Write a two column proof for the following statement.

“Given that 2( -x + 4) = 3 , prove that x = 5/2.

The quotient of of x³ + 1 divided by x² - x + 1 is x + 1

Given,

The polynomial function; x³ + 1 and x² - x + 1

We have to find the quotient of x³ + 1 divided by x² - x + 1

Polynomial function;-

In an equation such as the quadratic equation, cubic equation, etc., a polynomial function is a function that only uses non-negative integer powers or only positive integer exponents of a variable. For instance, the polynomial 2x+5 has an exponent of 1.

Here,

Divide x³ + 1  by x² - x + 1

x² - x + 1 ) x³ + 1        (x + 1 ⇒ Quotient

               x³ - x² + x

             0 + 1 + x² - x

            0 + 1 + x² - x

                          0   ⇒ Remainder

That is,

The quotient of of x³ + 1 divided by x² - x + 1 is x + 1

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Some other measurements used to measure length are meter, centimeter millimeter, decameter, decimeter, hectometer, foot, etc

What is length of an object?

Length of an object is the distance between one end point to another end point of the object.

The concept of length is a very important tool in mathematics, especially in geometry and mensuration. Concept of length is very important in physics also

Kilometer and inches are some measurements used to measure length.

Some other measurements used to measure length are meter, centimeter millimeter, decameter, decimeter, hectometer, foot, micrometer, nanometer, angstrom, picometer, Fermi, etc

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Answer:

Step-by-step explanation:

a

2/2 x^2 = 24/2

x^2 = 12

x = √12

x = 3.464102 = 3.46

x = -√12 = -3.46

b

x^2 = 14 + 4

x^2 = 18

x = √18

x = 4.242641 = 4.24

x = -√18 = -4.24

As per the given differential equation, the minimum radius of convergence R of power series solutions about the ordinary point x = 0 is 9 and that about the ordinary point x = 1 is 10 units.

Differential equation:

Differential equation refers the equation which contains one or more terms and the derivatives of one variable with respect to the other variable

And it can be written as,

=> dy/dx = f(x)

Given,

The given differential equation is  (x² - 81)y"+ 7xy' + y = 0 R = 9 (x = 0) R = (x = 1).

Now we have to find the minimum radius of convergence R of power series solutions about the ordinary point x = 0 and x = 1.

Here we know that the minimum radius of convergence is defined as the distance between the ordinary point and the singularity of the differential equation.

And the Singularity point is the root of the polynomial attached with the second derivative. So, the singularity points will be calculated as,

=> x² - 81

=> (x + 9) (x - 9)

=> x = 9, - 9

So, the singularity points are 9, - 9.

So, the minimum radius of convergence can be calculated as,

Now, the ordinary points are 1 + 0i  and 1 - 0i.

So,

=> r1 = | 1 + 0i - 9|

=> r1 = 9

=> r2 = | 1 + 0i + 9|

=> r2 = 10

Therefore, the minimum radius of convergence R of power series solutions about the ordinary point x = 0 is 9 and that about the ordinary point x = 1 is 10 units.

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The experimental probability of landing on a 6 is 1/10.

Probability is used to calculate the odds that a random event would happen. The probability the event occurs is 1 and the probability that the event does not occur is 0.

Experimental probability is based on the result of an experiment that has been carried out multiples times.

Result of the experiment:

Face Frequency

1          2

2         1

3        0

4        5

5          1

6          1

Probability of landing on a 6 = 1 /10

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The decay rate for the resale price of laser cutting tool will be -121000[tex]e^{-0.22t}[/tex]

Let R is the resale value of a particular type of laser cutting tool after years

Where, R(t) = 550000[tex]e^{-0.22t}[/tex]

Rate at which the resale value is changing will be dR/dt = 550000[tex]e^{-0.22t}[/tex](-0.22)= -121000[tex]e^{-0.22t}[/tex]

Initial value of the laser cutting tool, at time ‘t’ =0 will be 550000[tex]e^{-0.22t}[/tex]x0 = 550000

At time ‘t’ resale value is 550000 [tex]e^{-0.22t}[/tex]

Therefore, the decay in resale price = 550000 - 550000 [tex]e^{-0.22t}[/tex]

Therefore, the decay rate = (550000 - 550000 [tex]e^{-0.22t}[/tex])/t = 550000(1-[tex]e^{-0.22t}[/tex])/t

As we have dR/dt = R’ = -121000[tex]e^{-0.22t}[/tex]

Therefore, R’(10) = -121000[tex]e^{-2.2}[/tex] = -13407.18

Therefore, after 10 years the resale price will be -13407.18

R’(20) = -121000[tex]e^{-4.4}[/tex]= -1485.55

Therefore, after 20 years the resale price will be -1485.55

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We have proved that the set of all points (x, y) that are equidistant from the point (f, k) and the line x.

What is a parabola?

A parabola is a plane curve that is mirror-symmetrical and roughly U-shaped in mathematics. It fits several seemingly disparate mathematical descriptions, all of which can be shown to define the same curves. A point and a line are two ways to describe a parabola.

That’s a parabola in the usual orientation. y=1 is the directrix. (f, k) is the focus. Squared distance, as usual, is the fundamental quantity.

If (x,y) is a point in the set, the squared distance to y=1 is (y−1)². The squared distance to (f, k).

     (y - 1)²=(x - f)²+(y - k)²

Hence, we can say that the set of all points (x, y) is equidistant from the point (f, k) and the line x.

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Plastic, wood, and rubber are the three materials that do not conduct electricity.

Conductors are materials that conduct heat or electricity. Insulators are the materials that do not allow to conduct heat or electricity.

Electrical insulators are materials that do not allow to pass electricity  through them.

Glass,  wood, rubber and plastic are examples of these materials. Air also acts as an insulator.

Thus,these non-conductors/insulators are useful in a variety of applications.

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[tex]2^{10}\cdot 2^{15}\implies 2^{10+15}\implies 2^{25}\implies 2^{5\cdot 5}\implies (2^5)^5\implies {\Large \begin{array}{llll} 32^5 \end{array}}[/tex]

The problem is saying there are two distinct zeros, and both are strictly lie between 0 and 1. The least positive integer value of "a" is 5 for which such a polynomial exists.

Let f(x) = ax² - bx + c , be our polynomial.

where a,b are integers coefficient and c constant. It has two distinct real roots lying between 0 and 1. f(x) can be written as, f(x) = a(x−α)(x−β)

where α and β are the roots of the equation.

we can be observed that f(0)f(1) > 0

As a,b and c are integers f(0)f(1) > 1

f(0) = aαβ , f(1)=a(1−α)(1−β)

=> a²(α)(1−α)(β)(1−β) > 0 -------(1)

Using A.M ≥ G.M, we get

(α+1−α+β+1−β)/4 > (αβ(1−α)(1−β))¹/⁴

Arithmetic Mean and Geometric Mean cannot be equal because the equation has distinct roots.

a²/16 > a²(αβ(1−α)(1−β)).

=> a²/16 > 1

=> a > 4

Therefore, the minimum value of 'a' is 5

The discriminant of the equation,

Δ = b²−4ac ≥ 0

=> b² >20c

consider the minimum value of c = 1,

we get b² >20

=> Minimum Value of b = 5.

Hence, the least positive integer of 'a' is 5.

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The rate of change of area is given as 27 π. The time at which radius of circle is increasing is 1.386 sec and change in radius is 3 cm.

What is rate of change?

The rate of change function is defined as the rate at which one quantity is changing with respect to another quantity. In simple terms, in the rate of change, the amount of change in one item is divided by the corresponding amount of change in another.

a) Given, radius of circle = 9 cm

rate of change of radius = 3/2

We know area of circle = πr^2

change in area =2πrr'

we get  2×π×9×2/3=27π

B) Given,

g(t) = 12 - 12e^0.5t

derivate it

g'(t) =-12×-1/2e^t/2

g'(t)=3

3= 6e^t/2

t = 1.386

c) f1(t) = t2-4t+4.

Integrate it we get

radius =3.

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The main advantage of using a graphic calculator is that it speeds up the process of solving more difficult arithmetic problems for students.

A graphing calculator, often known as a graphics calculator or graphic display calculator, is a portable computer that can plot graphs, solve many equations at once, and carry out other variable-related activities. The most common graphing calculators are programmable calculators, which let the user design unique programs, generally for scientific, technical, or educational purposes. They feature big screens that show computations and several lines of text.

Numerous graphing calculators may be connected to sensors such as accelerometers, electronic thermometers, pH meters, weather stations, decibels, light meters, and other devices to operate as data recorders and as monitoring, polling, and communication tools with teachers. The use of such data in student laboratory tasks improves their understanding of math, particularly statistics and mechanics. 

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The closest object that she is going to be able to see clearly is going to be situated at a distance of 20 cm from her eyes.

Here, it has been given that the lens-to-retina distance of her eye is = 2.00 cm (i)

The normal power for distant vision is 50.00 Dioptres. This woman has the ability to increase lens strength by 10%. The new power for this is going to be -

= P1 = 50.00 + 10% of 50

= P1 = 55.00 D (ii)

Now, the power of a lens, P = 1/f = 1/u + 1/v where f = focal length of the lens, u = object distance from eye lens, and v = image distance from eye lens.

We are aware of the formula = P1 = 1/u + 1/v

Using the values of (i) and (ii) and the formula, we get -

= 1/u = P'- 1/v

= 1/u = 55.0 D - 1/0.02 m

= 1/u = 5.0 m⁻¹

= u = 1/5.0 m⁻¹

= u = 20 cm

Hence, we have found out that the woman can see objects at a distance of 20 cm.

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Answer:360ml

Step-by-step explanation:

400/10=10%=40

400-40=360ml

True average percentage of organic matter in such soil is different from 3%.

What is Statistics?

Statistics is the study of data collection, analysis, presentation, and interpretation. Much of the early push for the subject of statistics came from government demands for census data as well as information about a range of economic operations.

Solution:

We are given that the values of the sample mean and sample standard deviation are 2.481 and 1.611 respectively.

Suppose we know the population distribution is normal, we have to test the hypothesis that does this data suggest that the true average percentage of organic matter in such soil is something other than 3%.

Let μ = true average percentage of organic matter in such soil

SO, Null Hypothesis : μ = 3%   {means that the true average percentage of organic matter in such soil is equal to 3%}

Alternate Hypothesis  : μ ≠ 3%   {means that the true average percentage of organic matter in such soil is different than 3%}

The test statistics that will be used here is One-sample t test statistics because we don't know about the population standard deviation;

TS = (0.02481 - 0.03)/(0.01611)/√30

TS = 4.6% (approx)

Therefore, we conclude that the true average percentage of organic matter in such soil is different from 3%.

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Any two triangles must meet a number of requirements in order for congruence to be established. In order for two triangles to be similar:

Investigation is required to determine whether the triangles in the preceding diagram, ABC and CDA, are congruent. Considering the diagram

The two triangles share side AC.

Side BC equals Side AD (given)

Angle DAC equals angle ACD (alternate angles as depicted in the diagram)

Therefore, ABC and CDA are congruent because both triangles' two sides and an included angle (SAS) are equivalent.

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The question is incomplete. Completed question is given below;

Based on the given information in the figure at the right, how can you justify that triangle ABC is congruent to triangle CDA?ASA, SSS, AAS, SAS

Figure is attached

The percentage of 2000 from 50000 is 4%.

A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100.

The percentage therefore refers to a component per hundred. Per 100 is what the word percent means. It is represented by %.

The percentage of 2000 from 50000 will be:

= 2000/50000 × 100

= 4%

The percentage is 4%.

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The loop will be written in Matlab.

Simply setting the years' range will allow the for loop to run its course and complete the task. To allow the loop to iterate from 1 to the number of years, we add the expression 1:number years.

function savingsBalance = CalculateBalance(numberYears,savingsBalance)

% numberYears: Number of years that interest is applied to savings

% savingsBalance: Initial savings in dollars

interestRate = 0.025;  

% 2.5 percent annual interest rate

% Complete the for loop to iterate from 1 to numberYears (inclusive)

for n = 1:numberYears

savingsBalance = savingsBalance + (savingsBalance * interestRate);

end

end

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The four ways to care for our nervous system are exercise, meditation, proper sleep and healthy food.

There are many ways to care and protect our nervous system. But some of the basics and the most important ways to care for our nervous system are to follow a basic routine which includes regular exercise, practicing meditation daily, having proper sleep for at least seven hours a day and having healthy food mostly.

Now, doing exercise means that we are constantly engaged in some kind of physical activity and healthy food also means to have a balanced lifestyle.

Hence, the four ways to care for our nervous system are exercise, meditation, proper sleep and healthy food.

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The probability that the sample contains 1 or more rotten oranges be, 1/8.

Given, a customer at cavallaro's fruit stand picks a sample of 3 oranges at random from a crate containing 40 oranges, of which 5 are rotten.

we have to find the probability that the sample contains 1 or more rotten oranges,

first we will find the probability that the sample contains no rotten oranges,

P(E) = 35/40

P(E) = 7/8

Now, the probability that the sample contains 1 or more rotten oranges be,

1 - 7/8

P(E) = 1/8

Hence, the probability that the sample contains 1 or more rotten oranges be, 1/8.

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The maximum absolute values of the shear and bending moments are

Cy = 1800 lbs

Ay = 4800 lbs

Max absolute BM = 14400 lbs-feet

Given,

Draw the shear and bending moment diagrams for the beam and the indicated loading, and then figure out the highest absolute values of the shear and bending moment.

(a) The shears and bending-moment diagrams,

Ay + Cy = 800 * 9 - 600

Ay + Cy = 6600 lbs

Ay

∑Ma = 0

Cy * 15 + 600 * 9 = 800 * 9²/2

Cy = 1800 lbs

Ay = 4800 lbs

(b) Determine the maximum absolute values of the shear and bending moment,

Map absolute B.M:

Map positive B.M occurs, when SF = 0 is at 'x'

4800 - 800x = 0

     x = 6ft

Positive map = 4800 * 6 = 800 * 6²/2

Max absolute BM = 14400 lbs-feet and it developed at 6ft from support A.

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Option A, which states that the ordered pair of point P' will be, is the appropriate response (-8.75, 5.25).

Given,

A scaling factor of 7/4 is used to enlarge the pentagon in the OPQRS.

In order to generate O'P'Q'R'S, we must identify the ordered pair of point P'.

Here,

To obtain the coordinates of P', multiply the coordinates of point P by the scale factor 7/4.

P(-5,3)

P' (x, y) = P (-5 x 7/4, 3 x 7/4).

= P(-8.75, 5.25) 

As a result, option A, which states that the ordered pair of point P' will be, is the appropriate response (-8.75, 5.25).

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The question is improper. Proper question is given below;

If pentagon OPQRS is dilated by a scale factor of 7/4 from the origin, to create O’P’Q’R’S’, what is the ordered pair of point P’?

A. (-8.75, 5.25)

B. (-1.25, 0.75)

C. (-3.5, 8.75)

D. (-1.75, 3.5)

If the area of a triangle is 96 sq. inches. its altitude is 2 inches greater than five times its base then the altitude is 32 inch

The area of a triangle is 96 sq. inches

Let base be b

Its altitude is 2 inches greater than five times its base.

a=5b+2

ab/2=A

(5b+2)b/2=96

(5b+2)b=192

5b²+2b=192

5b²2+2b-192=0

On solving the quadrartic equation,

we get

b=6

a=32

Therefore, if the area of a triangle is 96 sq. inches. its altitude is 2 inches greater than five times its base then the altitude is 32 inchs

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the  measure that captures the variation in responses is mean and it is the weighted average

What is the meaning of mean in math?

In mathematics and statistics, the mean refers to the average of a set of values. The mean can be computed in a number of ways, including the simple arithmetic mean (add up the numbers and divide the total by the number of observations), the geometric mean, and the harmonic mean.

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The length of the tail if it's half the insects total length as described in the task content is; 7/16 inches.

It follows from the task content that the length of the tail of the insect whose length is as given is to be determined.

Since the given length of the insect is; 7/8 inches.

Also, the tail of the insect is half the insects to tak length.

Consequently, the expression which holds true for the length of the tail is;

(1/2) × (7/8)

= 7/16.

The length of the insect's tail is therefore; 7/16 feet.

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Maura's driveway is 4.7 meters long and the distance from the garage into the lot is 3.6 meters

The length of Maura's drive way is solved using SOH CAH TOA

Sin = opposite / hypotenuse - SOH

Cos = adjacent / hypotenuse - CAH

Tan = opposite / adjacent - TOA

The information in the question describes a right angle triangle of

opposite = 3 meters

adjacent = distance into the lot = ?

hypotenuse = Maura's drive way = ?

angle of depression = 40

The length of the drive way

sin 40 = 3 / hypotenuse

hypotenuse = 3 / sin 40

hypotenuse = 4.667 m

The distance from the garage to the lot

tan 40 = 3 / adjacent

adjacent = 3 / tan 40

adjacent = 3.575 m

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The rule of obtuse angle is that one angle measures more than 90° and total sum og angle should be 180°.

Acute, obtuse, right, straight, reflex, and complete rotation are some of the terms used to describe fundamental angles. Two rays are joined at their termini to form the geometric shape known as an angle. The most common way to express an angle is in degrees.

A measure of an angle or the amount of turning required to bring one line or plane into coincidence with or parallel to another angled is represented by two lines extending from the same point.

A triangle is said to have an obtuse angle if one of the inner angles is greater than 90 degrees. If one of the angles of an obtuse triangle is greater than 90°, then the total of the other two angles is also less than 90°.

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By using the formula for the distance between the two points, it is found that S could lie on the perpendicular bisector QR.

Distance between points:

The distance between two points, (x1,y1) and (x2,y2) is calculated as,

D = √(x2 - x1)² + (y2 - y1)²

Given,

Here we have the coordinates: Q(-5, 4), R(8, -3), S(-2,-5)

Now, we have to determine if S could lie on the perpendicular bisector of QR with the given coordinates.

Here we have the endpoints are: Q(-5, -1), R(3, 7)

Point S is (4,-2).

The distance of S to Q is calculated as,

D = √(4 - (-5))² + (-2 - (-1))²

D = √82

Now, the distance of S to R is calculated as,

D = √(4 - 3)² + (-2 - 7)²

D = √82

Both have the Same distance, thus states the S could lie on the perpendicular bisector QR.

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The correct equation is tan θ = 6 / 13, and the value of angle θ will be 24.775°. Then the correct option is C.

Trigonometric functions examine the interaction between the dimensions and angles of a triangular form.

The right-angle triangle is shown below.

The length of the perpendicular is 6 cm and the length of the base is 13 cm. Then the angle θ is written as,

tan θ = P / B

tan θ = 6 / 13

Simplify the equation to get the value of the angle θ will be given as,

tan θ = 6 / 13

tan θ = 0.4615

θ = 24.775°

The correct equation is tan θ = 6 / 13, and the value of angle θ will be 24.775°. Then the correct option is C.

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Sales revenue will be recorded at $4,000 and yang Yang would be required to Recognize sales tax liability of $360.

What is  sales revenue?

Sales = $4,000

State sales tax rate = 9%

State sales tax = Sales x State sales tax rate

= 4,000 x 9%

= $360

Sales revenue will be recorded at $4,000 and sales revenue is not affected by the sales tax amount.

Hence, Yang would be required to Recognize sales tax liability of $360.

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