, the third test needs to be: 273 - 174 = 99

Your score needs to be between 87 and 99

Answer:

C

Step-by-step explanation:

Credit Score is a numerical expression which analyzes a person's credit level by looking at this financial conditions. Will he/she be worthy of loan or not.

payment history comprises 35% of a person's credit score. This is a huge factor. If you consistently make your payments on time, your credit score increases.

length of credit history tells how secure you will be to lenders. Usually 7 years+ is a great length of credit history. This pretty much affects credit score.

marital status doesn't affect credit score. Lenders assess a person based on their financial condition and past activity, NOT whether or not he/she is married or not. That's personal agenda.

debt ratio is the ratio of total debt to total assets. If this is high, it means a person owes money to banks/individuals and is more likely to be not given credit. It affects credit score highly.

THus, the correct answer is C

Answer:

C

Step-by-step explanation:

Answer:

B. 63.8 mph

Step-by-step explanation:

IN order to solve this problem we just have to keep in mind that the miles per hour is calulcated by the number of revolutions that the tire is doing multiplied by the circumference of the tire, so the circumference of the tire is given by the first set of number in the tire the first one would be:

[tex]first:\frac{215}{60} \\Second: \frac{235}{70}[/tex]

So we just have to multiply it by the first one and then divide it by the second:

[tex]\frac{215}{60}*60=215\\215*\frac{235}{70}=63.8[/tex]

SO when the speedometer is on 60 mph, he will actually be going at 63.8 mph.

Firstly we calculate the height(h)...,

by using trigonometry ratios

;tan 45° = h/5

;where.. h = 5 tan 45°

;h = 5

Then we calculate the side x...using the height h and the angle of 30°

by also using trigonometry ratios

; sin 30° = 5/x

;where.. x = 5/(sin 30°)

Hence...., x = 10

Answer: x = 10

Answer:

Step-by-step explanation:

note : an equation is : y - b = m(x - a)    ... the point is : A ( a , b) and slope : m

in this exercice :  m= - 8     and  a= - 3    and b= -6

an equation is : y+6 = -8(x+3)

you can write :   y+6 = - 8x -24

8x+y = - 30 .....answer : C

Answer:

C. 8x + y = -30

Step-by-step explanation:

-6 = -8[-3] + b

24

-30 = b

y = -8x - 30

+8x +8x

___________

8x + y = -30 >> Standard Linear Equation

* This is the fastest way to do it.

I am joyous to assist you anytime.

Answer:

The mammoth died 17.190 years ago.

Step-by-step explanation:

Given information:

For every 5700 years, the elephant's bones losses a half-life of C-14. And supposing that the living mammoth bones had the same percentage of C-14.

If the mammoth when living has 40%, then:

Answer:

[tex]\large \boxed{\text{17 190 yr ago}}[/tex]

Step-by-step explanation

The amount of C-14 has declined from 40 % to 5 % of the original, that is, to ⅛ of the original amount.

The half-life of C-14 (5730 yr) is the time it takes for half of the isotope to decay.

We can make a table of the amount remaining after each successive half-life.

[tex]\begin{array}{ccc}\textbf{Number of} && \textbf{Fraction} \\\textbf{half-lives} & \textbf{Years} & \textbf{remaining} \\0 &0 & 1 \\1 &5730 &\frac{1}{2 } \\\\2 &11460 &\frac{1}{4 } \\\\3 &17190 &\frac{1}{8} \\\\4 & 22920& \frac{1}{16} \\\end{array}\\\text{We see that the fraction of C-14 is reduced to $\frac{1}{8}$ after three half-lives.}\\\text{The mammoth died $\large \boxed{\textbf{17 190 yr ago}}$}[/tex]

Answer:

The value of the car in very nice condition will be $35500.

Step-by-step explanation:

Let the value of car in a very nice condition be = x

The value $62125 is [tex]1\frac{3}{4}[/tex] or [tex]\frac{7}{4}[/tex] or 1.75 times of x.

Now, we can calculate for x:

[tex]1.75x=62125[/tex]

[tex]x=\frac{62125}{1.75}[/tex]

x = 35500

Hence, the value of the car in very nice condition will be $35500.

The value of the car in very nice condition is $35,500. This is determined by dividing the value of the car in excellent condition by 1.75.

To determine the value of the car in very nice condition, you need to use the information that a car in excellent condition is worth 1 3/4 times (or 1.75 times) as much as a car in very nice condition.

Given the value of the car in excellent condition is $62,125, you can set up the equation:

Substitute the known value:

To find the Value in Very Nice Condition, divide both sides by 1.75:

Hence, the value of the car in very nice condition is $35,500.

The average distance of Mercury from the Sun is found using the formula E(x) = 0.2x^3/2, where E(x) = 88 days, which is the year length for Mercury. Solving for x, the distance comes out to be approximately 57.9 million kilometers.

The question asks us to determine the average distance of Mercury from the center star, which can also be understood as the Sun, given that its year (the period it takes to orbit the Sun) is approximately 88 Earth days. We are provided with the formula E(x) = 0.2x3/2, where E(x) represents the number of Earth days in a planet's year and x is the planet's average distance from the Sun in millions of kilometers.

To solve for x, we set E(x) equal to 88 days and solve for x.

E(x) = 88
0.2x3/2 = 88
x3/2 = 88 / 0.2
x3/2 = 440
Now we take both sides to the power of 2/3 to solve for x.
x = 4402/3

To find the value of x, we calculate 440 raised to the 2/3 power.

x = 4402/3
≈ 57.9

Therefore, the average distance of Mercury from the Sun is approximately 57.9 million kilometers.

Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. More about this later.

If you are given, say, the polynomial equation y = x2 + 5x + 6, you can factor the polynomial as y = (x + 3)(x + 2). Then you can find the zeroes of y by setting each factor equal to zero and solving. You will find that x = –2 and x = –3 are the two zeroes of y.

You can, however, also work backwards from the zeroes to find the originating polynomial. For instance, if you are given that x = –2 and x = –3 are the zeroes of a quadratic, then you know that x + 2 = 0, so x + 2 is a factor, and x + 3 = 0, so x + 3 is a factor. Therefore, you know that the quadratic must be of the form y = a(x + 3)(x + 2).

(The extra number "a" in that last sentence is in there because, when you are working backwards from the zeroes, you don't know toward which quadratic you're working. For any non-zero value of "a", your quadratic will still have the same zeroes. But the issue of the value of "a" is just a technical consideration; as long as you see the relationship between the zeroes and the factors, that's all you really need to know for this lesson.)

Anyway, the above is a long-winded way of saying that, if x – n is a factor, then x = n is a zero, and if x = n is a zero, then x – n is a factor. And this is the fact you use when you do synthetic division.

Let's look again at the quadratic from above: y = x2 + 5x + 6. From the Rational Roots Test, you know that ± 1, 2, 3, and 6 are possible zeroes of the quadratic. (And, from the factoring above, you know that the zeroes are, in fact, –3 and –2.) How would you use synthetic division to check the potential zeroes? Well, think about how long polynomial divison works. If we guess that x = 1 is a zero, then this means that x – 1 is a factor of the quadratic. And if it's a factor, then it will divide out evenly; that is, if we divide x2 + 5x + 6 by x – 1, we would get a zero remainder. Let's check:

As expected (since we know that x – 1 is not a factor), we got a non-zero remainder. What does this look like in synthetic division? Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

First, write the coefficients ONLY inside an upside-down division symbol:

Answer:

Step-by-step explanation:

[tex]A=\frac{1}{2} r^{2} \theta\\\frac{dA}{d\theta} =r \\\\when r=2\\rate of change at r=2 is  2[/tex]

We want to find the rate of change of a given function of two variables when we fix one of the two, and then we want to evaluate it at r = 2.

We will get that the rate of change is:

[tex]\frac{dA(r, \theta)}{d\theta} = 12*r^2[/tex]

And when r = 2, the rate is 48.

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We define the rate of change of a function with respect to some variable as the differentiation of the function with respect to that variable.

Here we have the function:

[tex]A(r, \theta) = 12*r^2*\theta[/tex]

We need to differentiate it with respect to θ, we will get:

[tex]\frac{dA(r, \theta)}{d\theta} = 1*12*r^2*\theta^0 = 12*r^2[/tex]

Where I used the general rule to derive functions with exponents:

[tex]f(x) = x^n\\\\\frac{df(x)}{dx} = n*x^{n -1}[/tex]

Now that we know the rate of change, we want to evaluate it in r = 2, we will get:

[tex]\frac{dA(2, \theta)}{d\theta} = 12*2^2 = 48[/tex]

Notice that this does not depend on the value of θ.

If you want to learn more, you can read:

https://brainly.com/question/18904995